Regularity theorem for totally nonnegative flag varieties
RREGULARITY THEOREM FOR TOTALLY NONNEGATIVEFLAG VARIETIES
PAVEL GALASHIN, STEVEN N. KARP, AND THOMAS LAM
Abstract.
We show that the totally nonnegative part of a partial flag variety
G/P (inthe sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. Inparticular, the closure of each positroid cell inside the totally nonnegative Grassmannian ishomeomorphic to a ball, confirming a conjecture of Postnikov.
Contents
1. Introduction 12. Overview of the proof 53. Topological results 74.
G/P : preliminaries 135. Subtraction-free parametrizations 226. Bruhat projections and total positivity 297. Affine Bruhat atlas for the projected Richardson stratification 358. From Bruhat atlas to Fomin–Shapiro atlas 409. The case G = SL n Introduction
Let G be a semisimple algebraic group, split over R , and let P ⊂ G be a parabolicsubgroup. Lusztig [Lus94] introduced the totally nonnegative part of the partial flag variety G/P , denoted (
G/P ) ≥ , which he called a “remarkable polyhedral subspace”. He conjecturedand Rietsch proved [Rie99] that ( G/P ) ≥ has a decomposition into open cells. We prove thefollowing conjecture of Williams [Wil07]: Theorem 1.1.
The cell decomposition of ( G/P ) ≥ forms a regular CW complex. Thus theclosure of each cell is homeomorphic to a closed ball. A special case of particular interest is when
G/P is the Grassmannian Gr( k, n ) of k -dimensional linear subspaces of R n . In this case, ( G/P ) ≥ becomes the totally nonnegative Date : May 8, 2019.2010
Mathematics Subject Classification.
Primary: 14M15. Secondary: 05E45, 15B48, 20G20.
Key words and phrases.
Total positivity, algebraic group, partial flag variety, Richardson varieties, totallynonnegative Grassmannian, positroid cells, affine Kac–Moody group.T.L. was supported by a von Neumann Fellowship from the Institute for Advanced Study and by grantDMS-1464693 from the National Science Foundation. a r X i v : . [ m a t h . C O ] M a y PAVEL GALASHIN, STEVEN N. KARP, AND THOMAS LAM
Grassmannian Gr ≥ ( k, n ), introduced by Postnikov [Pos07] as the subset of Gr( k, n ) whereall Pl¨ucker coordinates are nonnegative. He gave a stratification of Gr ≥ ( k, n ) into positroidcells according to which Pl¨ucker coordinates are zero and which are strictly positive, andconjectured that the closure of each positroid cell is homeomorphic to a closed ball. Post-nikov’s conjecture follows as a special case of Theorem 1.1: Corollary 1.2.
The decomposition of Gr ≥ ( k, n ) into positroid cells forms a regular CWcomplex. Thus the closure of each positroid cell is homeomorphic to a closed ball. When k = 1, Gr ≥ (1 , n ) is the standard ( n − n − ⊂ P n − , aprototypical example of a regular CW complex.1.1. History and motivation.
A matrix is called totally nonnegative if all its minors arenonnegative. The theory of such matrices originated in the 1930’s [Sch30, GK37]. Later,Lusztig [Lus94] was motivated by a question of Kostant to consider connections betweentotally nonnegative matrices and his theory of canonical bases for quantum groups [Lus90].This led him to introduce the totally nonnegative part G ≥ of a split semisimple G . Inspiredby a result of Whitney [Whi52], he defined G ≥ to be generated by exponentiated Chevalleygenerators with positive real parameters, and generalized many classical results for G = SL n to this setting. He introduced the totally nonnegative part ( G/P ) ≥ of a partial flag variety G/P , and showed [Lus98b, §
4] that G ≥ and ( G/P ) ≥ are contractible.Fomin and Shapiro [FS00] realized that Lusztig’s work may be used to address a long-standing problem in poset topology. Namely, the Bruhat order of the Weyl group W of G had been shown to be shellable by Bj¨orner and Wachs [BW82], and by general results ofBj¨orner [Bj¨o84] it followed that there exists a “synthetic” regular CW complex whose faceposet coincides with ( W, ≤ ). The motivation of [FS00] was to answer a natural question dueto Bernstein and Bj¨orner of whether such a regular CW complex exists “in nature”. Let U ⊂ G be the unipotent radical of the standard Borel subgroup, and let U ≥ := U ∩ G ≥ be its totally nonnegative part. For G = SL n , U ≥ is the semigroup of upper-triangularunipotent matrices with all minors nonnegative. The work of Lusztig [Lus94] implies that U ≥ has a cell decomposition whose face poset is ( W, ≤ ). The space U ≥ is not compact,but Fomin and Shapiro [FS00] conjectured that taking the link of the identity element in U ≥ , which also has ( W, ≤ ) as its face poset, gives the desired regular CW complex. Theirconjecture was confirmed by Hersh [Her14]. Hersh’s theorem also follows as a corollary toour proof of Theorem 1.1, see Remark 3.13. Corollary 1.3 ([Her14]) . The link of the identity in U ≥ is a regular CW complex. For recent related developments, see [DHM19].Meanwhile, Postnikov [Pos07] defined the totally nonnegative Grassmannian Gr ≥ ( k, n ),decomposed it into positroid cells, and showed that each positroid cell is homeomorphic toan open ball. Motivated by work of Fomin and Zelevinsky [FZ99] on double Bruhat cells, heconjectured [Pos07, Conjecture 3.6] that this decomposition forms a regular CW complex. Itwas later realized (see (9.16)) that the space Gr ≥ ( k, n ) and its cell decomposition coincidewith the one studied by Lusztig and Rietsch in the special case that G/P = Gr( k, n ).Williams [Wil07, §
7] extended Postnikov’s conjecture from Gr ≥ ( k, n ) to ( G/P ) ≥ .There has been much progress towards proving these conjectures. Williams [Wil07] showedthat the face poset of ( G/P ) ≥ (and hence of Gr ≥ ( k, n )) is graded, thin, and shellable, andtherefore by [Bj¨o84] is the face poset of some regular CW complex. Postnikov, Speyer, and EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 3
Williams [PSW09] showed that Gr ≥ ( k, n ) is a CW complex, and their result was generalizedto ( G/P ) ≥ by Rietsch and Williams [RW08]. Rietsch and Williams [RW10] also showedthat the closure of each cell in ( G/P ) ≥ is contractible. In previous work [GKL17, GKL18],we showed that the spaces Gr ≥ ( k, n ) and ( G/P ) ≥ are homeomorphic to closed balls, whichis the special case of Theorem 1.1 for the top-dimensional cell of ( G/P ) ≥ . We remark thatour proof of Theorem 1.1 uses different methods than those employed in [GKL17, GKL18],in which we relied on the existence of a vector field on G/P contracting (
G/P ) ≥ to a pointin its interior. Singularities of lower-dimensional positroid cells give obstructions to theexistence of a continuous vector field with analogous properties.Totally positive spaces have attracted a lot of interest due to their appearances in othercontexts such as cluster algebras [FZ02] and the physics of scattering amplitudes [AHBC + amplituhedra of Arkani-Hamed and Trnka [AHT14], and moregenerally the Grassmann polytopes of the third author [Lam16]. A Grassmann polytope is ageneralization of a convex polytope in the Grassmannian Gr( k, n ). For example, the totallynonnegative Grassmannian Gr ≥ ( k, n ) is a generalization of a simplex, while amplituhedrageneralize cyclic polytopes [Stu88]. The faces of a Grassmann polytope are linear projectionsof closures of positroid cells, and therefore it is essential to understand the topology of theseclosures in order to develop a theory of Grassmann polytopes.1.2. Stars, links, and the Fomin–Shapiro atlas.
Rietsch [Rie99, Rie06] defined a certainposet ( Q J , (cid:22) ), and established the decomposition ( G/P ) ≥ = (cid:70) g ∈ Q J Π > g into open ballsΠ > g indexed by g ∈ Q J . She showed that for h ∈ Q J , the closure Π ≥ h of Π > h is given byΠ ≥ h = (cid:70) g (cid:22) h Π > h . When ( G/P ) ≥ is the totally nonnegative Grassmannian Gr ≥ ( k, n ), thisis the positroid cell decomposition of [Pos07].Given g ∈ Q J , define the star of g in ( G/P ) ≥ by(1.1) Star ≥ g := (cid:71) h (cid:23) g Π > h . In Section 3.1, we define another space Lk ≥ g (the link of g ) stratified as Lk ≥ g = (cid:70) h (cid:31) g Lk > g,h .We later show in Theorem 3.12 that Lk ≥ g is a regular CW complex homeomorphic to aclosed ball.At the core of our approach is a collection of (stratification-preserving) homeomorphisms(1.2) ¯ ν g : Star ≥ g ∼ −→ Π > g × Cone(Lk ≥ g ) , one for each g ∈ Q J . Here Cone( A ) := ( A × R ≥ ) / ( A × { } ) denotes the open cone over A .The homeomorphisms { ¯ ν g | g ∈ Q J } are part of the data of what we call a Fomin–Shapiroatlas , cf. Definition 2.3. Our construction is inspired by similar maps introduced in [FS00]for the unipotent radical U ≥ . Example 1.4.
When G = SL n and P = B is the standard Borel subgroup, G/B is the complete flag variety consisting of flags in R n , and the Weyl group W is the group S n ofpermutations of n elements. The face poset Q J of ( G/B ) ≥ is the set { ( v, w ) ∈ S n × S n | v ≤ w } of Bruhat intervals in S n , and the cell Π > v,w ) ⊂ ( G/B ) ≥ indexed by ( v, w ) ∈ Q J has dimension (cid:96) ( w ) − (cid:96) ( v ). For example, when n = 3, this gives a cell decomposition ofa 3-dimensional ball, see Figure 1 (left). For g := ( s , s s ), Π > g is an open line segment, PAVEL GALASHIN, STEVEN N. KARP, AND THOMAS LAM id s s s s s s w Π > g ¯ ν g −→ s s s Π > g w s id s s Figure 1.
The map ¯ ν g for the case G = SL and P = B from Example 1.4.and Star ≥ g consists of 4 cells: a line segment Π > g = Π > s ,s s ) , two open square faces Π > s ,w ) and Π > ,s s ) , and an open 3-dimensional ball Π > ,w ) . This union is indeed homeomorphicto Π > g × Cone(Lk ≥ g ) shown in Figure 1 (right). Here Lk ≥ g is a closed line segment whoseendpoints are Lk > g, ( s ,w ) and Lk > g, (id ,s s ) , and whose interior is Lk > g, (id ,w ) .In Definition 2.1, we introduce the abstract notion of a totally nonnegative space , whichcaptures several known combinatorial and geometric properties of ( G/P ) ≥ used in ourproof. This includes the shellability of Q J due to Williams [Wil07], and some topologicalresults [Rie06, KLS14] on Richardson varieties.In Section 3, we prove (Theorem 2.4) that every totally nonnegative space that admitsa Fomin–Shapiro atlas is a regular CW complex. Our argument proceeds by induction onthe dimension of Lk > g,h , and depends on a delicate interplay between objects in smooth andtopological categories. We use crucially that the maps (1.2) in a Fomin–Shapiro atlas arerestrictions of smooth maps. On the topological level, we rely on the generalized Poincar´econjecture [Sma61, Fre82, Per02] combined with some general results on poset topology.The bulk of the paper is devoted to the construction of the Fomin–Shapiro atlas. For each g ∈ Q J we give an isomorphism ¯ ϕ u between an open dense subset O g ⊂ G/P and a certainsubset of the affine flag variety G / B of the loop group G associated with G . The map ¯ ϕ u ,which we call an affine Bruhat atlas , sends the projected Richardson stratification [KLS14]of G/P to the affine Richardson stratification of its image inside G / B . The hardest part ofthe proof consists of showing that the subset O g ⊂ G/P contains Star ≥ g . See Section 2.2for a more in-depth overview of the construction of ¯ ϕ u . Remark 1.5.
The map ¯ ϕ u generalizes the map of Snider [Sni10] from Gr( k, n ) to all G/P ,see Remark 9.9. A different approach to give such a generalization is due to He, Knutson,and Lu [HKL], which led them to the notion of a
Bruhat atlas . See [Ele16] for the definition.We call our map ¯ ϕ u an affine Bruhat atlas since its target space is always an affine flagvariety, while the Bruhat atlases of [HKL] necessarily involve more general Kac–Moody flagvarieties . We understand that Daoji Huang has also made progress towards the constructionof a similar map.1.3.
Outline.
In Section 2, we introduce totally nonnegative spaces and define Fomin–Shapiro atlases. We state in Theorem 2.4 that every totally nonnegative space that admits
EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 5 a Fomin–Shapiro atlas is a regular CW complex, and prove it in Section 3. We give back-ground on
G/P in Section 4, and study subtraction-free Marsh–Rietsch parametrizations inSection 5. We then apply our results on such parametrizations to prove Theorem 6.4, thatwill later imply that the above open subset O g contains Star ≥ g . We introduce affine Bruhatatlases in Section 7 and use them to construct a Fomin–Shapiro atlas for G/P in Section 8.Theorem 2.5 (which implies our main result Theorem 1.1) is proved in Section 8.3. Section 9is devoted to specializing our construction to type A (when G = SL n ), with a special focus onthe totally nonnegative Grassmannian Gr ≥ ( k, n ). We illustrate many of our constructionsby examples in Section 9, and we encourage the reader to consult this section while studyingother parts of the paper. We discuss some conjectures and further directions in Section 10.Finally, we give additional background on Kac–Moody flag varieties in Appendix A. Acknowledgments.
We thank Sergey Fomin, Patricia Hersh, Alex Postnikov, and LaurenWilliams for stimulating discussions. We are also grateful to George Lusztig and KonniRietsch for their comments on the first version of this manuscript.2.
Overview of the proof
We formulate our results in the abstract language of totally nonnegative spaces , since weexpect that they can be applied in other contexts, see Section 10.2.1.
Totally nonnegative spaces.
We refer the reader to Section 3.2 for background onposets and regular CW complexes. For a finite poset ( Q, (cid:22) ), we denote by (cid:98) Q := Q (cid:116) { ˆ0 } the poset obtained from Q by adjoining a minimum ˆ0. Bj¨orner showed [Bj¨o84, Proposi-tion 4.5(a)] that if (cid:98) Q is graded , thin , and shellable , then Q is isomorphic to the face poset ofsome regular CW complex. If (cid:98) Q is a graded poset, we let dim : Q → Z ≥ denote the rankfunction of Q . Definition 2.1.
We say that a triple ( Y , Y ≥ , Q ) is a totally nonnegative space (or TNNspace for short) if the following conditions are satisfied.(TNN1) ( (cid:98) Q, (cid:22) ) is graded, thin, and shellable, and it contains a unique maximal element ˆ1.(TNN2) Y is a smooth manifold, stratified into embedded submanifolds Y = (cid:70) g ∈ Q ◦ Y g , andfor each h ∈ Q , ◦ Y h has dimension dim( h ) and closure Y h := (cid:70) g (cid:22) h ◦ Y g .(TNN3) Y ≥ is a compact subset of Y .(TNN4) For g ∈ Q , Y > g := ◦ Y g ∩Y ≥ is a connected component of ◦ Y g diffeomorphic to R dim( g ) > .(TNN5) The closure of Y > h inside Y equals Y ≥ h := (cid:70) g (cid:22) h Y > g .For the case Y = G/P , the smooth submanifolds ◦ Y g are the open projected Richardsonvarieties of [KLS14]. Definition 2.2.
Let N ≥
0, and denote by (cid:107) · (cid:107) the Euclidean norm on R N . We saythat a pair ( Z, ϑ ) is a smooth cone if Z ⊂ R N is a closed embedded submanifold and ϑ : R > × R N → R N a smooth map such that(SC1) ϑ gives an ( R > , · )-action on R N that restricts to an ( R > , · )-action on Z .(SC2) ∂∂t (cid:107) ϑ ( t, x ) (cid:107) > t ∈ R > and x ∈ R N \ { } .The map ϑ is a smooth analog of a contractive flow of [GKL17], see Lemma 3.4.For g ∈ Q , define Star g := (cid:70) h (cid:23) g ◦ Y h and Star ≥ g := Star g ∩Y ≥ = (cid:70) h (cid:23) g Y > h , cf. (1.1). PAVEL GALASHIN, STEVEN N. KARP, AND THOMAS LAM
Definition 2.3.
We say that a TNN space ( Y , Y ≥ , Q ) admits a Fomin–Shapiro atlas iffor each g ∈ Q , there exists an open subset O g ⊂ Star g , a smooth cone ( Z g , ϑ g ), and adiffeomorphism(2.1) ¯ ν g : O g ∼ −→ ( ◦ Y g ∩ O g ) × Z g satisfying the following conditions.(FS1) For all h (cid:23) g , we are given ◦ Z g,h ⊂ Z g such that Z g = (cid:70) h (cid:23) g ◦ Z g,h and ◦ Z g,g = { } .(FS2) For all h (cid:23) g and t ∈ R > , we have ϑ g ( t, ◦ Z g,h ) = ◦ Z g,h .(FS3) For all h (cid:23) g , we have ¯ ν g ( ◦ Y h ∩ O g ) = ( ◦ Y g ∩ O g ) × ◦ Z g,h .(FS4) For all y ∈ ◦ Y g ∩ O g , we have ¯ ν g ( y ) = ( y, ≥ g ⊂ O g .We will prove the following result in Section 3.3, using link induction . Theorem 2.4.
Suppose that a TNN space ( Y , Y ≥ , Q ) admits a Fomin–Shapiro atlas. Then Y ≥ = (cid:70) h ∈ Q Y > h is a regular CW complex. In particular, for each h ∈ Q , Y ≥ h is homeo-morphic to a closed ball of dimension dim( h ) . Thus Theorem 1.1 follows as a corollary of Theorem 2.4 and the following result:
Theorem 2.5. ( G/P, ( G/P ) ≥ , Q J ) is a TNN space that admits a Fomin–Shapiro atlas. Plan of the proof.
We give a brief outline of the proof of Theorem 2.5. See Section 4for background on
G/P , and see Section 7 and Appendix A for background on G / B . Wededuce that ( G/P, ( G/P ) ≥ , Q J ) is a TNN space from known results in Corollary 4.20. Inorder to construct a Fomin–Shapiro atlas, we consider the (infinite-dimensional) affine flagvariety G / B associated to G . It is stratified into (finite-dimensional) affine Richardsonvarieties G / B = (cid:70) ˜ h ≤ ˜ f ∈ ˜ W ◦ R ˜ f ˜ h , where ˜ W is the affine Weyl group and ≤ denotes its Bruhatorder. There exists an order-reversing injective map ψ : Q J → ˜ W , defined by [HL15],see (7.7). The set of minimal elements of Q J equals { ( u, u ) | u ∈ W J } , where W J is the setof minimal length parabolic coset representatives of the Weyl group, see Section 4.6. Foreach minimal element f := ( u, u ) ∈ Q J , ψ identifies the interval [ f, ˆ1] of Q J with (the dualof) a certain interval [ τ Jλ , τ uλ ] ⊂ ˜ W . For the case G/P = Gr( k, n ), elements of Q J are inbijection with L -diagrams of [Pos07], and ψ sends a L-diagram indexing a positroid cell tothe corresponding bounded affine permutation of [KLS14], see Example 9.6.In Section 7.3, we lift ψ to the geometric level: given a minimal element f := ( u, u ) ∈ Q J ,we introduce a map ¯ ϕ u : C ( J ) u → G / B defined on an open dense subset C ( J ) u ⊂ G/P . Weshow in Theorem 7.2 that for g ∈ Q J such that g (cid:23) f , ¯ ϕ u sends C ( J ) u ∩ ◦ Π g isomorphically tothe affine Richardson cell ◦ R ψ ( f ) ψ ( g ) .For every ˜ g ∈ ˜ W , we consider an open dense subset C ˜ g ⊂ G / B defined by C ˜ g := ˜ g ·B − ·B / B ,as well as affine Schubert and opposite Schubert cells ◦ X ˜ g = (cid:70) ˜ h ≤ ˜ g ◦ R ˜ g ˜ h , ◦ X ˜ g = (cid:70) ˜ g ≤ ˜ f ◦ R ˜ f ˜ g . InProposition 8.2, we give a natural isomorphism(2.2) C ˜ g ∼ −→ ◦ X ˜ g × ◦ X ˜ g , which restricts to ( C ˜ g ∩ ◦ R ˜ f ˜ h ) ∼ −→ ◦ R ˜ f ˜ g × ◦ R ˜ g ˜ h for all ˜ h ≤ ˜ g ≤ ˜ f .A finite-dimensional analog of this map is due to [KWY13], and similar maps have beenconsidered by [KL79, FS00]. The action of ϑ on ◦ X ˜ g essentially amounts to multiplying byan element of the affine torus, and thus preserves ◦ R ˜ g ˜ h for all ˜ h ≤ ˜ g . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 7
Let us now fix g ∈ Q J , and choose some minimal element f := ( u, u ) ∈ Q J such that f (cid:22) g . Then the map ¯ ϕ u is defined on an open dense subset C ( J ) u ⊂ G/P , and let us denoteby O g ⊂ C ( J ) u the preimage of C ψ ( g ) under ¯ ϕ u . The diffeomorphism (2.1) is obtained byconjugating the isomorphism (2.2) by the map ¯ ϕ u . The smooth cone ( Z g , ϑ g ) is extractedfrom the corresponding structure on ◦ X ψ ( g ) . As we have already mentioned, the hardeststep in the proof consists of showing (FS5). To achieve this, we study subtraction-freeparametrizations of partial flag varieties in Section 5, and then use them to show that somegeneralized minors of a particular group element ζ ( J ) u,v ( x ) from Section 6 do not vanish for all x ∈ Star ≥ g . The definition of ζ ( J ) u,v ( x ) is quite technical, but we conjecture in Section 9 that inthe Grassmannian case, these generalized minors specialize to simple functions on Gr( k, n )that we call u -truncated minors . We complete the proof of Theorem 2.5 in Section 8.3.3. Topological results
Throughout this section, we assume that ( Y , Y ≥ , Q ) is a TNN space that admits a Fomin–Shapiro atlas. Thus for each g ∈ Q , we have the objects O g , Z g , ϑ g , and ¯ ν g from Defini-tion 2.3. Additionally, we assume some familiarity with basic theory of smooth manifolds,see e.g. [Lee13].3.1. Links.
Throughout, we denote the two components of the map ¯ ν g from (2.1) by ¯ ν g =(¯ ν g, , ¯ ν g, ), where ¯ ν g, : O g → ◦ Y g ∩ O g and ¯ ν g, : O g → Z g . We set Star ≥ g,h := Y ≥ h ∩ Star ≥ g = (cid:70) g (cid:22) g (cid:48) (cid:22) h Y > g (cid:48) . Let N g be the integer from Definition 2.2 such that Z g ⊂ R N g . Definition 3.1.
Let g (cid:22) h ∈ Q . Denote Z ≥ g := ¯ ν g, (cid:0) Star ≥ g (cid:1) , Z ≥ g,h := ¯ ν g, (cid:0) Star ≥ g,h (cid:1) , Z > g,h := Z ≥ g ∩ ◦ Z g,h ,S g := { x ∈ R N g : (cid:107) x (cid:107) = 1 } , Lk ≥ g,h := Z ≥ g,h ∩ S g , Lk > g,h := Z > g,h ∩ S g . Note that by (FS3), we have Z ≥ g,h = (cid:71) g (cid:22) g (cid:48) (cid:22) h Z > g,g (cid:48) , Lk ≥ g,h = (cid:71) g ≺ g (cid:48) (cid:22) h Lk > g,g (cid:48) . (3.1)In the latter disjoint union, we have Lk > g,g = ∅ since ◦ Z g,g = { } by (FS1). Lemma 3.2.
Let g ≺ h ∈ Q . (i) For all x ∈ O g , we have x ∈ Y > h if and only if ¯ ν g ( x ) ∈ Y > g × Z > g,h . (ii) Z > g,h is an embedded submanifold of Z g of dimension dim( h ) − dim( g ) that intersects S g transversely. For all t ∈ R > and x ∈ Z > g,h , we have ϑ ( t, x ) ∈ Z > g,h . (iii) Lk > g,h is a contractible smooth manifold of dimension dim( h ) − dim( g ) − . (iv) Lk ≥ g,h is a compact subset of Z g . Before we prove these properties, let us state some preliminary results on smooth manifolds.Given smooth manifolds
A, B and a smooth map f : A → B , a point a ∈ A is called a regular point of f if the differential of f at a is surjective. Similarly, b ∈ B is called a regularvalue of f if f − ( b ) consists of regular points. In this case f − ( b ) is a closed embeddedsubmanifold of A of dimension dim( A ) − dim( B ) [Lee13, Corollary 8.10]. Lemma 3.3.
Suppose that
A, B are smooth manifolds and B (cid:48) ⊂ B is such that A × B (cid:48) isan embedded submanifold of A × B . Then B (cid:48) is an embedded submanifold of B . PAVEL GALASHIN, STEVEN N. KARP, AND THOMAS LAM
Proof.
Choose a ∈ A . Clearly a is a regular value of the projection A × B (cid:48) → A , so { a } × B (cid:48) is an embedded submanifold of A × B (cid:48) , and hence of { a } × B . (cid:3) We also recall some facts about ϑ from [GKL17]. Lemma 3.4.
Let ϑ : R > × R N → R N be a smooth map satisfying (SC1) and (SC2) . (i) We have ϑ ( t,
0) = 0 for all t ∈ R > . (ii) We have lim t → ϑ ( t, x ) = 0 for all x ∈ R N . (iii) For all x ∈ R N \ { } , there exists a unique t ∈ R > such that (cid:107) ϑ ( t, x ) (cid:107) = 1 , which wedenote by t ( x ) . The function t : R N \ { } → R > is continuous.Proof. The function f : R × R N → R N defined by f ( t, x ) = ϑ ( e − t , x ) is a contractiveflow, as defined in [GKL17, Definition 2.1]. Therefore the statements follow from [GKL17,Lemma 2.2] and the claim in the proof of [GKL17, Lemma 2.3]. (cid:3) Proof of Lemma 3.2. (i): We prove this more generally for g (cid:22) h . The set Star ≥ g is con-nected since it contains a connected dense subset Y > . Therefore ¯ ν g, (Star ≥ g ) is a connectedsubset of ◦ Y g ∩ O g . By (FS4), it contains Y > g , therefore ¯ ν g, (Star ≥ g ) = Y > g by (TNN4).By definition, ¯ ν g, (Star ≥ g,h ) = Z ≥ g,h , thus ¯ ν g (Star ≥ g,h ) ⊂ Y > g × Z ≥ g,h . By (FS3), we get¯ ν g ( Y > h ) ⊂ Y > g × Z > g,h . In particular, Z > g,h = ¯ ν g, ( Y > h ) is a connected subset of ◦ Z g,h . Let C be the connected component of ◦ Z g,h containing Z > g,h . By (FS3), ¯ ν − g ( Y > g × C ) is a connectedsubset of ◦ Y h ∩ O g , which contains Y > h as we have just shown. Therefore we must have¯ ν − g ( Y > g × C ) = Y > h by (TNN4), which shows that Z > g,h = C is a connected component of ◦ Z g,h . Thus indeed ¯ ν g ( Y > h ) = Y > g × Z > g,h .(ii): By (TNN4) and (TNN2), Y > h is an embedded submanifold of Y . Applying ¯ ν g andusing (i), we get that Y > g × Z > g,h is an embedded submanifold of Y > g × Z g , of dimensiondim( h ) − dim( g ). By Lemma 3.3, Z > g,h is an embedded submanifold of Z g . Moreover, it followsfrom (FS2) that ϑ g ( t, Z > g,h ) = Z > g,h for all t ∈ R > , since Z > g,h is a connected component of ◦ Z g,h . Thus 1 is a regular value of the restriction (cid:107) · (cid:107) : Z > g,h → R > , so the manifolds S g and Z > g,h intersect transversely inside R N g .(iii): By (ii), Lk > g,h = Z > g,h ∩ S g is an embedded submanifold of Z g of dimension dim( h ) − dim( g ) −
1. To show that it is contractible, we use the fact that a retract of a contractiblespace is contractible [Hat02, Exercise 0.9]. Since Y > h is contractible (by (TNN4)), so is¯ ν g ( Y > h ) = Y > g × Z > g,h . Then { x } × Z > g,h is a retract of Y > g × Z > g,h for any x ∈ Y > g , so Z > g,h is contractible. Finally, by (ii) and Lemma 3.4(iii), the map sending x (cid:55)→ ϑ g ( t ( x ) , x ) givesa retraction Z > g,h → Lk > g,h .(iv): By (FS5), Star ≥ g,h = Y ≥ h ∩ Star ≥ g = Y ≥ h ∩ O g is a closed subset of O g . Thus¯ ν g (Star ≥ g,h ) is a closed subset of Y > g × Z g . Since ¯ ν g (Star ≥ g,h ) = Y > g × Z ≥ g,h (by (i) and (3.1)),we get that Z ≥ g,h is a closed subset of Z g . It follows that Lk ≥ g,h = Z ≥ g,h ∩ S g is a closed andbounded subset of Z g , which is closed in R N g by Definition 2.2. (cid:3) Recall that Cone( A ) := ( A × R ≥ ) / ( A × { } ) is the open cone over A . We denote by c := ( ∗ , ∈ Cone( A ) its cone point . Proposition 3.5.
Let g ≺ h ∈ Q . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 9 (i)
We have a homeomorphism Z ≥ g,h ∼ −→ Cone(Lk ≥ g,h ) sending to the cone point c , andsending Z > g,g (cid:48) to Lk > g,g (cid:48) × R > for all g ≺ g (cid:48) (cid:22) h . (ii) We have a homeomorphism
Star ≥ g,h ∼ −→ Y > g × Cone(Lk ≥ g,h ) sending Y > g to Y > g × { c } .Proof. (i): Define a map ξ : Z ≥ g,h → Cone(Lk ≥ g,h ) sending 0 (cid:55)→ c and x (cid:55)→ (cid:16) ϑ g ( t ( x ) , x ) , t ( x ) (cid:17) for x ∈ Z ≥ g,h \ { } , where t ( x ) is defined in Lemma 3.4(iii) and ϑ g ( t ( x ) , x ) ∈ Lk ≥ g,h by Lemma 3.2(ii). We claim that ξ is a homeomorphism. Note that ξ has an inverse ξ − , which sends c (cid:55)→ y, t ) (cid:55)→ ϑ g ( t, y ) for ( y, t ) ∈ Cone(Lk ≥ g,h ) \ { c } = Lk ≥ g,h × R > .By Lemma 3.4(iii), ξ is continuous on Z ≥ g,h \ { } and ξ − is continuous on Lk ≥ g,h × R > . Itremains to show that ξ is continuous at 0 and that ξ − is continuous at c .Suppose that ( x n ) n ≥ is a sequence in Z ≥ g,h \{ } converging to 0. We claim that t ( x n ) → ∞ as n → ∞ . Otherwise, after passing to a subsequence, we may assume that there exists R ∈ R > such that t ( x n ) ≤ R for all n ≥
0. Then (SC2) implies that (cid:107) ϑ g ( R, x n ) (cid:107) ≥(cid:107) ϑ g ( t ( x n ) , x n ) (cid:107) = 1 for all n ≥
0. Taking n → ∞ gives (cid:107) ϑ g ( R, (cid:107) ≥
1, contradict-ing Lemma 3.4(i). This shows that ξ is continuous at 0.Suppose now that (( y n , t n )) n ≥ is a sequence in Lk ≥ g,h × R > converging to c , i.e. t n → D ( t ) := max x ∈ S g (cid:107) ϑ g ( t, x ) (cid:107) is increasing in t , by compactness of S g and (SC2).We have lim t → D ( t ) = 0 by Lemma 3.4(ii) and compactness of S g (more precisely, by Dini’s theorem ). Therefore ξ − ( y n , t n ) = ϑ g ( t n , y n ) converges to 0 as n → ∞ , showing that ξ − is continuous at c .(ii): By Lemma 3.2(i), ¯ ν g restricts to a homeomorphism Star ≥ g,h ∼ −→ Y > g × Z ≥ g,h , whichby (FS4) sends Y > g to Y > g × { } . The result follows from (i). (cid:3) Our next aim is to analyze the local structure of the space Lk ≥ g,h . For two topologicalspaces A and B and a ∈ A , b ∈ B , a local homeomorphism between ( A, a ) and ( B, b ) is ahomeomorphism from an open neighborhood of a in A to an open neighborhood of b in B which sends a to b . Lemma 3.6.
Let g ≺ p (cid:22) h ∈ Q , x p ∈ Lk > g,p , and set d := dim( p ) − dim( g ) − . Then thereexists a local homeomorphism between (cid:0) Lk ≥ g,h , x p (cid:1) and (cid:0) Z ≥ p,h × R d , (0 , (cid:1) . Proof.
Choose some x g ∈ Y > g and consider the open subset H p ⊂ Z g defined by H p := { x ∈ Z g | ¯ ν − g ( x g , x ) ∈ O p } . Introduce a map θ g,p : H p ∩ S g → Z p , x (cid:55)→ ¯ ν p, (¯ ν − g ( x g , x )) . Since x p ∈ Lk > g,p ⊂ Z > g,p and x g ∈ Y > g , we get ¯ x p := ¯ ν − g ( x g , x p ) ∈ Y > p by Lemma 3.2(i).By (FS5), we have Y > p ⊂ Star ≥ p ⊂ O p , thus x p ∈ H p . Since H p is open in Z g , H p ∩ S g isan open subset of Z g ∩ S g , which is nonempty since it contains x p . We have θ g,p ( x p ) = 0by (FS4).We claim that x p is a regular point of θ g,p . By (FS4), the differential of ¯ ν p, : O p → Z p issurjective at ¯ x p , and its kernel is the tangent space of ◦ Y p at ¯ x p . By (TNN4) and (FS5), Y > p is a connected component of ◦ Y p ∩ O p , and it contains ¯ x p = ¯ ν − g ( x g , x p ) as we have shownabove. Therefore x p is a regular point of θ g,p if and only if the manifolds Y > p and F :=¯ ν − g ( { x g } × ( H p ∩ S g )) intersect transversely at ¯ x p . By Lemma 3.2(i), we have ¯ ν g ( Y > p ) = Y > g × Z > g,p , and clearly ¯ ν g ( F ) = { x g } × ( H p ∩ S g ). These two manifolds intersect transverselyat ( x g , x p ) by Lemma 3.2(ii). We have shown that x p is a regular point of θ g,p .By the submersion theorem (see e.g. [Kos93, Corollary A(1.3)]), there exist local coordi-nates centered at x p ∈ H p ∩ S g and at 0 ∈ Z p in which θ g,p is just the canonical projection R dim( H p ∩ S g ) → R dim( Z p ) . Recall that Q contains a unique maximal element ˆ1, and by (2.1)we have dim( Z g ) = codim( g ) := dim(ˆ1) − dim( g ). Thus dim( H p ∩ S g ) = codim( g ) − Z p ) = codim( p ), and dim( H p ∩ S g ) − dim( Z p ) = d . We have shown that there exist openneighborhoods U of x p in H p ∩ S g and V of 0 in Z p and a diffeomorphism β : U ∼ −→ V × R d send-ing x p to (0 ,
0) such that the first component of β coincides with the restriction θ g,p : U → V .In order to complete the proof, we need to show that the image β ( U ∩ Lk ≥ g,h ) equals( V ∩ Z ≥ p,h ) × R d . We may assume that U is connected. Suppose we are given x ∈ U andlet r ∈ Q be such that x (cid:48) := ¯ ν − g ( x g , x ) ∈ ◦ Y r . Since U ⊂ H p , x (cid:48) belongs to O p ⊂ Star p byDefinition 2.3, and therefore p (cid:22) r . By Lemma 3.2(i), we have x ∈ U ∩ Lk > g,r if and onlyif x (cid:48) ∈ Y > r . On the other hand, ¯ ν p, (¯ ν − g ( { x g } × U )) is a connected subset of ◦ Y p ∩ O p thatcontains ¯ ν p, (¯ x p ) ∈ Y > p . Thus ¯ ν p, (¯ ν − g ( x g , U )) ⊂ Y > p by (TNN4). It follows that x (cid:48) ∈ Y > r if and only if θ g,p ( x ) = ¯ ν p, ( x (cid:48) ) belongs to Z > p,r . The result follows by taking the union overall p (cid:22) r (cid:22) h , using (3.1). (cid:3) Topological background.
Regular CW complexes.
We refer to [Hat02, LW69] for background on CW complexes.
Definition 3.7.
Let X be a Hausdorff space. We call a finite disjoint union X = (cid:70) α ∈ Q X α a regular CW complex if it satisfies the following two properties.(CW1) For each α ∈ Q , there exists a homeomorphism from the closure X α to a closed ball B which sends X α to the interior of B .(CW2) For each α ∈ Q , there exists Q (cid:48) ⊂ Q such that X α = (cid:70) β ∈ Q (cid:48) X β .The face poset of X is the poset ( Q, (cid:22) ), where β (cid:22) α if and only if X β ⊂ X α .The condition (CW2) is often omitted from the definition of a regular CW complex, but isnecessary in order to apply the arguments of [Bj¨o84]. We remark that the cell decompositionof Y ≥ satisfies (CW2) by (TNN5).3.2.2. Posets.
We review the definitions of graded , thin , and shellable for finite posets,though we will not need to work with them in our arguments. We refer to [Bj¨o80, Sta12]for background.A finite poset ( Q, (cid:22) ) is called graded if every maximal chain in Q has the same length (cid:96) , in which case we denote rank( Q ) := (cid:96) . For x ≤ z ∈ Q , we denote by [ x, z ] := { y ∈ Q | x ≤ y ≤ z } the corresponding interval . Note that the intervals in a graded poset Q are alsograded, and we call Q thin if every interval of rank 2 has exactly 4 elements.The order complex of a graded poset Q is the pure (rank( Q ) − Q , and whose faces are the chains in Q . We saythat Q is shellable if its order complex is shellable, i.e. its maximal faces can be ordered as F , . . . , F n so that for 2 ≤ k ≤ n , F k ∩ (cid:0)(cid:83) ≤ i Proposition 3.8 ([Bj¨o80, Proposition 4.2]) . If a graded poset is shellable, then so are eachof its intervals. See [Bj¨o84, §§ 2, 3] for the proof of the following result. Theorem 3.9 ([LW69, DK74, Bj¨o84]) . Suppose that X is a regular CW complex with faceposet Q . If Q (cid:116) { ˆ0 , ˆ1 } (obtained by adjoining a minimum ˆ0 and a maximum ˆ1 to Q ) isgraded, thin, and shellable, then X is homeomorphic to a sphere of dimension rank( Q ) − . Poincar´e conjecture. Recall that an n -dimensional topological manifold with boundary is a Hausdorff space C such that every point x ∈ C has an open neighborhood homeomorphiceither to R n , or to R ≥ × R n − via a homeomorphism which takes x to a point in { } × R n − .In the latter case, we say that x belongs to the boundary of C , denoted ∂C .The following is a well known consequence of the (generalized) Poincar´e conjecture dueto Smale [Sma61], Freedman [Fre82], and Perelman [Per02]. We refer to [Dav08, Theo-rem 10.3.3(ii)] for this formulation. Theorem 3.10 ([Sma61, Fre82, Per02]) . Let C be a compact contractible n -dimensionaltopological manifold with boundary, such that its boundary ∂C is homeomorphic to an ( n − -dimensional sphere. Then C is homeomorphic to an n -dimensional closed ball. For n ≥ 6, Theorem 3.10 can be proved using the topological h -cobordism theorem [Mil65,KS77]. We sketch another standard argument for deducing Theorem 3.10 from the Poincar´econjecture when n is arbitrary. The boundary of C is collared by [Bro62, Theorem 2], i.e.there exists a homeomorphism from an open neighborhood of ∂C in C to ∂C × [0 , ∂C to ∂C × { } . Thus we can attach the (collared) boundary of a closed n -dimensionalball to the (collared) boundary of C , obtaining a topological manifold C (cid:48) . By van Kampen’stheorem, C (cid:48) is simply connected. It is easy to see from the Mayer–Vietoris sequence that C (cid:48) isa homology sphere. Thus C (cid:48) must be homeomorphic to a sphere by the Poincar´e conjecture.Therefore C is homeomorphic to a closed ball by Brown’s Schoenflies theorem [Bro60].The following is also a consequence of Brown’s collar theorem [Bro62, Theorem 2]. Proposition 3.11. Suppose that C is a topological manifold with boundary ∂C . Then C ishomotopy equivalent to its interior C \ ∂C . Link induction.Theorem 3.12. Let ( Y , Y ≥ , Q ) be a TNN space that admits a Fomin–Shapiro atlas, and let g ≺ h ∈ Q . Then Lk ≥ g,h = (cid:70) g ≺ g (cid:48) (cid:22) h Lk > g,g (cid:48) (cf. (3.1) ) is a regular CW complex homeomorphicto a closed ball of dimension dim( h ) − dim( g ) − .Proof. We proceed by induction on d := dim( h ) − dim( g ) − 1. For the base case d = 0, wehave Lk ≥ g,h = Lk > g,h , which is a 0-dimensional contractible manifold by Lemma 3.2(iii). ThusLk ≥ g,h is a point, and we are done with the base case. Assume now that d > d (cid:48) < d . We need to verify (CW1) and (CW2) when X α = Lk > g,h (the othercases follow from the induction hypothesis).We claim that Lk ≥ g,h is a topological manifold with boundary ∂ Lk ≥ g,h , where ∂ Lk ≥ g,h = (cid:71) g ≺ g (cid:48) ≺ h Lk > g,g (cid:48) . (3.2) Let x ∈ Lk ≥ g,h . By (3.1), we have x ∈ Lk > g,g (cid:48) for a unique g ≺ g (cid:48) (cid:22) h . If g (cid:48) = h , then x hasan open neighborhood in Lk ≥ g,h homeomorphic to R d by Lemma 3.2(iii). If g (cid:48) ≺ h , then byLemma 3.6 we have a local homeomorphism (Lk ≥ g,h , x ) ∼ −→ ( Z ≥ g (cid:48) ,h × R d (cid:48) , (0 , d (cid:48) :=dim( g (cid:48) ) − dim( g ) − 1. By Proposition 3.5(i), we have a homeomorphism Z ≥ g (cid:48) ,h ∼ −→ Cone(Lk ≥ g (cid:48) ,h )which sends 0 to the cone point c . By the induction hypothesis, Lk ≥ g (cid:48) ,h is homeomorphic toa ( d − d (cid:48) − ≥ g (cid:48) ,h ) ∼ −→ R ≥ × R d − d (cid:48) − which sends c to (0 , ≥ g,h , x ) ∼ −→ ( R ≥ × R d − d (cid:48) − × R d (cid:48) , (0 , , ≥ g,h is a topological manifold with boundarygiven by (3.2).By Lemma 3.2(iv), Lk ≥ g,h is compact. By Lemma 3.2(iii) and Proposition 3.11, Lk ≥ g,h iscontractible. Thus Lk ≥ g,h is a compact contractible topological manifold with boundary. Inaddition, the boundary ∂ Lk ≥ g,h is a regular CW complex by the induction hypothesis. Itsface poset is the interval ( g, h ) := [ g, h ] \ { g, h } in Q . The interval [ g, h ] is graded, thin,and shellable by (TNN1) and Proposition 3.8, thus ∂ Lk ≥ g,h is homeomorphic to a ( d − ≥ g,h to a d -dimensional closed ball B . By (3.2), it sends the interior Lk > g,h to the interior of B .This proves (CW1), and (CW2) follows from (3.2). This completes the induction. (cid:3) Proof of Theorem 2.4. The proof follows the same structure as the proof of Theorem 3.12.We proceed by induction on dim( h ). If dim( h ) = 0, then Y ≥ h = Y > h is a point by (TNN4),which finishes the base case.Let dim( h ) > 0. We want to show that Y ≥ h is a topological manifold with boundary(3.3) ∂ Y ≥ h = (cid:71) g ≺ h Y > g . Let x ∈ Y ≥ h . By (TNN5), we have x ∈ Y > g for a unique g (cid:22) h . If g = h , then x has anopen neighborhood in Y ≥ h homeomorphic to R dim( h ) by (TNN4). If g ≺ h , then Star ≥ g is anopen subset of Y ≥ (its complement is (cid:83) g (cid:48) (cid:54)(cid:23) g Y ≥ g (cid:48) , which is closed by (TNN5)). Thus Star ≥ g,h is an open neighborhood of x in Y ≥ h . By Proposition 3.5(ii), (TNN4), and Theorem 3.12,we get a homeomorphism Star ≥ g,h ∼ −→ R ≥ × R dim( h ) − whose first component sends x ∈ Y > g to 0 ∈ R ≥ . This shows that Y ≥ h is a topological manifold with boundary given by (3.3).By (TNN3) and (TNN5), Y ≥ h is compact. By (TNN4) and Proposition 3.11, Y ≥ h iscontractible. Thus Y ≥ h is a compact contractible topological manifold with boundary. Inaddition, the boundary ∂ Y ≥ h is a regular CW complex by the induction hypothesis. Itsface poset is the interval (ˆ0 , h ) in (cid:98) Q . The interval [ˆ0 , h ] is graded, thin, and shellableby (TNN1) and Proposition 3.8, thus ∂ Y ≥ h is homeomorphic to a ( d − (cid:3) Remark 3.13. We note that Theorems 2.5 and 3.12 imply the result of Hersh [Her14](see Corollary 1.3) that the link of the identity in the Bruhat decomposition of U ≥ is aregular CW complex. (Recall that U is the unipotent radical of the standard Borel subgroup B ⊂ G .) Indeed, let B − ⊂ G denote the opposite Borel subgroup. Then the natural inclusion U (cid:44) → G/B − sends U to the opposite Schubert cell Star (id , id) indexed by id ∈ W , and the EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 13 composition of this map with ¯ ν (id , id) sends the link of the identity in U w> homeomorphicallyto Lk ≥ , id) , (id ,w ) for all w ∈ W . 4. G/P : preliminaries We give some background on partial flag varieties. Throughout, K denotes an algebraicallyclosed field of characteristic 0, and K ∗ := K \ { } denotes its multiplicative group.4.1. Pinnings. We recall some standard notions that can be found in e.g. [Lus94, § G is a simple and simply connected algebraic group over K , with T ⊂ G amaximal torus. Let B, B − be opposite Borel subgroups satisfying T = B ∩ B − . We identify G with its set of K -valued points. When K = C , we assume that G and T are split over R , and denote by G ( R ) ⊂ G and T ( R ) ⊂ T the sets of their R -valued points. (Thus whatwas denoted by G in Section 1 is from now on denoted by G ( R ). We are also assuming that G is a simple algebraic group, rather than semisimple; our results for the case of a generalsemisimple group reduce to the simple case by taking products.)Let X ( T ) := Hom( T, K ∗ ) be the weight lattice , and for a weight γ ∈ X ( T ) and a ∈ T , wedenote the value of γ on a by a γ . Let Φ ⊂ X ( T ) be the set of roots . We have a decompositionΦ = Φ + (cid:116) Φ − of Φ as a union of positive and negative roots corresponding to the choice of B , see [Hum75, § α ∈ Φ, we write α > α ∈ Φ + and α < α ∈ Φ − . Let { α i } i ∈ I be the simple roots corresponding to the choice of Φ + . For every i ∈ I , we have ahomomorphism φ i : SL → G , and denote(4.1) x i ( t ) := φ i (cid:18) t (cid:19) , y i ( t ) := φ i (cid:18) t (cid:19) , ˙ s i := φ i (cid:18) − 11 0 (cid:19) = y i (1) x i ( − y i (1) . The data ( T, B, B − , x i , y i ; i ∈ I ) is called a pinning for G . Let W := N G ( T ) /T be the Weylgroup, and for i ∈ I , let s i ∈ W be represented by ˙ s i above. Then W is generated by { s i } i ∈ I ,and ( W, { s i } i ∈ I ) is a finite Coxeter group. For w ∈ W , the length (cid:96) ( w ) is the minimal n suchthat w = s i · · · s i n for some i , . . . , i n ∈ I . When n = (cid:96) ( w ), we call i := ( i , . . . , i n ) a reducedword for w . The representatives { ˙ s i } i ∈ I satisfy the braid relations [Spr98, Proposition 9.3.2],so we set ˙ w := ˙ s i · · · ˙ s i n ∈ G , and this representative does not depend on the choice of i .Let Y ( T ) := Hom( K ∗ , T ) be the coweight lattice . For i ∈ I , we have a simple coroot α ∨ i ( t ) := φ i (cid:18) t t − (cid:19) ∈ Y ( T ). Denote by (cid:104)· , ·(cid:105) : Y ( T ) × X ( T ) → Z the natural pairing sothat for γ ∈ X ( T ), β ∈ Y ( T ), and t ∈ K ∗ , we have ( β ( t )) γ = t (cid:104) β,γ (cid:105) . Let { ω i } i ∈ I ⊂ X ( T ) bethe fundamental weights . They form a dual basis to { α ∨ i } i ∈ I : (cid:104) α ∨ j , ω i (cid:105) = δ ij for i, j ∈ I .The Weyl group W acts on T by conjugation, which induces an action on Y ( T ), X ( T ),and Φ. For γ ∈ X ( T ), t ∈ K ∗ , a ∈ T , and w ∈ W , we have [FZ99, (1.2) and (2.5)](4.2) ( ˙ w − a ˙ w ) γ = a wγ , ax i ( t ) a − = x i ( a α i t ) , ay i ( t ) a − = y i ( a − α i t ) . Following [BZ97, (1.6) and (1.7)] (see also [FZ99, (2.1) and (2.2)]), we define two involutiveanti-automorphisms x (cid:55)→ x T and x (cid:55)→ x ι of G by( x i ( t )) T = y i ( t ) , ( y i ( t )) T = x i ( t ) , ˙ w T = ˙ w − , a T = a (4.3) ( x i ( t )) ι = x i ( t ) , ( y i ( t )) ι = y i ( t ) , ˙ w ι = ˙ z, a ι = a − , (4.4)for all i ∈ I , t ∈ K ∗ , a ∈ T , and w ∈ W , where z := w − . We note that when z = w − ∈ W and i = ( i , . . . , i n ) is a reduced word for w then ˙ w − = ˙ s − i n · · · ˙ s − i while ˙ z = ˙ s i n · · · ˙ s i . Subgroups of U . We say that a subset Θ ⊂ Φ is bracket closed if whenever α, β ∈ Θare such that α + β ∈ Φ, we have α + β ∈ Θ. For a bracket closed subset Θ ⊂ Φ + ,define U (Θ) ⊂ U to be the subgroup generated by { U α | α ∈ Θ } , where U α is a rootsubgroup of G , see [Hum75, Theorem 26.3]. For a bracket closed subset Θ ⊂ Φ − , let U − (Θ) := U ( − Θ) T ⊂ U − .Given closed subgroups H , . . . , H n of an algebraic group H , we say that H , · · · , H n directly span H if the multiplication map H × · · · × H n → H is a biregular isomorphism. Lemma 4.1 ([Hum75, Proposition 28.1]) . Let Θ ⊂ Φ + be a bracket closed subset. (i) If Θ = (cid:70) ni =1 Θ i and Θ , Θ , . . . , Θ n ⊂ Φ + are bracket closed then U (Θ) is directlyspanned by U (Θ ) , . . . , U (Θ n ) . (ii) In particular, U (Θ) is directly spanned by { U α | α ∈ Θ } in any order, and therefore U (Θ) ∼ = K | Θ | . For α ∈ Φ and w ∈ W , we have ˙ wU α ˙ w − = U wα . For w ∈ W , define Inv( w ) :=( w − Φ − ) ∩ Φ + . The subsets Inv( w ) and Φ + \ Inv( w ) are bracket closed [Hum75, § U (Inv( w )) = ˙ w − U − ˙ w ∩ U. Bruhat projections. Let Θ ⊂ Φ + be bracket closed, and let w ∈ W . Define Θ :=Θ ∩ Inv( w ) and Θ := Θ \ Inv( w ). Thus both sets are bracket closed and˙ wU (Θ) ˙ w − ∩ U − = U − ( w Θ ) , ˙ wU (Θ) ˙ w − ∩ U = U ( w Θ ) . Denote U := U − ( w Θ ) and U := U ( w Θ ). Then by Lemma 4.1(i), the multiplication mapgives isomorphisms µ : U × U → ˙ wU (Θ) ˙ w − and µ : U × U → ˙ wU (Θ) ˙ w − . Denoteby ν : ˙ wU (Θ) ˙ w − → U and ν : ˙ wU (Θ) ˙ w − → U the second component of µ − and µ − ,respectively. In other words, given g ∈ ˙ wU (Θ) ˙ w − , ν ( g ) is the unique element in U ∩ U g and ν ( g ) is the unique element in U ∩ U g . Lemma 4.2 ([KWY13, Lemma 2.2]) . The map ( ν , ν ) : ˙ wU (Θ) ˙ w − → U × U is a biregularisomorphism. Commutation relations. Let a, b ∈ W be such that (cid:96) ( ab ) = (cid:96) ( a ) + (cid:96) ( b ). Then(4.6) Inv( b ) ⊂ Inv( ab ) , b − Inv( a ) ⊂ Φ + , and Inv( ab ) = (cid:0) b − Inv( a ) (cid:1) (cid:116) Inv( b ) . Thus by Lemma 4.1(i), the multiplication map gives an isomorphism(4.7) ˙ b − U (Inv( a ))˙ b × U (Inv( b )) ∼ −→ U (Inv( ab )) . We will later need the following consequences of (4.7): if (cid:96) ( ab ) = (cid:96) ( a ) + (cid:96) ( b ) then˙ b − · ( U − ∩ ˙ a − U ˙ a ) ⊂ ( U − ∩ ( ˙ a ˙ b ) − U ˙ a ˙ b ) · ˙ b − , (4.8) ( U ∩ ˙ a − U − ˙ a ) · ˙ b ⊂ ˙ b · ( U ∩ ( ˙ a ˙ b ) − U − ˙ a ˙ b ) . (4.9)Multiplying both sides of (4.9) by ˙ b − on the left, we get ˙ b − U (Inv( a ))˙ b ⊂ U (Inv( ab )), whichfollows from (4.6). We obtain (4.8) from (4.9) by applying the map x (cid:55)→ x T , see (4.3). Lemma 4.3. Let α ∈ Φ + and i ∈ I be such that α (cid:54) = α i . Let Ψ ⊂ Φ denote the set of allroots of the form mα − rα i for integers m > , r ≥ . Then Ψ is a bracket closed subset of Φ + , and for all t ∈ K we have y i ( t ) U α y i ( − t ) ⊂ U (Ψ) . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 15 Proof. Let x ∈ U α and x (cid:48) := ˙ s − i x ˙ s i ∈ U s i α . By [BB05, Lemma 4.4.3], s i permutes Φ + \ { α i } (in particular, s i α > y i ( t ) · x · y i ( − t ) = ˙ s i x i ( − t ) ˙ s − i · x · ˙ s i x i ( t ) ˙ s − i = ˙ s i x i ( − t ) · x (cid:48) · x i ( t ) ˙ s − i . Denote by Ψ (cid:48) ⊂ Φ the set of all roots of the form ms i α + rα i for integers m, r > 0. It isclear that Ψ (cid:48) ⊂ Φ + \ { α i , s i α } is a bracket closed subset. By [Hum75, Lemma 32.5], we have x i ( − t ) x (cid:48) x i ( t ) x (cid:48)− ∈ U (Ψ (cid:48) ), so x i ( − t ) x (cid:48) x i ( t ) ∈ U (Ψ (cid:48) ) x (cid:48) . Thus Ψ (cid:48)(cid:48) := s i Ψ (cid:48) is also a bracketclosed subset of Φ + \ { α i , α } , and we have ˙ s i U (Ψ (cid:48) ) x (cid:48) ˙ s − i = U (Ψ (cid:48)(cid:48) ) x . Clearly, Ψ = Ψ (cid:48)(cid:48) (cid:116) { α } .We thus have y i ( t ) U α y i ( − t ) ⊂ U (Ψ (cid:48)(cid:48) ) U α = U (Ψ). (cid:3) Flag variety and Bruhat decomposition. Let G/B be the flag variety of G (over K ). We recall some standard properties of the Bruhat decomposition that can be found ine.g. [Hum75, § ◦ X w = B ˙ wB/B, ◦ X v = B − ˙ vB/B, ◦ R v,w := ◦ X v ∩ ◦ X w (for v ≤ w ∈ W ) . Recall the Bruhat and Birkhoff decompositions: G = (cid:71) w ∈ W B ˙ wB = (cid:71) v ∈ W B − ˙ vB, where(4.11) B − ˙ vB ∩ B ˙ wB = ∅ and ◦ X v ∩ ◦ X w = ∅ for v (cid:54)≤ w ∈ W . (4.12)Let X v denote the (Zariski) closure of ◦ X v . Similarly, let X w denote the closure of ◦ X w ,and then R v,w = X v ∩ X w is the closure of ◦ R v,w in G/B . We have X v = (cid:71) v ≤ v (cid:48) ◦ X v (cid:48) , X w = (cid:71) w (cid:48) ≤ w ◦ X w (cid:48) , (4.13) G/B = (cid:71) v ≤ w ◦ R v,w , R v,w = (cid:71) v ≤ v (cid:48) ≤ w (cid:48) ≤ w ◦ R v (cid:48) ,w (cid:48) . (4.14)For any w ∈ W , i ∈ I , and t ∈ K ∗ , we have x i ( t ) ∈ B − ˙ s i B − , y i ( t ) ∈ B ˙ s i B, (4.15) B ˙ s i B · B ˙ wB ⊂ (cid:40) B ˙ s i ˙ wB, if s i w > w , B ˙ s i ˙ wB (cid:116) B ˙ wB, if s i w < w ,(4.16) B − ˙ s i B − · B − ˙ wB ⊂ (cid:40) B − ˙ s i ˙ wB, if s i w < w , B − ˙ s i ˙ wB (cid:116) B − ˙ wB, if s i w > w ,(4.17) B ˙ vB · B ˙ wB ⊂ B ˙ v ˙ wB for v ∈ W such that (cid:96) ( vw ) = (cid:96) ( v ) + (cid:96) ( w ).(4.18)For t = ( t , . . . , t n ) ∈ ( K ∗ ) n and a reduced word i = ( i , . . . , i n ) for w ∈ W , define(4.19) x i ( t ) := x i ( t ) · · · x i n ( t n ) , and y i ( t ) := y i ( t ) · · · y i n ( t n ) . It follows from (4.15), (4.16), and (4.3) that(4.20) x i ( t ) ∈ B − ˙ wB − , y i ( t ) ∈ B ˙ wB. Parabolic subgroup W J of W . Let J ⊂ I , and denote by W J ⊂ W the subgroupgenerated by { s i } i ∈ J . Let W J be the set of minimal-length coset representatives of W/W J ,see [BB05, § w J be the longest element of W J , and w J := w w J be the maximalelement of W J . Let Φ J ⊂ Φ consist of roots that are linear combinations of { α i } i ∈ J . DefineΦ + J := Φ J ∩ Φ + , Φ − J := Φ J ∩ Φ − , Φ ( J )+ := Φ + \ Φ + J , Φ ( J ) − := Φ − \ Φ − J . The sets Φ + J , Φ ( J )+ , Φ − J , Φ ( J ) − are clearly bracket closed, so consider subgroups U J = U (Φ + J ) , U − J = U − (Φ − J ) , U ( J ) = U (Φ ( J )+ ) , U ( J ) − = U − (Φ ( J ) − ) . In fact, we have(4.21) Φ + J = Inv( w J ) , Φ ( J )+ = Inv( w J ) , ˙ w J U − J ˙ w − J = U J . Let W J max := { ww J | w ∈ W J } . By [BB05, Proposition 2.4.4], every w ∈ W admits aunique parabolic factorization w = w w for w ∈ W J and w ∈ W J , and this factorizationis length-additive. We state some standard facts on parabolic factorizations for later use. Lemma 4.4. (i) If u ∈ W J and s i u < u , then s i u ∈ W J . (ii) Given u ∈ W J and r, r (cid:48) ∈ W J , we have ur ≤ ur (cid:48) if and only if r ≤ r (cid:48) .Proof. For (i) suppose that s i u / ∈ W J , so that s i us j < s i u for some j ∈ J . Then s i us j 1. For (ii), see [BB05, Exercise 2.21]. (cid:3) Lemma 4.5. For any w ∈ W J , we have Inv( w ) ⊂ Φ ( J )+ . In particular, w Φ + J ⊂ Φ + , ˙ wU J ˙ w − ⊂ U , and ˙ wU − J ˙ w − ⊂ U − .Proof. Let α ∈ Φ + be a positive root. Then it can be written as α = (cid:80) i ∈ I c i α i for c i ∈ Z ≥ .Since w ∈ W J , we have wα i > i ∈ J . Thus if wα < 0, we must have c i (cid:54) = 0 for some i / ∈ J , so α ∈ Φ ( J )+ . (cid:3) Lemma 4.6 ([He09]) . Let x, y ∈ W . (i) The set { uv | u ≤ x, v ≤ y } contains a unique maximal element, denoted x ∗ y . Theset { xv | v ≤ y } contains a unique minimal element, denoted x (cid:47) y . (ii) There exist elements u (cid:48) ≤ x and v (cid:48) ≤ y such that x ∗ y = xv (cid:48) = u (cid:48) y , and thesefactorizations are both length-additive. (iii) If x (cid:48) ≤ x , then x (cid:48) ∗ y ≤ x ∗ y and x (cid:48) (cid:47) y ≤ x (cid:47) y . (iv) If xy is length-additive, then x ∗ y = xy and ( xy ) (cid:47) y − = x . The operations ∗ and (cid:47) are called the Demazure product and downwards Demazure product . Proof. The first three parts were shown in [He09, § (cid:47) is the‘mirror image’ of He’s (cid:46) . Part (iv) follows from the definitions of ∗ and (cid:47) . (cid:3) Definition 4.7. Let Q J = { ( v, w ) ∈ W × W J | v ≤ w } . We define an order relation (cid:22) on Q J as follows: for ( v, w ) , ( v (cid:48) , w (cid:48) ) ∈ Q J , we write ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) if and only if there exists r ∈ W J such that vr is length-additive and v (cid:48) ≤ vr ≤ wr ≤ w (cid:48) . For ( v, w ) ∈ Q J , denote Q (cid:23) ( v,w ) J := { ( v (cid:48) , w (cid:48) ) ∈ Q J | ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) } , Q (cid:22) ( v,w ) J := { ( v (cid:48) , w (cid:48) ) ∈ Q J | ( v (cid:48) , w (cid:48) ) (cid:22) ( v, w ) } . Lemma 4.8. (i) Let ( v, w ) , ( v (cid:48) , w (cid:48) ) ∈ Q J , r ∈ W J , and v (cid:48) ≤ vr ≤ wr ≤ w (cid:48) . Then ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 17 (ii) Let ( u, u ) , ( v, w ) , ( v (cid:48) , w (cid:48) ) ∈ Q J . Then ( u, u ) (cid:22) ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) if and only if v (cid:48) ≤ vr (cid:48) ≤ ur ≤ wr (cid:48) ≤ w (cid:48) for some r, r (cid:48) ∈ W J such that vr (cid:48) is length-additive . (4.22) Proof. (i): By Lemma 4.6, there exists r (cid:48) ≤ r such that v ∗ r = vr (cid:48) ≥ vr , and vr (cid:48) is length-additive. We have vr (cid:48) ≤ wr (cid:48) by Lemma 4.6(iii), and wr (cid:48) ≤ wr by Lemma 4.4(ii). Therefore v (cid:48) ≤ vr ≤ vr (cid:48) ≤ wr (cid:48) ≤ wr ≤ w (cid:48) , so ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ).(ii) ( ⇒ ): Suppose that ( u, u ) (cid:22) ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ). Then by Definition 4.7, there exist r (cid:48) , r (cid:48)(cid:48) ∈ W J such that vr (cid:48) is length-additive, v (cid:48) ≤ vr (cid:48) ≤ wr (cid:48) ≤ w (cid:48) , and v ≤ ur (cid:48)(cid:48) ≤ w . Define r ∈ W J by the equality ( ur (cid:48)(cid:48) ) ∗ r (cid:48) = ur . Then applying ∗ r (cid:48) on the right to v ≤ ur (cid:48)(cid:48) ≤ w , byLemma 4.6(iii)–(iv), we obtain vr (cid:48) ≤ ur ≤ wr (cid:48) . Therefore (4.22) holds.(ii) ( ⇐ ): Suppose that (4.22) holds. Then ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ). Define r (cid:48)(cid:48) ∈ W J by theequality ( ur ) (cid:47) r (cid:48)− = ur (cid:48)(cid:48) . Then applying (cid:47) ( r (cid:48) ) − on the right to vr (cid:48) ≤ ur ≤ wr (cid:48) , byLemma 4.6(iii)–(iv), we obtain v ≤ ur (cid:48)(cid:48) ≤ w . Therefore ( u, u ) (cid:22) ( v, w ). (cid:3) Remark 4.9. By Lemma 4.8(i), Definition 4.7 remains unchanged if we omit the conditionthat vr is length-additive. It follows that Q J coincides with the poset studied in [HL15, § Q J is also isomorphic to the posets studied in [Rie06,GY09, KLS13].4.7. Partial flag variety G/P . Fix J ⊂ I as before. Let P ⊂ G be the subgroup generatedby B and { y i ( t ) | t ∈ K ∗ , i ∈ J } . We denote by G/P the partial flag variety correspondingto J , and let π J : G/B → G/P be the natural projection map. Let L J ⊂ P be the Levisubgroup of P . It is generated by T and { x i ( t ) , y i ( t ) | i ∈ J, t ∈ K ∗ } . Let P − be theparabolic subgroup opposite to P , with L J = P ∩ P − .For ( v, w ) ∈ Q J we introduce ◦ Π v,w := π J ( ◦ R v,w ) ⊂ G/P , and define the projected Richard-son variety Π v,w ⊂ G/P to be the closure of ◦ Π v,w in the Zariski topology. By [KLS14,Proposition 3.6], we have(4.23) G/P = (cid:71) ( v,w ) ∈ Q J ◦ Π v,w , and Π v,w = (cid:71) ( v (cid:48) ,w (cid:48) ) ∈ Q (cid:22) ( v,w ) J ◦ Π v (cid:48) ,w (cid:48) . Let now K = C . The varieties ◦ X w , ◦ X v , X w , X v , ◦ R v,w , and R v,w are defined over R . Themap π J is defined over R as well, thus so are ◦ Π v,w and Π v,w . We let( G/B ) R := { gB | g ∈ G ( R ) } ⊂ G/B, ◦ R R v,w := ( G/B ) R ∩ ◦ R v,w , R R v,w := ( G/B ) R ∩ R v,w , ( G/P ) R := { xP | x ∈ G ( R ) } ⊂ G/P, ◦ Π R v,w := ◦ Π v,w ∩ ( G/P ) R , Π R v,w := Π v,w ∩ ( G/P ) R . It follows that the decomposition (4.23) is valid over R :( G/P ) R = (cid:71) ( v,w ) ∈ Q J ◦ Π R v,w , Π R v,w = (cid:71) ( v (cid:48) ,w (cid:48) ) ∈ Q (cid:22) ( v,w ) J ◦ Π R v (cid:48) ,w (cid:48) . (4.24)4.8. Total positivity. Assume K = C in this subsection. Recall from Section 4.1 that foreach i ∈ I , we have elements x i ( t ), y i ( t ) (for t ∈ C ) and α ∨ i ( t ) (for t ∈ C ∗ ). Definition 4.10 ([Lus94]) . Let G ≥ ⊂ G ( R ) be the submonoid generated by x i ( t ), y i ( t ), α ∨ ( t ) for t ∈ R > . Define ( G/B ) ≥ to be the closure of ( G ≥ /B ) ⊂ ( G/B ) R in the analytictopology. For v ≤ w ∈ W , let R ≥ v,w denote the closure of R > v,w := ◦ R v,w ∩ ( G/B ) ≥ inside( G/B ) ≥ . Definition 4.11 ([Lus98a, Rie99]) . Set ( G/P ) ≥ := π J (( G/B ) ≥ ). For ( v, w ) ∈ Q J , let Π ≥ v,w denote the closure of Π > v,w := π J ( R > v,w ) inside ( G/P ) ≥ .Thus we denote by Π > v,w what was denoted by Π > v,w ) in Example 1.4. We have decompositions( G/P ) ≥ = (cid:71) ( v,w ) ∈ Q J Π > v,w , Π ≥ v,w = (cid:71) ( v (cid:48) ,w (cid:48) ) ∈ Q (cid:22) ( v,w ) J Π > v (cid:48) ,w (cid:48) . (4.25)As a special case of (4.25) for J = ∅ , we have( G/B ) ≥ = (cid:71) v ≤ w R > v,w , R ≥ v,w = (cid:71) v ≤ v (cid:48) ≤ w (cid:48) ≤ w R > v (cid:48) ,w (cid:48) . (4.26) Lemma 4.12. (Assume K = C .) Let ( v, w ) ∈ Q J and r ∈ W J be such that vr is length-additive. Then ◦ Π v,w = π J ( ◦ R v,w ) = π J ( ◦ R vr,wr ) , Π > v,w = π J ( R > v,w ) = π J ( R > vr,wr ) , (4.27) Π v,w = π J ( R v,w ) = π J ( R vr,wr ) , Π ≥ v,w = π J ( R ≥ v,w ) = π J ( R ≥ vr,wr ) . (4.28) Proof. By [KLS13, Lemma 3.1], we have π J ( ◦ R v,w ) = π J ( ◦ R vr,wr ) = ◦ Π v,w , and π J restricts toisomorphisms ◦ R v,w ∼ −→ ◦ Π v,w , ◦ R vr,wr ∼ −→ ◦ Π v,w . Thus π J ( R > v,w ) = π J ( R > vr,wr ) = Π > v,w followsfrom the equality π J (( G/B ) ≥ ) = ( G/P ) ≥ , proving (4.27). To show (4.28), note that R a,b and R ≥ a,b are compact for any a ≤ b , and therefore their images under π J are closed. (cid:3) Recall the definition of x i ( t ) and y i ( t ) from (4.19). Choose a reduced word i = ( i , . . . , i n )for w ∈ W and define U > ( w ) := { x i ( t ) | t ∈ R n> } , U − > ( w ) := { y i ( t ) | t ∈ R n> } . Let U ≥ ⊂ U ( R ) (respectively, U −≥ ⊂ U − ( R )) be the submonoid generated by x i ( t ) (respec-tively, by y i ( t )) for t ∈ R > . Then U ≥ = (cid:70) w ∈ W U > ( w ) and U −≥ = (cid:70) w ∈ W U − > ( w ). We have U > ( w ) = U ≥ ∩ B − ˙ wB − and U − > ( w ) = U −≥ ∩ B ˙ wB , and these sets do not depend on thechoice of the reduced word i for w , see [Lus94, Prop. 2.7].4.9. MR-parametrizations. Assume that K is algebraically closed. Given w ∈ W , an expression w for w is a sequence w = ( w (0) , . . . , w ( n ) ) such that w (0) = id, w ( n ) = w , andfor j = 1 , . . . , n , either w ( j ) = w ( j − or w ( j ) = w ( j − s i j for some i j ∈ I . In the latter casewe require w ( j − < w ( j ) , unlike [MR04]. We denote J + w := { ≤ j ≤ n | w ( j − < w ( j ) } and J ◦ w := { ≤ j ≤ n | w ( j − = w ( j ) } so that J + w (cid:116) J ◦ w = { , , . . . , n } . Every reduced word i = ( i , . . . , i n ) for w gives rise to a reduced expression w = w ( i ) = ( w (0) , . . . , w ( n ) ) with w ( j ) = w ( j − s i j for j = 1 , . . . , n . Lemma 4.13 ([MR04, Lemma 3.5]) . Let v ≤ w ∈ W , and consider a reduced expression w = ( w (0) , . . . , w ( n ) ) for w corresponding to a reduced word i = ( i , . . . , i n ) . Then there existsa unique positive subexpression v for v inside w , i.e., an expression v = ( v (0) , . . . , v ( n ) ) for v such that for j = 1 , . . . , n , we have v ( j − < v ( j − s i j . This positive subexpression can beconstructed inductively by setting v ( n ) := v and (4.29) v ( j − := (cid:40) v ( j ) s i j , if v ( j ) s i j < v ( j ) , v ( j ) , otherwise , for j = n, . . . , . Corollary 4.14. In the above setting, if v (1) = s i for some i ∈ I then v (cid:54)≤ s i w . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 19 Proof. Indeed, if v ≤ s i w < w then there exists a positive subexpression v (cid:48) = ( v (cid:48) (0) , . . . , v (cid:48) ( n − )for v inside w ( i (cid:48) ), where i (cid:48) = ( i , . . . , i n ). By (4.29), we have v (cid:48) ( j ) = v ( j +1) for j =0 , , . . . , n − 1, which contradicts the fact that v (cid:48) (0) = 1 while v (1) = s i . (cid:3) For w ∈ W , let Red( w ) := { w | w is a reduced expression for w } . For v ≤ w ∈ W , letRed( v, w ) := { ( v , w ) | w ∈ Red( w ) , v is a positive subexpression for v inside w } . Thus for all v ≤ w , the sets Red( w ) and Red( v, w ) have the same cardinality. Let v ≤ w ∈ W and ( v , w ) ∈ Red( v, w ). Given a collection t = ( t k ) k ∈ J ◦ v ∈ ( K ∗ ) J ◦ v , define g v , w ( t ) := g · · · g n , where g k := (cid:40) y i k ( t k ) , if k ∈ J ◦ v ,˙ s i k , if k ∈ J + v .(4.30)4.9.1. MR-parametrizations of ( G/B ) ≥ . In this section, we assume K = C . Let v , w , v ,and w be as above. Define a subset G > v , w ⊂ G ( R ) by G > v , w := { g v , w ( t ) | t ∈ R J ◦ v > } . Theorem 4.15 ([MR04, Theorem 11.3]) . The map G ( R ) → ( G/B ) R sending g (cid:55)→ gB restricts to an isomorphism of real semialgebraic varieties G > v , w ∼ −→ R > v,w . Proposition 4.16 ([Lus94, Proposition 8.12]) . We have G ≥ · ( G/B ) ≥ ⊂ ( G/B ) ≥ . Lemma 4.17. Suppose that g ∈ G ≥ and x ∈ G are such that xB ∈ R > v,w for some v ≤ w ∈ W . Then gxB ∈ R > v (cid:48) ,w (cid:48) for some v (cid:48) ≤ v ≤ w ≤ w (cid:48) .Proof. By Proposition 4.16, we have gxB ∈ ( G/B ) ≥ , so it suffices to show that gx ∈ B ˙ w (cid:48) B ∩ B − ˙ v (cid:48) B for some v (cid:48) ≤ v ≤ w ≤ w (cid:48) . Note that we have x ∈ B ˙ wB ∩ B − ˙ vB . ByDefinition 4.10, it is enough to consider the cases g = x i ( t ) and g = y i ( t ) for i ∈ I and t ∈ R > .Suppose that g = y i ( t ). We clearly have gx ∈ B − ˙ vB . If s i w > w then by (4.16) wehave gx ∈ B ˙ s i ˙ wB . Thus we may assume that s i w < w . By Theorem 4.15, we can alsoassume x = g v , w ( t ) = g · · · g n for t ∈ R J ◦ v > and some choice of ( v , w ) ∈ Red( v, w ) such that w = ( w (0) , . . . w ( n ) ) satisfies w (1) = s i . Let v = ( v (0) , . . . , v ( n ) ). If v (1) (cid:54) = s i then g = y i ( t (cid:48) )so gx ∈ G > v , w and we are done. If v (1) = s i then by Corollary 4.14 we have v (cid:54)≤ s i w . Recallthat gx ∈ B − ˙ vB and by (4.16), gx ∈ B ˙ s i ˙ wB (cid:116) B ˙ wB . But B − ˙ vB ∩ B ˙ s i ˙ wB = ∅ by (4.12).Therefore we must have gx ∈ B ˙ wB , finishing the proof in this case.The case g = x i ( t ) follows similarly using a “dual” Marsh–Rietsch parametrization [Rie06, § v , w ) ∈ Red( v, w ), every element of R > ww ,vw is parametrized as g · · · g n ˙ w B, where g k := (cid:40) x i k ( t k ) , if k ∈ J ◦ v ,˙ s − i k , if k ∈ J + v . (cid:3) We will use the following consequence of Theorem 4.15 in Section 9.11. Corollary 4.18 (cf. [KLS14, Proposition 3.3]) . Let u ∈ W J , r ∈ W J , and v ∈ W be suchthat v ≤ ur . Then π J ( R > v,ur ) = π J ( R > v(cid:47)r − ,u ) = Π > v(cid:47)r − ,u . Proof. Let i = ( i , . . . , i n ) be a reduced word for w := ur , such that ( i (cid:96) ( u )+1 , . . . , i n ) is areduced word for r . Let ( v , w ) ∈ Red( v, w ) be such that w corresponds to i . Then it is clearfrom Lemma 4.13 that after setting v (cid:48) := ( v (0) , . . . , v ( (cid:96) ( u )) ) and u := ( w (0) , . . . , w ( (cid:96) ( u )) ), weget ( v (cid:48) , u ) ∈ Red( v (cid:47) r − , u ). Moreover, the indices i (cid:96) ( u )+1 , . . . , i n clearly belong to J , so if g . . . g n ∈ G > v , w then g . . . g (cid:96) ( u ) ∈ G > v (cid:48) , u and π J ( g . . . g n B ) = π J ( g . . . g (cid:96) ( u ) B ). We are doneby Theorem 4.15. (cid:3) G/P is a TNN space. We show that the triple (( G/P ) R , ( G/P ) ≥ , Q J ) is a TNNspace in the sense of Definition 2.1. We start by recalling several known results. Theorem 4.19. (i) The poset (cid:98) Q J := Q J (cid:116) { ˆ0 } is graded, thin, and shellable. (ii) ( G/P ) R is a smooth manifold. Each ◦ Π R v,w is a smooth embedded submanifold of ( G/P ) R . (iii) For ( v, w ) ∈ Q J , Π > v,w is a connected component of ◦ Π R v,w .Proof. Part (i) is due to Williams [Wil07]. For (ii), ( G/P ) R is a smooth manifold becauseit is a homogeneous space of a real Lie group. Each ◦ Π R v,w is a smooth embedded manifoldbecause it is the set of real points of a smooth algebraic subvariety ◦ Π v,w of G/P , see [KLS14,Corollary 3.2] or [Lus98a, Rie06]. Part (iii) is due to Rietsch [Rie99]. (cid:3) Corollary 4.20. (( G/P ) R , ( G/P ) ≥ , Q J ) is a TNN space.Proof. Let us check each part of Definition 2.1.(TNN1): This follows from Theorem 4.19(i). The maximal element ˆ1 ∈ Q J is given by(id , w J ), see Section 4.6.(TNN2): This follows from Theorem 4.19(ii) and (4.24).(TNN3): This holds since ( G/P ) R is compact and Π ≥ v,w ⊂ G/P is closed.(TNN4): This follows from Theorem 4.19(iii) combined with Theorem 4.15.(TNN5): This result is due to Rietsch [Rie06], see (4.25). (cid:3) Gaussian decomposition. Assume K is algebraically closed. Let us define G ∓ := B − B, G ± := BB − . For i ∈ I , let ∆ ∓ i : G ∓ → K and ∆ ± i : G ± → K be defined as follows. Given ( x − , x , x + ) ∈ U − × T × U , we have x − x x + ∈ G ∓ and x + x x − ∈ G ± , and we set ∆ ∓ i ( x − x x + ) := x ω i ,∆ ± i ( x + x x − ) := x w ω i . For a finite set A , let P A denote the ( | A | − K , with coordinates indexed by elements of A . Lemma 4.21. (i) The multiplication map gives biregular isomorphisms: U − × T × U ∼ −→ G ∓ , U × T × U − ∼ −→ G ± . (ii) The maps ∆ ∓ i and ∆ ± i extend to regular functions G → K . (iii) G ∓ = { x ∈ G | ∆ ∓ i ( x ) (cid:54) = 0 for all i ∈ I } , G ± = { x ∈ G | ∆ ± i ( x ) (cid:54) = 0 for all i ∈ I } . (iv) Fix i ∈ I and let W ω i := { wω i | w ∈ W } denote the W -orbit of the correspondingfundamental weight. Then there exists a regular map ∆ flag i : G/B → P W ω i such thatfor w ∈ W and x ∈ G , the wω i -th coordinate of ∆ flag i ( xB ) equals ∆ ∓ i ( ˙ w − x ) . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 21 Proof. For (i), see [Hum75, Proposition 28.5]. Parts (ii) and (iii) are well known when K = C , see [FZ99, Proposition 2.4 and Corollary 2.5]. We give a proof for arbitrary alge-braically closed K , using a standard argument that relies on representation theory. We referto [Hum75, § 31] for the necessary notation and background.We have G ± = ˙ w − G ∓ ˙ w and ∆ ± i ( ˙ w − g ˙ w ) = ∆ ∓ i ( g ) for all g ∈ G ∓ . Thus it suffices togive a proof for ∆ ∓ i and G ∓ . For i ∈ I , there exists a regular function c ω i : G → K thatcoincides with ∆ ∓ i on G ∓ , see [Hum75, § c ω i is given asfollows: consider the highest weight module V ω i for G , and let v + ∈ V ω i be its highest weightvector. We have a direct sum of vector spaces V ω i = K v + ⊕ V (cid:48) , where V (cid:48) is spanned byweight vectors of weight other than ω i . Let r + : V ω i → K denote the linear function suchthat r + ( v + ) = 1 and r + ( V (cid:48) ) = { } , then c ω i ( g ) := r + ( gv + ) for all g ∈ G . The decomposition V ω i = K v + ⊕ V (cid:48) is such that for ( x − , x , x + ) ∈ U − × T × U and w ∈ W , we have x + v + = v + , x v + = M v + for some M ∈ K ∗ , x − v + ∈ v + + V (cid:48) , x − V (cid:48) ⊂ V (cid:48) , and ˙ wv + ∈ V (cid:48) if wω i (cid:54) = ω i .Thus if g ∈ G ∓ then c ω i ( g ) (cid:54) = 0 for all i ∈ I . Conversely, if x / ∈ G ± then by (4.11), thereexists a unique w (cid:54) = id ∈ W such that g ∈ U − ˙ wT U . For i ∈ I such that wω i (cid:54) = ω i , weget c ω i ( g ) = 0. This proves (iii). For (iv), let V ω i = V ⊕ V where V is spanned by allweight vectors of weight in W ω i , and V is spanned by the remaining weight vectors. Let π : V ω i → V denote the projection along V . It follows that for all g ∈ G , π ( gv + ) (cid:54) = 0.Then ∆ flag i is the natural morphism G/B → P ( V ), sending gB to [ π ( gv + )]. (cid:3) Lemma 4.22. Define G ( J )0 := P − P . (i) We have G ( J )0 = P − B and P = (cid:70) r ∈ W J B ˙ rB . (ii) For p ∈ P , we have pU ( J ) p − = U ( J ) . Similarly, for p ∈ P − , we have pU ( J ) − p − = U ( J ) − .In particular, for p ∈ L J , we have pU ( J ) p − = U ( J ) and pU ( J ) − p − = U ( J ) − . (iii) The multiplication map gives a biregular isomorphism U ( J ) − × L J × U ( J ) ∼ −→ G ( J )0 . Inparticular, every element x ∈ G ( J )0 can be uniquely factorized as [ x ] ( J ) − · [ x ] J · [ x ] ( J )+ ∈ U ( J ) − · L J · U ( J ) . The map G ( J )0 → L J sending x (cid:55)→ [ x ] J satisfies [ p − xp + ] J = [ p − ] J [ x ] J [ p + ] J for all x ∈ G ( J )0 , p − ∈ P − , and p + ∈ P . (iv) The map b (cid:55)→ [ b ] J gives group homomorphisms U → U J and U − → U − J , sending x i ( t ) (cid:55)→ [ x i ( t )] J = (cid:40) x i ( t ) , if i ∈ J , , otherwise, y i ( t ) (cid:55)→ [ y i ( t )] J = (cid:40) y i ( t ) , if i ∈ J , , otherwise.Proof. By [Hum75, § U ( J ) is the unipotent radical (in particular, a normal subgroup)of P and U ( J ) − is the unipotent radical of P − . This shows (ii). It follows that P = L J U ( J ) = L J B , therefore G ( J )0 = P − B . By [Hum75, § P = (cid:70) r ∈ W J B ˙ rB , whichproves (i).By [Bor91, Proposition 14.21(iii)], the multiplication map gives a biregular isomorphism U ( J ) − × P → G ( J )0 . By [Hum75, § L J × U ( J ) → P . Thus we get a biregular isomorphism U ( J ) − × L J × U ( J ) ∼ −→ G ( J )0 . Itis clear from the definition that [ p − xp + ] J = [ p − ] J [ x ] J [ p + ] J , since we can factorize p − =[ p − ] ( J ) − [ p − ] J and p + = [ p + ] J [ p + ] ( J )+ . Thus we are done with (iii), and (iv) follows by repeatedlyapplying (iii). (cid:3) Affine charts. For u ∈ W J , denote C ( J ) u := ˙ uG ( J )0 /P ⊂ G/P . The following mapsare biregular isomorphisms for u ∈ W J and v, w ∈ W , see [Bor91, Proposition 14.21(iii)],[Spr98, Proposition 8.5.1(ii)], and [FH91, Corollary 23.60]:˙ uU ( J ) − ˙ u − ∼ −→ C ( J ) u , g ( J ) (cid:55)→ g ( J ) ˙ uP, (4.31) ˙ vU − ˙ v − ∩ U − ∼ −→ ◦ X v , g (cid:55)→ g ˙ vB, (4.32) ˙ wU − ˙ w − ∩ U ∼ −→ ◦ X w , g (cid:55)→ g ˙ wB. (4.33)As a consequence of (4.32) and (4.33), we get(4.34) B − ˙ vB = ( ˙ vU − ∩ U − ˙ v ) · B, B ˙ wB = ( ˙ wU − ∩ U ˙ w ) · B. The isomorphism in (4.31) identifies an open dense subset C ( J ) u of G/P with the group˙ uU ( J ) − ˙ u − . We now combine this with Lemma 4.2. Definition 4.23. Let U ( J )1 := ˙ uU ( J ) − ˙ u − ∩ U and U ( J )2 := ˙ uU ( J ) − ˙ u − ∩ U − . For x ∈ ˙ uG ( J )0 ,consider the element g ( J ) ∈ ˙ uU ( J ) − ˙ u − such that g ( J ) ˙ u ∈ xP ∩ ˙ uU ( J ) − , unique by (4.31).Further, let h ( J )1 , g ( J )1 ∈ U ( J )1 and h ( J )2 , g ( J )2 ∈ U ( J )2 be the elements such that h ( J )2 g ( J ) = g ( J )1 and h ( J )1 g ( J ) = g ( J )2 . By (4.31), the map x (cid:55)→ g ( J ) is regular, and the map g ( J ) (cid:55)→ ( g ( J )1 , g ( J )2 , h ( J )1 , h ( J )2 ) is regular by Lemma 4.2. Let us denote by κ : ˙ uG ( J )0 → U ( J )2 the map x (cid:55)→ κ x := h ( J )2 . It descends to a regular map κ : C ( J ) u → U ( J )2 sending xP (cid:55)→ κ x .5. Subtraction-free parametrizations We study subtraction-free analogs of Marsh–Rietsch parametrizations [MR04] of ( G/B ) ≥ .5.1. Subtraction-free subsets. Given some fixed collection t of variables of size | t | , let R [ t ] be the ring of polynomials in t , and R > [ t ] ⊂ R [ t ] be the semiring of nonzero polynomialsin t with positive real coefficients. Let F := R ( t ) be the field of rational functions in t .Define F ∗ sf := { R ( t ) /Q ( t ) | R ( t ) , Q ( t ) ∈ R > [ t ] } , F sf := { } (cid:116) F ∗ sf , F (cid:5) := { R ( t ) /Q ( t ) | R ( t ) ∈ R [ t ] , Q ( t ) ∈ R > [ t ] } . We call elements of F sf subtraction-free rational expressions in t . In this section, we assumethat K = F is the algebraic closure of F . Definition 5.1. Let T sf ⊂ T be the subgroup generated by α ∨ i ( t ) for i ∈ I and t ∈ F ∗ sf . Let G (cid:5) ⊂ G be the subgroup generated by { x i ( t ) , y i ( t ) | i ∈ I, t ∈ F (cid:5) } ∪ { ˙ w | w ∈ W } ∪ T sf . We define subgroups U (cid:5) := U ∩ G (cid:5) , U (cid:5)− := U − ∩ G (cid:5) , B (cid:5) := T sf U (cid:5) = U (cid:5) T sf and B (cid:5)− = T sf U (cid:5)− = U (cid:5)− T sf (cf. Lemma 5.2 below). We also put U (cid:5) (Θ) := U (cid:5) ∩ U (Θ) (respectively, U (cid:5)− (Θ) := U (cid:5)− ∩ U − (Θ)) for a bracket closed subset Θ of Φ + (respectively, of Φ − ). Given areduced word i for w ∈ W , define(5.1) U sf ( w ) := { x i ( t (cid:48) ) | t (cid:48) ∈ ( F ∗ sf ) n } , U − sf ( w ) := { y i ( t (cid:48) ) | t (cid:48) ∈ ( F ∗ sf ) n } . These subsets do not depend on the choice of i , see [BZ97, § H , H of G , we say that H commutes with H if H · H = H · H . Wesay that H commutes with g ∈ G if H · g = g · H . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 23 Lemma 5.2. T sf commutes with B (cid:5) , U , U − , U (cid:5) (Θ) , U (cid:5)− (Θ) , U sf ( w ) , U − sf ( w ) and ˙ w .Proof. It follows from (4.2) that T sf commutes with B (cid:5) , U , U − , U sf ( w ), U − sf ( w ) and ˙ w . For U (cid:5) (Θ), U (cid:5)− (Θ), we use a generalization of (4.2): for α ∈ Φ + , i ∈ I , and w ∈ W such that wα i = α , write x α ( t ) := ˙ wx i ( t ) ˙ w − ∈ U (cid:5) ( { α } ) and y α ( t ) := ˙ wy i ( t ) ˙ w − ∈ U (cid:5)− ( {− α } ) for t ∈ F (cid:5) . Then (4.2) implies ax α ( t ) a − = x α ( a α t ) and ay α ( t ) a − = y α ( a − α t ). (cid:3) Let us now introduce subtraction-free analogs of MR parametrizations. Let v ≤ w ∈ W and ( v , w ) ∈ Red( v, w ). Recall that for t (cid:48) = ( t (cid:48) k ) k ∈ J ◦ v ∈ ( K ∗ ) J ◦ v , g v , w ( t (cid:48) ) = g · · · g n is definedin (4.30). Define G sf v , w := { g v , w ( t (cid:48) ) | t (cid:48) ∈ ( F ∗ sf ) J ◦ v } ⊂ G (cid:5) . The following result is closelyrelated to [MR04, Lemma 11.8]. Lemma 5.3. Let v ≤ w ∈ W and ( v , w ) ∈ Red( v, w ) . Let g v , w ( t (cid:48) ) be as in (4.30) for t (cid:48) ∈ ( F ∗ sf ) J ◦ v . Then for each k = 0 , , . . . , n and for all x ∈ U (cid:5) ∩ ˙ v − k ) U − ˙ v ( k ) , we have (5.2) g · · · g k · x · g k +1 · · · g n ∈ g · · · g n · U (cid:5) . Proof. We prove this by induction on k . For k = n , the result is trivial, so suppose that k < n . Let x ∈ U (cid:5) ∩ ˙ v − k ) U − ˙ v ( k ) . If g k +1 = ˙ s i for some i ∈ I then (cid:96) ( v ( k +1) ) = (cid:96) ( v ( k ) ) + (cid:96) ( s i ),so we use (4.9) to show that x · g k +1 = g k +1 · x (cid:48) for some x (cid:48) ∈ U ∩ ˙ v − k +1) U − ˙ v ( k +1) . Since x (cid:48) = ˙ s − i x ˙ s i and each term belongs to G (cid:5) , we see that x (cid:48) ∈ U (cid:5) ∩ ˙ v − k +1) U − ˙ v ( k +1) , so we aredone by induction.Suppose now that g k +1 = y i ( t ) for some i ∈ I and t ∈ F ∗ sf . Write x · g k +1 = g k +1 · g − k +1 xg k +1 = g k +1 · y i ( − t ) xy i ( t ) . By (4.5), U (cid:5) ∩ ˙ v − k ) U − ˙ v ( k ) = U (cid:5) (Inv( v ( k ) )). Clearly again y i ( − t ) xy i ( t ) ∈ G (cid:5) , and we claimthat y i ( − t ) xy i ( t ) ∈ U (Inv( v ( k ) )) for all x ∈ U (Inv( v ( k ) )). First, using Lemma 4.1(ii), we canassume that x ∈ U α for some α ∈ Inv( v ( k ) ). Since v ( k ) s i > v ( k ) , we have α i / ∈ Inv( v ( k ) ), so α (cid:54) = α i . Let Ψ = { mα − rα i } ⊂ Φ + be the set of roots as in Lemma 4.3. Our goal is to showthat Ψ ⊂ Inv( v ( k ) ). Let γ := mα − rα i ∈ Ψ for some m > r ≥ 0. We now show that γ ∈ Inv( v ( k ) ), which is equivalent to saying that v ( k ) γ < 0. Indeed, v ( k ) γ = mv ( k ) α − rv ( k ) α i .Since α ∈ Inv( v ( k ) ), v ( k ) α < 0. Since α i / ∈ Inv( v ( k ) ), v ( k ) α i > 0. Thus v ( k ) γ < 0, because − v ( k ) γ is a positive linear combination of positive roots. We have shown that Ψ ⊂ Inv( v ( k ) ),thus by Lemma 4.3, we find y i ( − t ) xy i ( t ) ∈ U (Inv( v ( k ) )). Since v ( k ) = v ( k +1) , we get y i ( − t ) xy i ( t ) ∈ U (cid:5) (Inv( v ( k ) )) = U (cid:5) ∩ ˙ v − k ) U − ˙ v ( k ) = U (cid:5) ∩ ˙ v − k +1) U − ˙ v ( k +1) , and we are done by induction. (cid:3) Proposition 5.4. For v ≤ w ∈ W , the set G sf v , w · U (cid:5) ⊂ G (cid:5) does not depend on the choiceof ( v , w ) ∈ Red( v, w ) . In other words: let ( v , w ) , ( v , w ) ∈ Red( v, w ) . Then for any t ∈ ( F ∗ sf ) J ◦ v there exists t ∈ ( F ∗ sf ) J ◦ v and x ∈ U (cid:5) such that g v , w ( t ) = g v , w ( t ) · x .Proof. Recall that for each w ∈ Red( w ) there exists a unique positive subexpression v for v such that ( v , w ) ∈ Red( v, w ). We need to show that choosing a different reducedexpression w for w results in a subtraction-free coordinate change t (cid:55)→ t of the parametersin Theorem 4.15. Any two reduced expressions for w are related by a sequence of braid moves,so it suffices to assume that w and w differ in a single braid move.The explicit formulae for the corresponding coordinate transformations can be found inthe proof of [Rie08, Proposition 7.2], however, an extra step is needed to show that those formulae indeed give the correct coordinate transformations. More precisely, suppose thatΦ (cid:48) is a root subsystem of Φ of rank 2, and let W (cid:48) be its Weyl group. Then it was checkedin the proof of [Rie08, Proposition 7.2] that for any v (cid:48) ≤ w (cid:48) ∈ W (cid:48) , any ( v (cid:48) , w (cid:48) ) , ( v (cid:48) , w (cid:48) ) ∈ Red( v (cid:48) , w (cid:48) ), and any t (cid:48) ∈ ( F ∗ sf ) J ◦ v (cid:48) , there exists t (cid:48) ∈ ( F ∗ sf ) J ◦ v (cid:48) and x ∈ U such that g v (cid:48) , w (cid:48) ( t (cid:48) ) = g v (cid:48) , w (cid:48) ( t (cid:48) ) · x .Let us now complete the proof of Proposition 5.4 (as well as of [Rie08, Proposition 7.2]). Suppose that w and w differ in a braid move along a subword g k +1 · · · g k + m of g · · · g n .Here g k +1 · · · g k + m = g v (cid:48) , w (cid:48) ( t (cid:48) ) as above. Applying a move from [Rie08], we transform g k +1 · · · g k + m into g (cid:48) k +1 · · · g (cid:48) k + m x for some x ∈ U and g (cid:48) k +1 · · · g (cid:48) k + m = g v (cid:48) , w (cid:48) ( t (cid:48) ). Thus g · · · g n = g · · · g k · g (cid:48) k +1 · · · g (cid:48) k + m · x · g k + m +1 · · · g n . By [MR04, Proposition 5.2], the elements h := g · · · g k + m and h (cid:48) := g · · · g k · g (cid:48) k +1 · · · g (cid:48) k + m belong to U − ˙ v ( k + m ) . Since h = h (cid:48) x , we get x ∈ ˙ v − k + m ) U − ˙ v ( k + m ) . Moreover, since h, h (cid:48) ∈ G (cid:5) and x ∈ U , we must have x ∈ U (cid:5) . Thus by Lemma 5.3, we have g · · · g n ∈ g · · · g k · g (cid:48) k +1 · · · g (cid:48) k + m · g k + m +1 · · · g n · U (cid:5) . (cid:3) Definition 5.5. From now on we denote R sf v,w := G sf v , w B (cid:5) ⊂ G (cid:5) . By Proposition 5.4, theset R sf v,w does not depend on the choice of ( v , w ) ∈ Red( v, w ). As we discuss in Section 5.4, R sf v,w is the “subtraction-free” analog of R > v,w .5.2. Collision moves. Assume K = F . By [FZ99, (2.13)], for each t ∈ F ∗ sf there exist t + ∈ F ∗ sf , a + ∈ T sf , and t − ∈ F (cid:5) satisfying(5.3) ˙ s i x i ( t ) = a + x i ( t − ) y i ( t + ) , x i ( t ) ˙ s i = y i ( t + ) x i ( t − ) a + , (5.4) ˙ s − i y i ( t ) = a + y i ( t − ) x i ( t + ) , y i ( t ) ˙ s − i = x i ( t + ) y i ( t − ) a + . (Here, each of the four moves yields different t + , a + , t − .) By [FZ99, (2.11)], for each t, t (cid:48) ∈ F ∗ sf there exist t + , t (cid:48) + ∈ F ∗ sf and a + ∈ T sf satisfying(5.5) x i ( t ) y i ( t (cid:48) ) = y i ( t (cid:48) + ) x i ( t + ) a + , y i ( t (cid:48) ) x i ( t ) = x i ( t + ) y i ( t (cid:48) + ) a + . By [FZ99, (2.9)], we have(5.6) x i ( t ) y j ( t (cid:48) ) = y j ( t (cid:48) ) x i ( t ) , for i (cid:54) = j. As a direct consequence of (5.5), (5.6), and Lemma 5.2, for any v, w ∈ W we get(5.7) U sf ( v ) · U − sf ( w ) · T sf = U − sf ( w ) · U sf ( v ) · T sf . Lemma 5.6. (i) Let w ∈ W . Then B (cid:5)− · ˙ w − · U − sf ( w ) = B (cid:5)− · U sf ( w − ) and U − sf ( w ) · ˙ w − · B (cid:5)− = U sf ( w − ) · B (cid:5)− . (5.8)(ii) If v, w ∈ W are such that (cid:96) ( vw ) = (cid:96) ( v ) + (cid:96) ( w ) , then (5.9) ˙ w − ˙ v − · U − sf ( v ) ⊂ B (cid:5)− · ˙ w − · U sf ( v − ) . Alternatively, the proof of [Rie08, Proposition 7.2] can be completed using [MR04, Theorem 7.1]. Wethank Konni Rietsch for pointing this out to us. EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 25 (iii) Let w , . . . , w k ∈ W be such that (cid:96) ( w · · · w k ) = (cid:96) ( w ) + · · · + (cid:96) ( w k ) . Then for any h ∈ U − sf ( w · · · w k ) there exist b ∈ U sf ( w − ) , . . . , b k ∈ U sf ( w − k ) such that for each ≤ i ≤ k , we have (5.10) ˙ w − i · · · ˙ w − · h ∈ B (cid:5)− · b i · · · b . (iv) Let v ≤ w ∈ W . Then (5.11) ˙ v − · U − sf ( w ) ⊂ B (cid:5)− · U sf ( v − ) . Proof. Let us prove the following claim: if vv = w and (cid:96) ( w ) = (cid:96) ( v ) + (cid:96) ( v ), then(5.12) ˙ v − U − sf ( w ) ⊂ T sf · ( U (cid:5)− ∩ ˙ v − U ˙ v ) · U − sf ( v ) · U sf ( v − ) . We prove this by induction on (cid:96) ( v ). If (cid:96) ( v ) = 0 then v = id and (5.12) is trivial. Otherwisethere exists an i ∈ I such that v (cid:48) := s i v < v and thus w (cid:48) := s i w < w . Let y i ( t (cid:48) ) ∈ U − sf ( w ).Using (5.4), we see that for some t ∈ F ∗ sf , t + ∈ F ∗ sf and t − ∈ F (cid:5) ,˙ v − · y i ( t (cid:48) ) ∈ ˙ v (cid:48)− · ˙ s − i y i ( t ) · U − sf ( w (cid:48) ) ⊂ T sf ˙ v (cid:48)− · y i ( t − ) x i ( t + ) · U − sf ( w (cid:48) ) . By (5.7), x i ( t + ) · U − sf ( w (cid:48) ) ⊂ T sf · U − sf ( w (cid:48) ) · U sf ( s i ). Clearly s i v (cid:48) > v (cid:48) , so y (cid:48) := ˙ v (cid:48)− y i ( t − ) ˙ v (cid:48) ∈ U − .On the other hand, ˙ vy (cid:48) ˙ v − = ˙ s − i y i ( t − ) ˙ s i = x i ( − t − ) ∈ U . Thus y (cid:48) ∈ U − ∩ ˙ v − U ˙ v , and it isalso clear that y (cid:48) ∈ G (cid:5) . We have shown that(5.13) ˙ v − · y i ( t (cid:48) ) ∈ T sf · y (cid:48) · ˙ v (cid:48)− · U − sf ( w (cid:48) ) · U sf ( s i ) ⊂ T sf · ( U (cid:5)− ∩ ˙ v − U ˙ v ) · ˙ v (cid:48)− · U − sf ( w (cid:48) ) · U sf ( s i ) . We have v (cid:48) v = w (cid:48) , so by induction,˙ v (cid:48)− · U − sf ( w (cid:48) ) ⊂ T sf · ( U (cid:5)− ∩ ˙ v (cid:48)− U ˙ v (cid:48) ) · U − sf ( v ) · U sf ( v (cid:48)− ) . Since U sf ( v (cid:48)− ) · U sf ( s i ) = U sf ( v − ), we have shown that˙ v − y i ( t (cid:48) ) ∈ T sf · ( U (cid:5)− ∩ ˙ v − U ˙ v ) · ( U (cid:5)− ∩ ˙ v (cid:48)− U ˙ v (cid:48) ) · U − sf ( v ) · U sf ( v − ) . By (4.6) applied to a = s i , b = v (cid:48) , ab = v , we get Inv( v (cid:48) ) ⊂ Inv( v ), so ( U (cid:5)− ∩ ˙ v (cid:48)− U ˙ v (cid:48) ) ⊂ ( U (cid:5)− ∩ ˙ v − U ˙ v ), and we have finished the proof of (5.12).Combining (5.12) with (4.8), we obtain (5.9). Next, (5.10) can be shown by induction: thecase k = 0 is trivial. For k ≥ 1, we can write h = h · · · h k ∈ U − sf ( w ) · · · U − sf ( w k ). By (5.9),we have ˙ w − i · · · ˙ w − · h · · · h k ∈ B (cid:5)− · ˙ w − i · · · ˙ w − · b (cid:48) · h · · · h k for some b (cid:48) ∈ U sf ( w ) that does not depend on i . Using (5.7), we write b (cid:48) · h · · · h k = h (cid:48) · · · h (cid:48) k · b ∈ U − sf ( w ) · · · U − sf ( w k ) · U sf ( w ), and then proceed by induction.Let us state several further corollaries of (5.12):˙ w − · U − sf ( w ) ⊂ T sf · ( U (cid:5)− ∩ ˙ w − U ˙ w ) · U sf ( w − ) , (5.14) U − sf ( w ) · ˙ w − ⊂ U sf ( w − ) · ( U (cid:5)− ∩ ˙ wU ˙ w − ) · T sf , (5.15) ˙ w · U sf ( w − ) ⊂ ( U (cid:5) ∩ ˙ wU − ˙ w − ) · U − sf ( w ) · T sf . (5.16)Indeed, specializing (5.12) to v = w , we obtain (5.14). We obtain (5.15) from (5.14) byreplacing w with z := w − and then applying the involution x (cid:55)→ x ι of (4.4), while (5.16) isobtained from (5.15) by applying the involution x (cid:55)→ x T of (4.3).To show (5.8), observe that the inclusion B (cid:5)− · ˙ w − · U − sf ( w ) ⊂ B (cid:5)− · U sf ( w − ) followsfrom (5.14). To show the reverse inclusion, we use (5.16) to write B (cid:5)− · U sf ( w − ) = B (cid:5)− · ˙ w − · ˙ w · U sf ( w − ) ⊂ B (cid:5)− · ˙ w − · ( U (cid:5) ∩ ˙ wU − ˙ w − ) · U − sf ( w ) . Since ˙ w − · ( U (cid:5) ∩ ˙ wU − ˙ w − ) ⊂ U (cid:5)− ˙ w − , we obtain B (cid:5)− · ˙ w − · U − sf ( w ) = B (cid:5)− · U sf ( w − ), which isthe first part of (5.8). The second part follows by applying the involution x (cid:55)→ x ι of (4.4).It remains to show (5.11). We argue by induction on (cid:96) ( w ), and the base case (cid:96) ( w ) = 0 isclear. Suppose that v ≤ w , and let w (cid:48) := s i w < w for some i ∈ I . If v (cid:48) := s i v < v then bythe same argument as in the proof of (5.13), we get˙ v − · U − sf ( w ) ⊂ B (cid:5)− · ˙ v (cid:48)− · U − sf ( w (cid:48) ) · U sf ( s i ) . Since v (cid:48) ≤ w (cid:48) , we can apply the induction hypothesis to write ˙ v (cid:48)− · U − sf ( w (cid:48) ) ⊂ B (cid:5)− · U sf ( v (cid:48)− ).We thus obtain ˙ v − · U − sf ( w ) ⊂ B (cid:5)− · U sf ( v (cid:48)− ) · U sf ( s i ) = B (cid:5)− · U sf ( v − ) , finishing the induction step in the case s i v < v . But if s i v > v then ˙ v − y i ( t ) ˙ v ∈ U (cid:5)− , soin this case we have ˙ v − U − sf ( w ) ⊂ U (cid:5)− · ˙ v − · U − sf ( w (cid:48) ) , and the result follows by applying theinduction hypothesis to the pair v ≤ w (cid:48) . (cid:3) Alternative parametrizations for the top cell. The following two lemmas aresubtraction-free versions of [Rie06, Lemmas 4.2 and 4.3]. Lemma 5.7. Let v ∈ W . Then we have R sf v,w = U sf ( vw ) · ˙ w · B (cid:5) . Proof. Recall from Definition 5.5 that R sf v,w = G sf v , w · B (cid:5) . We have w = w , so choose areduced expression w for w that ends with v . With this choice, G sf v , w = U − sf ( w v − ) · ˙ v .Thus we can write R sf v,w = G sf v , w · B (cid:5) = U − sf ( w v − ) · ˙ v · B (cid:5) = U − sf ( w v − ) · ˙ v ˙ w − · ˙ w · B (cid:5) . Let z := w v − . Using (5.8) and B (cid:5)− · ˙ w = ˙ w · B (cid:5) , we have U − sf ( w v − ) · ˙ v ˙ w − · ˙ w · B (cid:5) = U − sf ( z ) · ˙ z − · ˙ w · B (cid:5) = U sf ( z − ) · ˙ w · B (cid:5) . Combining the above equations, we find R sf v,w = U sf ( z − ) · ˙ w · B (cid:5) , and it remains to notethat z − = vw − = vw . (cid:3) Lemma 5.8. Let v ≤ w ∈ W . Then we have (5.17) U sf ( v − ) · U − sf ( w w − ) · R sf v,w = R sfid ,w = U − sf ( w ) · B (cid:5) . Proof. It follows from the definition of G sf v , w that if w (cid:48) w is length-additive then U − sf ( w (cid:48) ) R sf v,w = R sf v,w (cid:48) w . Applying this to w (cid:48) = w w − , we get U − sf ( w w − ) · R sf v,w = R sf v,w . By Lemma 5.7, wehave R sf v,w · B (cid:5) = U sf ( vw ) · ˙ w · B (cid:5) . Thus U sf ( v − ) · U sf ( vw ) · ˙ w · B (cid:5) = U sf ( w ) · ˙ w · B (cid:5) , soapplying Lemma 5.7 again, we find U sf ( w ) · ˙ w · B (cid:5) = R sfid ,w · B (cid:5) . The result follows since R sfid ,w = U − sf ( w ) · B (cid:5) . (cid:3) Evaluation. We explain the relationship between R sf v,w and R > v,w . Given t (cid:48) ∈ R | t | > ,we denote by eval t (cid:48) : F sf → R > the evaluation homomorphism (of semifields) sending f ( t ) → f ( t (cid:48) ). It extends to a well defined group homomorphism eval t (cid:48) : G (cid:5) → G ( R ), and it EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 27 follows from Theorem 4.15 that { eval t (cid:48) ( g ) B | g ∈ R sf v,w } = R > v,w as subsets of ( G/B ) R . It isclear that the following diagram is commutative.(5.18) F G (cid:5) F R G ( R ) R eval t (cid:48) ∆ ± i ∆ ∓ i eval t (cid:48) eval t (cid:48) ∆ ± i ∆ ∓ i Here solid arrows denote regular maps, and dashed arrows denote maps defined on a subset F (cid:48) ⊂ F given by F (cid:48) := { R ( t ) /Q ( t ) | R ( t ) , Q ( t ) ∈ R [ t ] , Q ( t (cid:48) ) (cid:54) = 0 } . Since the diagram (5.18)is commutative, it follows that the images ∆ ∓ i ( G (cid:5) ) and ∆ ± i ( G (cid:5) ) belong to F (cid:48) .Let t = ( t (cid:48) , t (cid:48)(cid:48) ). Observe that any f ( t (cid:48) , t (cid:48)(cid:48) ) ∈ F ∗ sf gives rise to a continuous function R | t (cid:48) | > × R | t (cid:48)(cid:48) | > → R > . Lemma 5.9. Suppose that f ( t (cid:48) , t (cid:48)(cid:48) ) ∈ F ∗ sf is such that the corresponding function R | t (cid:48) | > × R | t (cid:48)(cid:48) | > → R > extends to a continuous function R | t (cid:48) | > × R | t (cid:48)(cid:48) |≥ → R ≥ . Then lim t (cid:48)(cid:48) → f ( t (cid:48) , t (cid:48)(cid:48) ) canbe represented (as a function R | t (cid:48) | > → R ≥ ) by a subtraction-free rational expression in t (cid:48) .Proof. By induction, it is enough to prove this when | t (cid:48)(cid:48) | = 1, where t (cid:48)(cid:48) = t (cid:48)(cid:48) is a singlevariable. In this case, f ( t (cid:48) , t (cid:48)(cid:48) ) = R ( t (cid:48) , t (cid:48)(cid:48) ) /Q ( t (cid:48) , t (cid:48)(cid:48) ) where R and Q have positive coefficients.Let us consider R and Q as polynomials in t (cid:48)(cid:48) only. After dividing R and Q by ( t (cid:48)(cid:48) ) k for some k , we may assume that one of them is not divisible by t (cid:48)(cid:48) . Then Q cannot be divisible by t (cid:48)(cid:48) ,otherwise f would not give rise to a continuous function R | t (cid:48) | > × R | t (cid:48)(cid:48) |≥ → R ≥ . We can write Q ( t (cid:48) , t (cid:48)(cid:48) ) = Q ( t (cid:48) , t (cid:48)(cid:48) ) t (cid:48)(cid:48) + Q ( t (cid:48) ) and R ( t (cid:48) , t (cid:48)(cid:48) ) = R ( t (cid:48) , t (cid:48)(cid:48) ) t (cid:48)(cid:48) + R ( t (cid:48) ), where R , R , Q , Q are polynomials with nonnegative coefficients and Q ( t (cid:48) ) (cid:54) = 0. Thus lim t (cid:48)(cid:48) → f ( t (cid:48) , t (cid:48)(cid:48) ) can berepresented by R ( t (cid:48) ) /Q ( t (cid:48) ), which is a subtraction-free rational expression in t (cid:48) . (cid:3) Lemma 5.10. (Assume K = C .) Suppose that a ≤ b ≤ c ∈ W . Then ∆ ∓ (˙ b − x ) (cid:54) = 0 forsome x ∈ G ( R ) such that xB ∈ R > a,b .Proof. Suppose that ∆ ∓ (˙ b − x ) = 0 for all x ∈ G ( R ) such that xB ∈ R > a,b . Consider the map∆ flag i : G/B → P W ω i from Lemma 4.21(iv). We get that the bω i -th coordinate of ∆ flag i isidentically zero on R > a,c . Therefore it must be zero on the Zariski closure of R > a,c inside G/B ,which is R a,c . By (4.14), R a,c contains ˙ bB = ◦ R b,b , thus ∆ ∓ i (˙ b − ˙ b ) must be zero. We get acontradiction since by definition ∆ ∓ i (˙ b − ˙ b ) = 1. (cid:3) Applications to the flag variety. We use the machinery developed in the previoussections to obtain some natural statements about ( G/B ) ≥ . Lemma 5.11. (Assume K = F .) Suppose that a ≤ c ∈ W and b ∈ W . Then for any x ∈ R sf a,c and i ∈ I , (5.19) ∆ ∓ i (˙ b − x ) ∈ F sf . Moreover, if a ≤ b ≤ c then (5.20) ∆ ∓ i (˙ b − x ) ∈ F ∗ sf , and x ∈ ˙ bB − B. Proof. Let t = ( t , t , t ) for | t | = (cid:96) ( a ), | t | = (cid:96) ( w ) − (cid:96) ( c ), | t | = (cid:96) ( c ) − (cid:96) ( a ). Choose reducedwords i for a − and j for w c − , and let ( a , c ) ∈ Red( a, c ). Suppose that x ∈ g a , c ( t ) B (cid:5) andlet g := x i ( t ) · y j ( t ) · g a , c ( t ) ∈ U sf ( a − ) · U − sf ( w c − ) · R sf a,c . By Lemma 5.8, g ∈ U − sf ( w ) · B (cid:5) = U − sf ( b ) · U − sf ( b − w ) · B (cid:5) . By (5.8), we have ˙ b − · U − sf ( b ) ⊂ B (cid:5)− · U sf ( b − ). Therefore ˙ b − g ∈ B (cid:5)− · U sf ( b − ) · U − sf ( b − w ) · B (cid:5) . By (5.7), we get ˙ b − g ∈ B (cid:5)− · U − sf ( b − w ) · U sf ( b − ) · B (cid:5) = B (cid:5)− · B (cid:5) , and by definition, ∆ ∓ i ( y ) ∈ F ∗ sf for any y ∈ B (cid:5)− · B (cid:5) . Since ∆ ∓ i is a regular function on G by Lemma 4.21(ii), the function f ( t , t , t ) := ∆ ∓ i (˙ b − g ) ∈ F ∗ sf extends to a continuous function on R | t |≥ × R | t |≥ × R | t | > .Therefore by Lemma 5.9, lim t , t → f ( t , t , t ) is a subtraction-free rational expression in t . Since lim t , t → g = g a , c ( t ), we get that ∆ ∓ i (˙ b − g a , c ( t )) ∈ F sf . Since x ∈ g a , c ( t ) B (cid:5) ,(5.19) follows.Suppose now that a ≤ b ≤ c . We would like to show (5.20), thus assume that for some i ∈ I and for x ∈ R sf a,c , we have ∆ ∓ i (˙ b − x ) = 0. Let t (cid:48) ∈ ( F ∗ sf ) | t | and ( a , c ) ∈ Red( a, c ) besuch that x ∈ g a , c ( t (cid:48) ) B (cid:5) , and let y ( t ) := g a , c ( t ). Then we have ∆ ∓ i (˙ b − y ( t )) ∈ F sf by (5.19).If ∆ ∓ i (˙ b − y ( t )) was a nonzero rational function in t then clearly substituting t (cid:55)→ t (cid:48) for t (cid:48) ∈ ( F ∗ sf ) | t | would also produce a nonzero rational function. Since substituting t (cid:55)→ t (cid:48) yields∆ ∓ i (˙ b − x ) = 0, we must have ∆ ∓ i (˙ b − y ( t )) = 0. Therefore ∆ ∓ i (˙ b − x (cid:48) ) = 0 for all x (cid:48) ∈ R sf a,c .Now let t (cid:48) ∈ R | t | > . Recall from Section 5.4 that the image of R sf a,c in ( G/B ) R under the mapeval t (cid:48) equals R > a,c . Thus by (5.18), ∆ ∓ i (˙ b − x (cid:48) ) = 0 for all x (cid:48) ∈ G ( R ) such that x (cid:48) B ∈ R > a,c ,which contradicts Lemma 5.10. Hence ∆ ∓ i (˙ b − x ) ∈ F ∗ sf , and therefore x ∈ ˙ bB − B followsfrom Lemma 4.21(iii), finishing the proof of (5.20). (cid:3) Corollary 5.12. (Assume K = C .) Suppose that a ≤ c ∈ W and b ∈ W . Then for any ( a , c ) ∈ Red( a, c ) and t (cid:48) ∈ R J ◦ a > , we have (5.21) ∆ ∓ i (˙ b − g a , c ( t (cid:48) )) ≥ . Moreover, if a ≤ b ≤ c then (5.22) ∆ ∓ i (˙ b − g a , c ( t (cid:48) )) > , and R > a,c ⊂ ˙ bB − B/B. Proof. By (5.19), we know that ∆ ∓ i (˙ b − g a , c ( t )) ∈ F sf for all i ∈ I . Evaluating at t = t (cid:48) (cf.Section 5.4), we find that ∆ ∓ i (˙ b − g a , c ( t (cid:48) )) ≥ i ∈ I , showing (5.21). Similarly, (5.22)follows from (5.20). (cid:3) Proposition 5.13. (Assume K = F .) For all v, w, v (cid:48) , w (cid:48) ∈ W and x ∈ U sf ( v (cid:48) ) · T sf · U − sf ( w (cid:48) ) ,we have ∆ ± i ( ˙ vx ˙ w − ) ∈ F sf .Proof. Let t = ( t , t , t (cid:48) , t (cid:48) ) with | t | = (cid:96) ( v (cid:48) ), | t | = (cid:96) ( w (cid:48) ), | t (cid:48) | = (cid:96) ( w ) − (cid:96) ( v (cid:48) ), and | t (cid:48) | = (cid:96) ( w ) − (cid:96) ( w (cid:48) ). Let t v := ( t (cid:48) , t ) and t w := ( t , t (cid:48) ). Choose reduced words i , j for w such that i ends with a reduced word for v (cid:48) and j starts with a reduced word for w (cid:48) . Set g = g ( t , t , t v , t w ) := x i ( t v ) · a · y j ( t w ) for some arbitrary element a ∈ T sf . We get˙ vg ˙ w − ∈ ˙ v · U sf ( w ) · T sf · U − sf ( w ) · ˙ w − ⊂ ˙ v · U sf ( v − ) · U sf ( vw ) · T sf · U − sf ( w w − ) · U − sf ( w ) · ˙ w − . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 29 By (5.16), (5.7), and (5.8), we get ˙ vg ˙ w − ∈ B (cid:5) · U − sf ( v ) · U sf ( w − ) · B (cid:5)− . By (5.7), we canpermute U − sf ( v ) and U sf ( w − ), showing ˙ vg ˙ w − ∈ B (cid:5) · B (cid:5)− . Thus ∆ ± i ( ˙ vg ˙ w − ) ∈ F ∗ sf . It givesrise to a continuous function on R | t | > × R | t | > × R | t (cid:48) |≥ × R | t (cid:48) |≥ , so sending t (cid:48) , t (cid:48) → t , t , and a , we get ∆ ± i ( ˙ vx ˙ w − ) ∈ F sf for all x ∈ U sf ( v (cid:48) ) · T sf · U − sf ( w (cid:48) ). (cid:3) Bruhat projections and total positivity In this section, we prove a technical result (Theorem 6.4) which later will be used to finishthe proof of Theorem 2.5. Assume K is algebraically closed and fix u ∈ W J .6.1. The map ζ ( J ) u,v . Retain the notation from Definition 4.23. Given v ∈ W and u ∈ W J ,let us introduce a subset(6.1) G ( J ) u,v := { x ∈ ˙ uG ( J )0 | κ x x ∈ ˙ vG ( J )0 } ⊂ G. Note that if x ∈ G ( J ) u,v then xP ⊂ G ( J ) u,v , see Lemma 6.2(iii) below. Definition 6.1. Define a map η : G ( J ) u,v → L J sending x ∈ G ( J ) u,v to η ( x ) := [ ˙ v − κ x x ] J . Alsodefine a map π ˙ uP − : ˙ uG ( J )0 → ˙ uP − sending x ∈ ˙ uG ( J )0 to the unique element π ˙ uP − ( x ) ∈ ˙ uP − ∩ xU ( J ) . Explicitly (cf. Lemma 4.22(iii)), we put(6.2) π ˙ uP − ( x ) := ˙ u [ ˙ u − x ] ( J ) − [ ˙ u − x ] J = x · ([ ˙ u − x ] ( J )+ ) − . Finally, define ζ ( J ) u,v : G ( J ) u,v → G by ζ ( J ) u,v ( x ) := π ˙ uP − ( x ) · η ( x ) − . Lemma 6.2. (i) The maps κ and π ˙ uP − are regular on ˙ uG ( J )0 . (ii) The maps η and ζ ( J ) u,v are regular on G ( J ) u,v ⊂ ˙ uG ( J )0 . (iii) If x ∈ ˙ uG ( J )0 and x (cid:48) ∈ xP then κ x (cid:48) = κ x . (iv) If x ∈ G ( J ) u,v and x (cid:48) ∈ xP then ζ ( J ) u,v ( x ) = ζ ( J ) u,v ( x (cid:48) ) .Proof. Parts (i) and (ii) are clear since each map is a composition of regular maps. Part (iii)follows from Definition 4.23, since by construction the map κ starts by applying the iso-morphism in (4.31), which gives a regular map C ( J ) u → ˙ uU ( J ) − ˙ u − . To prove (iv), supposethat x ∈ G ( J ) u,v and x (cid:48) ∈ xP is given by x (cid:48) = xp for p ∈ P . Then π ˙ uP − ( x (cid:48) ) = π ˙ uP − ( x )[ p ] J byLemma 4.22(iii). By (iii), κ x (cid:48) = κ x , and η ( x (cid:48) ) = [ ˙ v − κ x (cid:48) x (cid:48) ] J = [ ˙ v − κ x x ] J [ p ] J = η ( x )[ p ] J , thus ζ ( J ) u,v ( x (cid:48) ) = π ˙ uP − ( x (cid:48) ) · η ( x (cid:48) ) − = π ˙ uP − ( x )[ p ] J · [ p ] − J η ( x ) − = ζ ( J ) u,v ( x ) . (cid:3) Lemma 6.3. Let x ∈ ˙ uP − . (i) We have π ˙ uP − ( x ) = x . (ii) If x ∈ G ( J ) u,v then ζ ( J ) u,v ( x ) = xη ( x ) − .Proof. Both parts are clear from Definition 6.1. (cid:3) The ultimate goal of this section is to prove the following result. Theorem 6.4. (Assume K = C .) Let ( u, u ) (cid:22) ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) ∈ Q J and x ∈ G be suchthat xB ∈ R > v (cid:48) ,w (cid:48) . Then x ∈ G ( J ) u,v and ζ ( J ) u,v ( x ) ∈ BB − ˙ w . Properties of κ . We further investigate the element κ x x . Denote ˜ u := uw J ∈ W J max . Lemma 6.5. The groups U ( J ) , U ( J )1 , U ( J )2 from Definition 4.23 satisfy ˙ uU ( J ) − ˙ u − = ˙˜ uU ( J ) − ˙˜ u − , (6.3) U ( J )1 = ˙ uU ( J ) − ˙ u − ∩ U = ˙ uU − ˙ u − ∩ U, (6.4) U ( J )2 = ˙ uU ( J ) − ˙ u − ∩ U − = ˙˜ uU − ˙˜ u − ∩ U − . (6.5) Proof. By Lemma 4.22(ii), we see that ˙ w J U ( J ) − ˙ w J − = U ( J ) − , which shows (6.3). For (6.4), U ( J )1 = ˙ uU ( J ) − ˙ u − ∩ U is just the definition. By Lemma 4.5, we have ˙ uU − J ˙ u − ⊂ U − , so (6.4)follows from (4.5). For (6.5), observe that w J Φ + J = Φ − J , so ˜ u Φ + J ⊂ Φ − by (4.6). We thushave ˙˜ uU − ˙˜ u − = ( ˙˜ uU − J ˙˜ u − ) · ( ˙˜ uU ( J ) − ˙˜ u − ) where ( ˙˜ uU − J ˙˜ u − ) ⊂ U , and hence ˙˜ uU − ˙˜ u − ∩ U − =˙˜ uU ( J ) − ˙˜ u − ∩ U − = U ( J )2 by the definition of U ( J )2 . (cid:3) Lemma 6.6. For x ∈ ˙ uG ( J )0 , there exists a unique element h ∈ U ( J )2 such that hx ∈ U ( J )1 ˙ uP ,and we have h = κ x .Proof. Let g ( J ) ∈ U ( J ) and p ∈ P be such that g ( J ) ˙ u = xp . We first show that such an h ∈ U ( J )2 exists. By Definition 4.23, κ x is an element of U ( J )2 such that κ x g ( J ) ∈ U ( J )1 . Inparticular, κ x x = κ x g ( J ) ˙ up − ∈ U ( J )1 ˙ uP , which shows existence. To show uniqueness, observethat the action of ˙ uU ( J ) − ˙ u − on ˙ uG ( J )0 /P ⊂ G/P is free by (4.31), and in particular the actionof U ( J )2 is also free. (cid:3) Lemma 6.7. If x ∈ ˙ uG ( J )0 ∩ B ˙ u ˙ rB for some r ∈ W J , then κ x = 1 .Proof. By Lemma 6.6, it suffices to show that B ˙ u ˙ rB ⊂ U ( J )1 uP . Write B ˙ u ˙ rB ⊂ B ˙ uP ⊂ ( B ˙ uB ) · P. By (4.34), B ˙ uB ⊂ ( ˙ uU − ∩ U ˙ u ) · B , therefore we find B ˙ u ˙ rB ⊂ ( ˙ uU − ∩ U ˙ u ) · P = ( ˙ uU − ˙ u − ∩ U ) ˙ uP = U ( J )1 ˙ uP, where the last equality follows from (6.4). (cid:3) Lemma 6.8. Let a ∈ T . (i) The subgroups ˙ uU ( J ) ˙ u − , U ( J )1 , and U ( J )2 are preserved under conjugation by a . (ii) If x ∈ ˙ uG ( J )0 , then ax ∈ ˙ uG ( J )0 and κ ax ax = aκ x x . (iii) (Assume K = C .) For each w ∈ W , there exists ρ ∨ w ∈ Y ( T ) such that for all x ∈ ˙ wB − B , lim t → ρ ∨ w ( t ) · xB = ˙ wB in G/B . If w ∈ W J , then for all x ∈ ˙ wG ( J )0 , lim t → ρ ∨ w ( t ) · xP = ˙ wP in G/P .Proof. Since ˙ u ∈ N G ( T ), there exists b ∈ T such that a ˙ u = ˙ ub . Thus a ˙ uU ( J ) ˙ u − a − =˙ ubU ( J ) b − ˙ u − = ˙ uU ( J ) ˙ u − , which shows (i), and (ii) is a simple consequence of (i). Toshow (iii), assume K = C and choose ρ ∨ ∈ Y ( T ) such that (cid:104) ρ ∨ , α i (cid:105) < i ∈ I . Thenlim t → ρ ∨ ( t ) yρ ∨ ( t ) − = 1 for all y ∈ U − , and in particular for all y ∈ U ( J ) − . Set ρ ∨ w := w − ρ ∨ ,thus for t ∈ C ∗ , ρ ∨ w ( t ) = ˙ wρ ∨ ( t ) ˙ w − by (4.2). Every x ∈ ˙ wB − B belongs to ˙ wyB for some y ∈ U − , so ρ ∨ w ( t ) · x · B = ˙ wρ ∨ ( t ) yρ ∨ ( t ) − · B → ˙ wB as t → 0. Similarly, if w ∈ W J then every x ∈ ˙ wG ( J )0 belongs to ˙ wyP for some y ∈ U ( J ) − by (4.31), so ρ ∨ w ( t ) · xP → ˙ wP as t → (cid:3) EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 31 Lemma 6.9. Suppose that v (cid:48)(cid:48) ≤ ur ≤ w (cid:48)(cid:48) for some v (cid:48)(cid:48) , w (cid:48)(cid:48) ∈ W and r ∈ W J , and let x ∈ G . (i) (Assume K = F .) If x ∈ R sf v (cid:48)(cid:48) ,w (cid:48)(cid:48) , then x ∈ ˙ uG ( J )0 . (ii) (Assume K = C .) If xB ∈ R > v (cid:48)(cid:48) ,w (cid:48)(cid:48) , then x ∈ ˙ uG ( J )0 and κ x xB ∈ R > v (cid:48)(cid:48) ,ur w for some r w ∈ W J such that r w ≥ r .Proof. When K = F , (5.20) shows that R sf v (cid:48)(cid:48) ,w (cid:48)(cid:48) ⊂ ˙ u ˙ rB − B ⊂ ˙ uP − B , and by Lemma 4.22(i), P − B = G ( J )0 , which shows (i). Similarly (for K = C ), by Corollary 5.12, we have x ∈ ˙ u ˙ rB − B for any x ∈ R > v (cid:48)(cid:48) ,w (cid:48)(cid:48) , so R > v (cid:48)(cid:48) ,w (cid:48)(cid:48) ⊂ ˙ uG ( J )0 .Assume now that K = C and xB ∈ R > v (cid:48)(cid:48) ,w (cid:48)(cid:48) . Let p ∈ P and g ( J ) ∈ ˙ uU ( J ) − ˙ u − be suchthat xp = g ( J ) ˙ u . Then κ x xp = g ( J )1 ˙ u for g ( J )1 ∈ U ( J )1 . By (6.4), U ( J )1 ˙ u ⊂ U ˙ u ⊂ B ˙ uB . ByLemma 4.22(i), we have p − ∈ B ˙ r w B for some r w ∈ W J . We get κ x x = g ( J )1 ˙ u · p − ∈ B ˙ uB · B ˙ r w B ⊂ B ˙ u ˙ r w B by (4.18). On the other hand, κ x ∈ U − and x ∈ B − v (cid:48)(cid:48) B , so κ x x ∈ B − v (cid:48)(cid:48) B . Therefore κ x xB ∈ ◦ R v (cid:48)(cid:48) ,ur w .We now show r w ≥ r . By (5.22), x ∈ ˙ u ˙ rB − B , so by Lemma 6.8(iii), we have ρ ∨ ur ( t ) · xB → ˙ u ˙ rB as t → G/B . Since ˙ u ˙ r ∈ ˙ uG ( J )0 , κ is regular at ˙ u ˙ rB , and by Lemma 6.7, we have κ ˙ u ˙ r = 1. Thus κ ρ ∨ ur ( t ) x ρ ∨ ur ( t ) xB → ˙ u ˙ rB as t → 0. By Lemma 6.8(ii), κ ρ ∨ ur ( t ) x ρ ∨ ur ( t ) xB = ρ ∨ ur ( t ) · κ x xB , which belongs to ◦ R v (cid:48)(cid:48) ,ur w for all t ∈ C ∗ . We see that the closure of ◦ R v (cid:48)(cid:48) ,ur w contains ˙ u ˙ rB , thus v (cid:48)(cid:48) ≤ ur ≤ ur w by (4.14), so r ≤ r w by Lemma 4.4(ii).Finally, we show κ x xB ∈ ( G/B ) ≥ . First, clearly the map κ is defined over R , thus κ x xB ∈ ( G/B ) R . Consider the subset R > v (cid:48)(cid:48) , [˜ u,w ] := (cid:70) w (cid:48)(cid:48) ≥ ˜ u R > v (cid:48)(cid:48) ,w (cid:48)(cid:48) ⊂ ( G/B ) ≥ . It contains R > v (cid:48)(cid:48) ,w as an open dense subset, and therefore R > v (cid:48)(cid:48) , [˜ u,w ] is connected. We have alreadyshown that for any x (cid:48) ∈ R > v (cid:48)(cid:48) , [˜ u,w ] , κ x (cid:48) x (cid:48) B ∈ ◦ R R v (cid:48)(cid:48) , ˜ u (because we have r w ≥ r = w J ). Thusthe image of the set R > v (cid:48)(cid:48) , [˜ u,w ] under the map x (cid:48) (cid:55)→ κ x (cid:48) x (cid:48) must lie inside a single connectedcomponent of ◦ R R v (cid:48)(cid:48) , ˜ u . However, if x (cid:48) ∈ R > v (cid:48)(cid:48) , ˜ u ⊂ R > v (cid:48)(cid:48) , [˜ u,w ] then κ x (cid:48) = 1 by Lemma 6.7, soin this case κ x (cid:48) x (cid:48) ∈ R > v (cid:48)(cid:48) , ˜ u . We conclude that the image of R > v (cid:48)(cid:48) , [˜ u,w ] is contained inside R > v (cid:48)(cid:48) , ˜ u ⊂ ( G/B ) ≥ . It follows by continuity that for arbitrary v (cid:48)(cid:48) ≤ ur ≤ w (cid:48)(cid:48) and x ∈ R > v (cid:48)(cid:48) ,w (cid:48)(cid:48) ,we have κ x xB ∈ ( G/B ) ≥ . (cid:3) We will use the following consequence of Lemma 6.9(ii) in Section 9.11. Corollary 6.10. (Assume K = C .) In the notation of Lemma , we have κ x xP ∈ Π > v (cid:48)(cid:48) ,u for ¯ v (cid:48)(cid:48) := v (cid:48)(cid:48) (cid:47) r − w .Proof. Indeed, Lemma 6.9(ii) says that κ x xB ∈ R > v (cid:48)(cid:48) ,ur w , so applying Corollary 4.18, we findthat π J ( κ x xB ) = κ x xP ∈ Π > v (cid:48)(cid:48) ,u . (cid:3) Proof via subtraction-free parametrizations. In this section, we fix some set t ofvariables and assume K = F . Also fix u ∈ W J and recall that ˜ u = uw J ∈ W J max .By Definition 4.23, the map κ is defined on ˙ uG ( J )0 . By Lemma 6.9(i), we have R sf v (cid:48)(cid:48) ,w (cid:48)(cid:48) ⊂ ˙ uG ( J )0 whenever v (cid:48)(cid:48) ≤ ur ≤ w (cid:48)(cid:48) for some r ∈ W J . In particular, κ is defined on U − sf ( w (cid:48)(cid:48) ) ⊂ R sfid ,w (cid:48)(cid:48) for all w (cid:48)(cid:48) ≥ ˜ u . Proposition 6.11. Let q ∈ W be such that (cid:96) (˜ uq ) = (cid:96) (˜ u ) + (cid:96) ( q ) . Then for h ∈ U − sf (˜ uq ) , wehave κ h h ∈ U − sf (˜ u ) . Proof. Write h ∈ U − sf (˜ uq ) = U − sf (˜ u ) · U − sf ( q ). Using (5.8), we find h ∈ ˙˜ u · ˙˜ u − · U − sf (˜ u ) · U − sf ( q ) ⊂ ˙˜ u · B (cid:5)− · U sf (˜ u − ) · U − sf ( q ) . By (5.7), B (cid:5)− · U sf (˜ u − ) · U − sf ( q ) = B (cid:5)− · U − sf ( q ) · U sf (˜ u − ) ⊂ B (cid:5)− · U sf (˜ u − ). Writing B (cid:5)− ⊂ U − · T sf ,we get h ∈ ˙˜ u · U − · T sf · U sf (˜ u − ) = T sf · ˙˜ uU − ˙˜ u − · ˙˜ u · U sf (˜ u − ) . Applying (5.16), we find h ∈ T sf · ˙˜ uU − ˙˜ u − · T sf · ( U (cid:5) ∩ ˙˜ uU − ˙˜ u − ) · U − sf (˜ u ) ⊂ ˙˜ uU − ˙˜ u − · T sf · U − sf (˜ u ) . Let g ∈ ˙˜ uU − ˙˜ u − be such that h ∈ g · T sf · U − sf (˜ u ). Recall from (6.5) that U ( J )2 = ˙˜ uU − ˙˜ u − ∩ U − .By Lemma 4.1(i), there exists h (cid:48) ∈ U ( J )2 such that h (cid:48) g ∈ ˙˜ uU − ˙˜ u − ∩ U . Thus h (cid:48) h ∈ ( ˙˜ uU − ˙˜ u − ∩ U ) · T sf · U − sf (˜ u ) ⊂ U · T sf · U − sf (˜ u ) . But observe that both h and h (cid:48) belong to U − . Since the factorization of h (cid:48) h as an element of U · T · U − is unique by Lemma 4.21(i), it follows that h (cid:48) h ∈ U − sf (˜ u ). By (4.20), U − sf (˜ u ) ⊂ B ˙˜ uB .By Lemma 6.7, κ h (cid:48) h = 1, so κ h = h (cid:48) , and thus κ h h ∈ U − sf (˜ u ). (cid:3) Corollary 6.12. For q ∈ W such that (cid:96) (˜ uq ) = (cid:96) (˜ u ) + (cid:96) ( q ) and v ≤ ˜ u , we have R sfid , ˜ uq ⊂ G ( J ) u,v .Proof. As we have already mentioned, Lemma 6.9(i) shows that R sfid , ˜ uq ⊂ ˙ uG ( J )0 . Let x ∈ R sfid , ˜ uq = U − sf (˜ uq ) · B (cid:5) , and let b ∈ B (cid:5) and h ∈ U − sf (˜ uq ) be such that x = hb . By Lemma 6.2(iii),we have κ x = κ h . By Proposition 6.11, κ h h ∈ U − sf (˜ u ), therefore κ x x ∈ U − sf (˜ u ) · B (cid:5) = R sfid , ˜ u .By (5.20), we get κ x x ∈ ˙ vB − B . (cid:3) Corollary 6.12 shows that the map ζ ( J ) u,v is defined on the whole R sfid , ˜ uq . Lemma 6.13. Suppose that u ∈ W J and v ≤ ˜ u := u w J . Let h ∈ U − sf (˜ u ) , and let b u , b v ∈ U be such that ˙˜ u − h ∈ B − · b u and ˙ v − h ∈ B − · b v . Then [ b u b − v ] J ∈ U sf ( r ) for some r ∈ W J .Proof. First, recall from Lemma 4.21(i) and (5.11) that b u and b v are uniquely defined andsatisfy b u ∈ U sf (˜ u − ), b v ∈ U sf ( v − ). Let h = h h for h ∈ U − sf ( u ) and h ∈ U − sf ( w J ). Ourfirst goal is to show that [ b u ] J ∈ U J satisfies (and is uniquely defined by) ˙ w − J h ∈ B − · [ b u ] J .Letting b (cid:48) u ∈ U J be uniquely defined by ˙ w − J h ∈ B − · b (cid:48) u , we thus need to show that [ b u ] J = b (cid:48) u .By (5.9), there exists d ∈ U sf ( u − ) such that˙ w − J ˙ u − h ∈ B (cid:5)− · ˙ w − J · d. Since d ∈ U , we can use Lemma 4.22(iii) to factorize it as d = [ d ] J [ d ] ( J )+ . Since h ∈ U − J ⊂ L J ,Lemma 4.22(ii) shows that there exists d (cid:48) ∈ U ( J ) such that [ d ] ( J )+ h = h d (cid:48) . Since [ d ] J ∈ U J by Lemma 4.22(iv), (4.21) shows that ˙ w − J [ d ] J ∈ U − ˙ w − J . Combining the pieces together, weget ˙˜ u − h = ˙ w − J ˙ u − h h ∈ B (cid:5)− · ˙ w − J · [ d ] J [ d ] ( J )+ · h ⊂ B − · ˙ w − J h d (cid:48) = B − · b (cid:48) u d (cid:48) . On the other hand, ˙˜ u − h ∈ B − · b u , so b u = b (cid:48) u d (cid:48) , where b (cid:48) u ∈ U J and d (cid:48) ∈ U ( J ) . It followsthat [ b u ] J = b (cid:48) u , and thus we have shown that ˙ w − J h ∈ B − · [ b u ] J .We now prove the result by induction on (cid:96) ( u ). When (cid:96) ( u ) = 0, we have ˜ u = w J and v ∈ W J . Thus there exists v ∈ W J such that w J = v · v with (cid:96) ( w J ) = (cid:96) ( v ) + (cid:96) ( v ). We EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 33 have b u , b v ∈ U J , so [ b u b − v ] J = b u b − v by Lemma 4.22(iv). By (5.10), there exist b ∈ U sf ( v − )and b ∈ U sf ( v − ) such that ˙ v − h ∈ B (cid:5)− · b , ˙ w − J h ∈ B (cid:5)− · b b . In particular, we have b v = b and b u = b b . Thus [ b u b − v ] J = b ∈ U sf ( v − ), and we aredone with the base case.Assume (cid:96) ( u ) > 0, and let i ∈ I be such that u := s i u < u . By Lemma 4.4(i), u ∈ W J , so denote ˜ u := u w J ∈ W J max . Let h ∈ U − sf (˜ u ) be factorized as h = h i h (cid:48) h for h i = y i ( t ) ∈ U − sf ( s i ), h (cid:48) ∈ U − sf ( u ), and h ∈ U − sf ( w J ).Suppose that s i v > v , in which case we have v ≤ ˜ u . Let h (cid:48) := h (cid:48) h and b (cid:48) u ∈ U be defined by ˙˜ u − h (cid:48) ∈ B − · b (cid:48) u . Since s i v > v , we see that ˙ v − h i ∈ B − · ˙ v − , so ˙ v − h (cid:48) ∈ B − · ˙ v − h = B − · b v . By the induction hypothesis applied to v ≤ ˜ u and h (cid:48) ∈ U − sf (˜ u ), wehave [ b (cid:48) u b − v ] J ∈ U sf ( r ) for some r ∈ W J . On the other hand, we have shown above that [ b u ] J satisfies ˙ w − J h ∈ B − · [ b u ] J . But since h (cid:48) = h (cid:48) h for h ∈ U − sf ( w J ), we get that [ b (cid:48) u ] J satisfies˙ w − J h ∈ B − · [ b (cid:48) u ] J , thus [ b u ] J = [ b (cid:48) u ] J . Therefore using Lemma 4.22(iv), we get[ b u b − v ] J = [ b u ] J [ b − v ] J = [ b (cid:48) u ] J [ b − v ] J = [ b (cid:48) u b − v ] J ∈ U sf ( r ) , finishing the induction step in the case s i v > v .Suppose now that v := s i v < v . Let h = h i h (cid:48) h ∈ U − sf (˜ u ) be as above. By (5.8),˙ s − i h i ∈ B (cid:5)− · U sf ( s i ), so let d i ∈ U sf ( s i ) be such that ˙ s − i h i ∈ B (cid:5)− · d i . By (5.7), U sf ( s i ) · U − sf (˜ u ) = U − sf (˜ u ) · U sf ( s i ), so let b i ∈ U sf ( s i ) and h (cid:48) ∈ U − sf (˜ u ) be such that d i h (cid:48) h = h (cid:48) b i . We checkusing (5.9) that(6.6) ˙˜ u − h ∈ B (cid:5)− · ˙˜ u − h (cid:48) · b i , ˙ v − h ∈ B (cid:5)− · ˙ v − h (cid:48) · b i . Let b (cid:48) u , b (cid:48) v ∈ U be defined by ˙˜ u − h (cid:48) ∈ B − · b (cid:48) u and ˙ v − h (cid:48) ∈ B − · b (cid:48) v . Then by the inductionhypothesis applied to v ≤ ˜ u and h (cid:48) ∈ U − sf (˜ u ), we find [ b (cid:48) u b (cid:48)− v ] J ∈ U sf ( r ) for some r ∈ W J .But it is clear from (6.6) that b u = b (cid:48) u b i and b v = b (cid:48) v b i . Therefore [ b u b − v ] J ∈ U sf ( r ). (cid:3) Theorem 6.14. For all v ≤ ˜ u , w ∈ W J , i ∈ I , and x ∈ R sfid ,w , we have (6.7) ∆ ± i ( ζ ( J ) u,v ( x ) ˙ w − ) ∈ F sf . Proof. Let q ∈ W be such that w = ˜ uq , thus (cid:96) (˜ uq ) = (cid:96) (˜ u ) + (cid:96) ( q ). Let x ∈ R sfid ,w = U − sf ( w ) · B (cid:5) be written as x = h · b , where h = h h h ∈ U − sf ( w ) for h ∈ U − sf ( u ), h ∈ U − sf ( w J ), h ∈ U − sf ( q ), and b ∈ B (cid:5) . By (5.10), there exist b ∈ U sf ( u − ), b ∈ U sf ( w J ), and b ∈ U sf ( q − )such that(6.8) ˙ u − h ∈ B (cid:5)− · b , ˙˜ u − h ∈ B (cid:5)− · b b , ˙ w − h ∈ B (cid:5)− · b b b . Let x (cid:48) := hb − . We have x (cid:48) = xb − b − ∈ xB ⊂ xP , therefore x (cid:48) ∈ G ( J ) u,v and ζ ( J ) u,v ( x (cid:48) ) = ζ ( J ) u,v ( x )by Lemma 6.2(iv). On the other hand, by (6.8), x (cid:48) ∈ ˙ uB (cid:5)− ⊂ ˙ uP − , so Lemma 6.3(ii) implies ζ ( J ) u,v ( x (cid:48) ) = x (cid:48) η ( x (cid:48) ) − .Let us now compute η ( x (cid:48) ) = [ ˙ v − κ x (cid:48) x (cid:48) ] J . By Lemma 6.2(iii), κ x = κ x (cid:48) = κ h , and byProposition 6.11, κ h h ∈ U − sf (˜ u ). Thus by (5.11), ˙ v − κ h h ∈ B (cid:5)− · U sf ( v − ), so let d ∈ B (cid:5)− and b ∈ U sf ( v − ) be such that ˙ v − κ h h = d b . By definition, κ h ∈ U ( J )2 , so by (6.5),˙˜ u − κ h ˙˜ u ∈ U − , and therefore using (6.8) we find˙˜ u − κ h h = ˙˜ u − κ h ˙˜ u · ˙˜ u − h ∈ U − · ˙˜ u − h ⊂ B − · b b . We can now apply Lemma 6.13: we have v ≤ ˜ u , κ h h ∈ U − sf (˜ u ), ˙˜ u − κ h h ∈ B − · b b , and˙ v − κ h h ∈ B − · b . Let b u := b b ∈ U and b v := b ∈ U . By Lemma 6.13, [ b u b − v ] J =[ b b b − ] J ∈ U sf ( r ) for some r ∈ W J .Recall that ˙ v − κ h h = d b for d ∈ B (cid:5)− and b ∈ U sf ( v − ). Thus η ( x (cid:48) ) = [ ˙ v − κ x (cid:48) x (cid:48) ] J = [ ˙ v − κ h x (cid:48) ] J = [ ˙ v − κ h hb − ] J = [ d b b − ] J . By Lemma 4.22(iii), we get [ d b b − ] J = [ d ] J [ b b − ] J . Thus ζ ( J ) u,v ( x ) = ζ ( J ) u,v ( x (cid:48) ) = x (cid:48) η ( x (cid:48) ) − = x (cid:48) [ b b − ] − J [ d ] − J . By (6.8), we have ˙ w − x (cid:48) ∈ B (cid:5)− · b b , so x (cid:48) ∈ B (cid:5) ˙ w b b . Using Lemma 4.22(iv), we thus get ζ ( J ) u,v ( x ) = x (cid:48) [ b b − ] − J [ d ] − J ∈ B (cid:5) · ˙ w b [ b b b − ] J [ d ] − J . We are interested in the element ζ ( J ) u,v ( x ) ˙ w − . We know that d ∈ B (cid:5)− , thus [ d ] J ∈ T sf U − J ,and by Lemma 4.5, ˙ w [ d ] J ˙ w − ∈ T sf · U − . Hence ζ ( J ) u,v ( x ) ˙ w − ∈ B (cid:5) · ˙ w b [ b b b − ] J [ d ] − J ˙ w − ⊂ B (cid:5) · ˙ w b [ b b b − ] J ˙ w − · T sf · U − . In particular, ∆ ± i ( ζ ( J ) u,v ( x ) ˙ w − ) ∈ F sf if and only if ∆ ± i ( ˙ w b [ b b b − ] J ˙ w − ) ∈ F sf . Recall that b ∈ U sf ( q − ) and [ b b b − ] J ∈ U sf ( r ) for some r ∈ W J . Thus b [ b b b − ] J ∈ U sf ( q − r ), so weare done by Proposition 5.13. (cid:3) Proof of Theorem 6.4. Our strategy will be very similar to the one we used in the proof ofCorollary 5.12.Fix ( u, u ) (cid:22) ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) ∈ Q J . Let t = ( t , t , t ) with | t | = (cid:96) ( v (cid:48) ), | t | = (cid:96) ( w ) − (cid:96) ( w (cid:48) ), and | t | := (cid:96) ( w (cid:48) ) − (cid:96) ( v (cid:48) ), and assume K = F . Choose reduced words i for v (cid:48)− and j for w w (cid:48)− , and let ( v (cid:48) , w (cid:48) ) ∈ Red( v (cid:48) , w (cid:48) ). Suppose that x ∈ g v (cid:48) , w (cid:48) ( t ) · B (cid:5) . Then g ( t , t , t ) := x i ( t ) · y j ( t ) · g v (cid:48) , w (cid:48) ( t ) ∈ U sf ( v (cid:48)− ) · U − sf ( w w (cid:48)− ) · R sf v (cid:48) ,w (cid:48) . By Lemma 5.8, we have g ( t , t , t ) ∈ R sfid ,w . Thus by Theorem 6.14, for all i ∈ I wehave ∆ ± i ( ζ ( J ) u,v ( g ( t , t , t )) ˙ w − ) ∈ F sf . Denote by f ( t , t , t ) := ∆ ± i ( ζ ( J ) u,v ( g ( t , t , t )) ˙ w − )the corresponding subtraction-free rational expression, which yields a continuous function R | t | > × R | t | > × R | t | > → R ≥ . We claim that f extends to a continuous function R | t |≥ × R | t |≥ × R | t | > → R ≥ . Indeed, fix some ( t (cid:48) , t (cid:48) , t (cid:48) ) ∈ R | t |≥ × R | t |≥ × R | t | > and let K = C . Theelement x (cid:48) := g ( t (cid:48) , t (cid:48) , t (cid:48) ) (obtained by evaluating at ( t (cid:48) , t (cid:48) , t (cid:48) ), see Section 5.4) belongs to G ≥ · R > v (cid:48) ,w (cid:48) , and by Lemma 4.17 there exist v (cid:48)(cid:48) , w (cid:48)(cid:48) ∈ W such that v (cid:48)(cid:48) ≤ v (cid:48) ≤ w (cid:48) ≤ w (cid:48)(cid:48) and x (cid:48) ∈ R > v (cid:48)(cid:48) ,w (cid:48)(cid:48) . Recall from Lemma 4.8(ii) that we have v (cid:48)(cid:48) ≤ v (cid:48) ≤ vr (cid:48) ≤ ur ≤ wr (cid:48) ≤ w (cid:48) ≤ w (cid:48)(cid:48) for some r (cid:48) , r ∈ W J such that (cid:96) ( vr (cid:48) ) = (cid:96) ( v ) + (cid:96) ( r (cid:48) ). In particular, by Lemma 6.9(ii), x (cid:48) ∈ ˙ uG ( J )0 and κ x (cid:48) x (cid:48) ∈ R > v (cid:48)(cid:48) ,ur w for some r w ∈ W J such that r w ≥ r . By Corollary 5.12, κ x (cid:48) x (cid:48) ∈ ˙ v ˙ r (cid:48) B − B ⊂ ˙ vG ( J )0 , which shows that x (cid:48) ∈ G ( J ) u,v . The map ζ ( J ) u,v is therefore regular at x (cid:48) by Lemma 6.2(ii). The map ∆ ± i is regular on G by Lemma 4.21(ii), so in particular itis regular at ζ ( J ) u,v ( x (cid:48) ) ˙ w − . We have shown that the map x (cid:48)(cid:48) (cid:55)→ ∆ ± i ( ζ ( J ) u,v ( x (cid:48)(cid:48) ) ˙ w − ) is regularat x (cid:48) = g ( t (cid:48) , t (cid:48) , t (cid:48) ) for all ( t (cid:48) , t (cid:48) , t (cid:48) ) ∈ R | t |≥ × R | t |≥ × R | t | > . Thus the map f ( t , t , t )extends to a continuous function R | t |≥ × R | t |≥ × R | t | > → R ≥ . By Lemma 5.9, we findthat f (0 , , t ) := lim t , t → f ( t , t , t ) belongs to F sf , i.e., it can be represented by a EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 35 subtraction-free rational expression in the variables t . On the other hand, it is clear that f (0 , , t ) = ∆ ± i ( ζ ( J ) u,v ( g v (cid:48) , w (cid:48) ( t )) ˙ w − ).Our next goal is to show that f (0 , , t ) ∈ F ∗ sf . Indeed, suppose otherwise that f (0 , , t ) =0 (as an element of F ). By Lemma 6.2(iv), ζ ( J ) u,v descends to a regular map G ( J ) u,v /P → G (stillassuming K = C ). Therefore the map ¯ f : G ( J ) u,v /P → C sending x (cid:48) P (cid:55)→ ∆ ± i ( ζ ( J ) u,v ( x (cid:48) ) ˙ w − ) isalso regular. If f (0 , , t ) = 0 then ¯ f vanishes on π J ( R > v (cid:48) ,w (cid:48) ) = Π > v (cid:48) ,w (cid:48) , and therefore it vanisheson its Zariski closure, which is Π v (cid:48) ,w (cid:48) . We have π J ( R > v,w ) = Π > v,w ⊂ Π v (cid:48) ,w (cid:48) , thus ¯ f ( x ) = 0 forany x ∈ G ( J ) u,v such that xB ∈ R > v,w . Let us show that this leads to a contradiction.Let x ∈ G be such that xB ∈ R > v,w . By (4.27), there exists x (cid:48) ∈ xP such that x (cid:48) B ∈ R > vr (cid:48) ,wr (cid:48) . By Lemma 6.9(ii), we have x (cid:48) ∈ ˙ uG ( J )0 , and thus x ∈ ˙ uG ( J )0 . Having xB ∈ R > v,w implies x ∈ B − ˙ vB ∩ B ˙ wB . Since κ x ∈ U ( J )2 ⊂ U − , we have κ x x ∈ B − ˙ vB . By (4.34), B − ˙ vB = ( ˙ vU − ∩ U − ˙ v ) B ⊂ ˙ vB − B , thus κ x x ∈ ˙ vB − B , and therefore x ∈ G ( J ) u,v . Moreover,˙ v − κ x x ∈ B − B , thus η ( x ) = [ ˙ v − κ x x ] J ∈ U − J T U J . On the other hand, π ˙ uP − ( x ) ∈ xU ( J ) ⊂ xB ⊂ B ˙ wB , see Definition 6.1. Thus ζ ( J ) u,v ( x ) = π ˙ uP − ( x ) η ( x ) − ∈ B ˙ wB · U J T U − J = B ˙ wB · U − J . Recall that because w ∈ W J , we have U − J ˙ w − ⊂ ˙ w − U − by Lemma 4.5. Hence ζ ( J ) u,v ( x ) ˙ w − ∈ B ˙ wB · U − J · ˙ w − ⊂ B ˙ wB ˙ w − B − . By (4.34) (after taking inverses of both sides), B ˙ wB = B · ( U − ˙ w ∩ ˙ wU ), so ζ ( J ) u,v ( x ) ˙ w − ∈ B · ( U − ∩ ˙ wU ˙ w − ) · B − ⊂ B · B − . In particular, ∆ ± i ( ζ ( J ) u,v ( x ) ˙ w − ) (cid:54) = 0 for all i ∈ I . This gives a contradiction, showing f (0 , , t ) ∈ F ∗ sf . But then evaluating f at any t (cid:48) ∈ R (cid:96) ( w (cid:48) ) − (cid:96) ( v (cid:48) ) > yields a positive real number.We have shown that ∆ ± i ( ζ ( J ) u,v ( x ) ˙ w − ) (cid:54) = 0 for all x ∈ G such that xB ∈ R > v (cid:48) ,w (cid:48) . We are doneby Lemma 4.21(iii). (cid:3) Affine Bruhat atlas for the projected Richardson stratification In this section, we embed the stratification (4.23) of G/P inside the affine Richardsonstratification of the affine flag variety. Throughout, we work over K = C .7.1. Loop groups and affine flag varieties. Recall that G is a simple and simply con-nected algebraic group. Let A := C [ z, z − ] and A + , A − ⊂ A denote the subrings given by A + := C [ z ], A − := C [ z − ]. Then we have ring homomorphisms ¯ev : A + → C (respectively,¯ev ∞ : A − → C ), sending a polynomial in z (respectively, in z − ) to its constant term. Let G := G ( A ) denote the polynomial loop group of G . Remark 7.1. The group G is closely related to the (minimal) affine Kac–Moody group G min associated to G , introduced by Kac and Peterson [KP83, PK83]. Below we state manystandard results about G without proof. We refer the reader unfamiliar with Kac–Moodygroups to Appendix A, where we give some background and explain how to derive thesestatements from Kumar’s book [Kum02].We introduce opposite Iwahori subgroups B := { g ( z ) ∈ G ( A + ) | ¯ev ( g ) ∈ B } , B − := { g ( z − ) ∈ G ( A − ) | ¯ev ∞ ( g ) ∈ B − } of G , and denote by U := { g ( z ) ∈ G ( A + ) | ¯ev ( g ) ∈ U } , U − := { g ( z − ) ∈ G ( A − ) | ¯ev ∞ ( g ) ∈ U − } their unipotent radicals. There exists a tautological embedding G (cid:44) → G , and we treat G asa subset of G .We let T := C ∗ × T ⊂ C ∗ (cid:110) G be the affine torus, where C ∗ acts on G via loop rotation,see Section 8.2. The affine root system ∆ of G is the subset of X ( T ) := Hom( T , C ∗ ) ∼ = X ( T ) ⊕ Z δ given by∆ = ∆ re (cid:116) ∆ im , where ∆ re := { β + jδ | β ∈ Φ , j ∈ Z } , ∆ im := { jδ | j ∈ Z \ { }} are the real and imaginary roots, and the set of positive roots ∆ + ⊂ ∆ has the form(7.1) ∆ + = { jδ | j > } (cid:116) { β + jδ | β ∈ Φ , j > } (cid:116) { β | β ∈ Φ + } . We let ∆ +re := ∆ + ∩ ∆ re and ∆ − re := ∆ − ∩ ∆ re . For each α ∈ ∆ +re (respectively, α ∈ ∆ − re ), wehave a one-parameter subgroup U α ⊂ U (respectively, U α ⊂ U − ). The group U (respectively, U − ) is generated by {U α } α ∈ ∆ +re (respectively, {U α } α ∈ ∆ − re ), and for each α ∈ ∆ re , we fix agroup isomorphism x α : C ∼ −→ U α .Let Q ∨ Φ := (cid:76) i ∈ I Z α ∨ i denote the coroot lattice of Φ. The affine Weyl group ˜ W = W (cid:110) Q ∨ Φ is a semidirect product of W and Q ∨ Φ , i.e., as a set we have ˜ W = W × Q ∨ Φ , and the productrule is given by ( w , λ ) · ( w , λ ) := ( w w , λ + w λ ) . For λ ∈ Q ∨ Φ , we denote the element(id , λ ) ∈ ˜ W by τ λ . The group ˜ W is isomorphic to N C ∗ (cid:110) G ( T ) / T , and for f ∈ ˜ W , wechoose a representative ˙ f ∈ G of f in N C ∗ (cid:110) G ( T ), with the assumption that for w ∈ W , therepresentative ˙ w ∈ G ⊂ G is given by (4.1). Thus ˜ W is a Coxeter group with generators s (cid:116) { s i } i ∈ I , length function (cid:96) : ˜ W → Z ≥ , and affine Bruhat order ≤ . The group ˜ W actson ∆, and for α ∈ Φ, β ∈ ∆ re , λ ∈ Q ∨ Φ , and w ∈ W , we have(7.2) wτ λ w − = τ wλ , τ λ α = α + (cid:104) λ, α (cid:105) δ, τ λ δ = δ, ˙ τ λ U β ˙ τ − λ = U τ λ β . Let G / B denote the affine flag variety of G . This is an ind-variety that is isomorphic to theflag variety of the corresponding affine Kac–Moody group G min , see Appendix A.4. For each h, f ∈ ˜ W we have Schubert cells ◦ X f := B ˙ f B / B and opposite Schubert cells ◦ X h := B − ˙ h B / B .If h (cid:54)≤ f ∈ ˜ W then ◦ X h ∩ ◦ X f = ∅ . For h ≤ f , we denote ◦ R fh := ◦ X h ∩ ◦ X f . For all g ∈ ˜ W , wehave ◦ X g = (cid:71) h ≤ g ◦ R gh , ◦ X g = (cid:71) g ≤ f ◦ R fg , X g := (cid:71) h ≤ g ◦ X h , X g := (cid:71) g ≤ f ◦ X f . (7.3)For g ∈ ˜ W , let(7.4) C g := ˙ g B − B / B , U ( g ) := ˙ g U − ˙ g − ∩ U , and U ( g ) := ˙ g U − ˙ g − ∩ U − . As we explain in Appendix A.5, the map x (cid:55)→ x ˙ g B gives biregular isomorphisms(7.5) ˙ g U − ˙ g − ∼ −→ C g , U ( g ) ∼ −→ ◦ X g , U ( g ) ∼ −→ ◦ X g . Let U ( I ) ⊂ U be the subgroup generated by {U α } α ∈ ∆ +re \ Φ + . Similarly, let U ( I ) − ⊂ U − be thesubgroup generated by {U α } α ∈ ∆ − re \ Φ − . For x ∈ G ⊂ G , we have(7.6) x · U ( I ) · x − = U ( I ) , x · U ( I ) − · x − = U ( I ) − . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 37 Combinatorial Bruhat atlas for G/P . We fix an element λ ∈ Q ∨ Φ such that (cid:104) λ, α i (cid:105) =0 for i ∈ J and (cid:104) λ, α i (cid:105) ∈ Z < for i ∈ I \ J . Thus λ is anti-dominant and the stabilizer of λ in W is equal to W J . Following [HL15], define a map(7.7) ψ : Q J → ˜ W , ( v, w ) (cid:55)→ vτ λ w − . By [HL15, Theorem 2.2], the map ψ gives an order-reversing bijection between Q J and asubposet of ˜ W . More precisely, let τ Jλ := τ λ ( w J ) − , and recall from (7.2) that uτ λ u − = τ uλ .By [HL15, § v, w ) ∈ Q J we have(7.8) vτ λ w − = v · τ Jλ · w J w − , (cid:96) ( vτ λ w − ) = (cid:96) ( v ) + (cid:96) ( τ Jλ ) + (cid:96) ( w J w − ) , see Figure 2 for an example. By [HL15, Theorem 2.2], for all u ∈ W J we have ψ ( Q (cid:23) ( u,u ) J ) = { g ∈ ˜ W | τ Jλ ≤ g ≤ τ uλ } , (7.9) ψ ( Q J ) = { g ∈ ˜ W | τ Jλ ≤ g ≤ τ wλ for some w ∈ W J } . (7.10)7.3. Bruhat atlas for the projected Richardson stratification of G/P . Let u ∈ W J .Recall that λ ∈ Q ∨ Φ has been fixed. We further assume that the representatives ˙ τ λ and ˙ τ uλ satisfy the identity ˙ u ˙ τ λ ˙ u − = ˙ τ uλ .Our goal is to construct a geometric lifting of the map ψ . Recall the maps x (cid:55)→ g ( J )1 and x (cid:55)→ g ( J )2 from Definition 4.23. We define maps ϕ u : C ( J ) u → G , xP (cid:55)→ g ( J )1 ˙ u · ˙ τ λ · ( g ( J )2 ˙ u ) − = g ( J )1 · ˙ τ uλ · ( g ( J )2 ) − , and(7.11) ¯ ϕ u : C ( J ) u → G / B , xP (cid:55)→ ϕ u ( xP ) · B . (7.12)The main result of this section is the following theorem. Theorem 7.2. (1) The map ¯ ϕ u is a biregular isomorphism ¯ ϕ u : C ( J ) u ∼ −→ X τ Jλ ∩ ◦ X τ uλ = (cid:71) ( v,w ) ∈ Q (cid:23) ( u,u ) J ◦ R τ uλ vτ λ w − , and for all ( v, w ) (cid:23) ( u, u ) ∈ Q J , ¯ ϕ u restricts to a biregular isomorphism ¯ ϕ u : C ( J ) u ∩ ◦ Π v,w ∼ −→ ◦ R τ uλ vτ λ w − . (2) Suppose that ( u, u ) (cid:22) ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) ∈ Q J . Then ¯ ϕ u (cid:0) Π > v (cid:48) ,w (cid:48) (cid:1) ⊂ C vτ λ w − . The remainder of this section will be devoted to the proof of Theorem 7.2.7.4. An alternative definition of ¯ ϕ u . Recall the notation from Definition 4.23, and thatwe have fixed u ∈ W J and λ ∈ Q ∨ Φ satisfying (cid:104) λ, α i (cid:105) = 0 for i ∈ J and (cid:104) λ, α i (cid:105) ∈ Z < for i ∈ I \ J . We list the rules for conjugating elements of G ⊂ G by ˙ τ λ . Lemma 7.3. We have ˙ τ λ · p = p · ˙ τ λ for all p ∈ L J , (7.13) ˙ τ λ · U ( J ) · ˙ τ − λ ⊂ U ( I ) − , ˙ τ λ · U ( J ) − · ˙ τ − λ ⊂ U ( I ) , (7.14) ˙ τ − λ · U ( J ) · ˙ τ λ ⊂ U ( I ) , ˙ τ − λ · U ( J ) − · ˙ τ λ ⊂ U ( I ) − , (7.15) ˙ τ uλ · U ( J )2 · ˙ τ − uλ ⊂ U ( I ) , ˙ τ − uλ · U ( J )1 · ˙ τ uλ ⊂ U ( I ) − . (7.16) Proof. Recall that L J is generated by T , U J , and U − J , and since τ λ α = α for all α ∈ Φ J ,we see that (7.13) follows from (7.2). By (7.2), we find τ λ α ∈ ∆ +re \ Φ + for α ∈ Φ ( J ) − and τ λ α ∈ ∆ − re \ Φ − for α ∈ Φ ( J )+ , which shows (7.14). Similarly, τ − λ α ∈ ∆ +re \ Φ + for α ∈ Φ ( J )+ and τ − λ α ∈ ∆ − re \ Φ − for α ∈ Φ ( J ) − , which shows (7.15).To show (7.16), we use (7.6), (7.14), (7.15), and U ( J )1 , U ( J )2 ⊂ ˙ uU ( J ) − ˙ u − to get˙ τ uλ · U ( J )2 · ˙ τ − uλ = ˙ u ˙ τ λ ˙ u − · U ( J )2 · ˙ u ˙ τ − λ ˙ u − ⊂ ˙ u ˙ τ λ · U ( J ) − · ˙ τ − λ ˙ u − ⊂ ˙ u U ( I ) ˙ u − = U ( I ) , ˙ τ − uλ · U ( J )1 · ˙ τ uλ = ˙ u ˙ τ − λ ˙ u − · U ( J )1 · ˙ u ˙ τ λ ˙ u − ⊂ ˙ u ˙ τ − λ · U ( J ) − · ˙ τ λ ˙ u − ⊂ ˙ u U ( I ) − ˙ u − = U ( I ) − . (cid:3) The map ¯ ϕ u can alternatively be characterized as follows. Recall from Definition 4.23 thatwe have a regular map κ : ˙ uG ( J )0 → U ( J )2 that descends to a regular map κ : C ( J ) u → U ( J )2 byLemma 6.2(iii). Recall also from Lemma 4.22(i) that ˙ uG ( J )0 = ˙ uP − · B . Lemma 7.4. Let x ∈ ˙ uP − . Then (7.17) ¯ ϕ u ( xP ) = κ x x · ˙ τ λ · x − · B . Proof. We continue using the notation of Definition 4.23. Let p ∈ L J and g ( J ) ∈ ˙ uU ( J ) − ˙ u − besuch that xp = g ( J ) ˙ u . Note that g ( J )2 ˙ u = h ( J )1 g ( J ) ˙ u = h ( J )1 xp , and since h ( J )1 ∈ U ( J )1 ⊂ U ⊂ B ,we see that ( g ( J )2 ˙ u ) − · B = ( xp ) − · B . On the other hand, κ x xp = h ( J )2 g ( J ) ˙ u = g ( J )1 ˙ u . Since p commutes with ˙ τ λ by (7.13), we find¯ ϕ u ( xP ) = g ( J )1 ˙ u · ˙ τ λ · ( g ( J )2 ˙ u ) − · B = κ x xp · ˙ τ λ · ( xp ) − · B = κ x x · ˙ τ λ · x − · B . (cid:3) The affine Richardson cell of ¯ ϕ u .Lemma 7.5. We have (7.18) C ( J ) u = (cid:71) ( v,w ) ∈ Q (cid:23) ( u,u ) J ( C ( J ) u ∩ ◦ Π v,w ) . Proof. The torus T acts on G/P by left multiplication and preserves the sets C ( J ) u and ◦ Π v,w for all ( v, w ) ∈ Q J . By (4.23), Π v,w contains ˙ uP if and only if ( u, u ) (cid:22) ( v, w ). Suppose that xP ∈ C ( J ) u ∩ ◦ Π v,w for some ( v, w ) ∈ Q J . Then T xP/P ⊂ C ( J ) u , and by Lemma 6.8(iii), theclosure of this set contains ˙ uP . On the other hand, the closure of this set is contained insideΠ v,w , thus ( u, u ) (cid:22) ( v, w ). (cid:3) Lemma 7.6. Let ( v, w ) ∈ Q (cid:23) ( u,u ) J . Then (7.19) ¯ ϕ u ( C ( J ) u ∩ ◦ Π v,w ) ⊂ ◦ R τ uλ vτ λ w − . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 39 Proof. Let x ∈ ˙ uG ( J )0 be such that xP ∈ ◦ Π v,w . Let us first show that ¯ ϕ u ( xP ) ∈ ◦ X τ uλ .By (7.12), we have(7.20) ¯ ϕ u ( xP ) = g ( J )1 · ˙ τ uλ · ( g ( J )2 ) − · ˙ τ − uλ · ˙ τ uλ · B . Observe that g ( J )1 ∈ U ( J )1 ⊂ U , and by (7.16), ˙ τ uλ · ( g ( J )2 ) − · ˙ τ − uλ ∈ U ( I ) . We get(7.21) ϕ u ( xP ) · ˙ τ − uλ ∈ U , thus ϕ u ( xP ) ∈ B · ˙ τ uλ · B . This proves that ¯ ϕ u ( xP ) ∈ ◦ X τ uλ .We now show ¯ ϕ u ( xP ) ∈ ◦ X vτ λ w − . Recall that ◦ Π v,w = π J ( ◦ R v,w ), so assume that x ∈ B − ˙ vB ∩ B ˙ wB . Since ˙ uG ( J )0 = ˙ uP − B by Lemma 4.22(i), we may assume that x ∈ ˙ uP − , inwhich case ¯ ϕ u ( xP ) is given by (7.17). We have κ x x ∈ B − ˙ vB and x − ∈ B ˙ w − B , so it sufficesto show(7.22) B − ˙ vB · ˙ τ λ · B ˙ w − B ⊂ B − · ˙ v ˙ τ λ ˙ w − · B . Clearly we have B − ˙ vB · ˙ τ λ · B ˙ w − B ⊂ B − · ˙ v · U ( J ) · U J · ˙ τ λ · U ( J ) · U J · ˙ w − · B . By (7.13) and Lemma 4.22(ii), U J can be moved to the right past ˙ τ λ and U ( J ) . We can thenmove U ( J ) to the left past ˙ τ λ using (7.14), which gives B − ˙ vB · ˙ τ λ · B ˙ w − B ⊂ B − · ˙ v · U ( J ) · U ( I ) − · ˙ τ λ · U J · ˙ w − · B . By (7.6), U ( I ) − can be moved to the left past ˙ v · U ( J ) , and then U ( J ) can be moved to theright past ˙ τ λ using (7.15), yielding B − ˙ vB · ˙ τ λ · B ˙ w − B ⊂ B − · ˙ v · ˙ τ λ · U ( I ) · U J · ˙ w − · B . By (7.6), U ( I ) can be moved to the right past U J · ˙ w − . Since w ∈ W J , Lemma 4.5 impliesthat U J · ˙ w − ⊂ ˙ w − U , so (7.22) follows. (cid:3) Proof of Theorem 7.2(1). Observe that X τ Jλ ∩ ◦ X τ uλ = (cid:70) ( v,w ) ∈ Q (cid:23) ( u,u ) J ◦ R τ uλ vτ λ w − by (7.3)and (7.9). By (7.19), ¯ ϕ u ( C ( J ) u ) ⊂ X τ Jλ ∩ ◦ X τ uλ . Let us identify ◦ X τ uλ with the affine variety U ( τ uλ ) via (7.5), and denote by ¯ ϕ † u : C ( J ) u → U ( τ uλ ) the composition of (7.5) and ¯ ϕ u .We claim that ¯ ϕ † u gives a biregular isomorphism between C ( J ) u and a closed subvariety of U ( τ uλ ). Let x ∈ ˙ uG ( J )0 and let g ( J ) , g ( J )1 , g ( J )2 be as in Definition 4.23. Let y := ϕ u ( xP ) · ˙ τ − uλ ,so ¯ ϕ u ( xP ) = y · ˙ τ uλ · B . Thus ¯ ϕ † u ( xP ) = y if and only if y ∈ U ( τ uλ ). By (7.21), we have y ∈ U . Hence in order to prove y ∈ U ( τ uλ ), we need to show y ∈ ˙ τ uλ U − ˙ τ − uλ . Conjugatingboth sides by ˙ τ uλ , we get ˙ τ − uλ · y · ˙ τ uλ = ˙ τ − uλ g ( J )1 ˙ τ uλ · ( g ( J )2 ) − , which belongs to U − since ( g ( J )2 ) − ∈ U ( J )2 ⊂ U − by definition and ˙ τ − uλ g ( J )1 ˙ τ uλ ∈ U ( I ) − by (7.16).Thus y ∈ U ( τ uλ ) and ¯ ϕ † u ( xP ) = y . By Lemma 4.2, we may identify C ( J ) u with U ( J )1 × U ( J )2 ,so let ¯ ϕ ‡ u : U ( J )1 × U ( J )2 → U ( τ uλ ) be the map sending ( g ( J )1 , g ( J )2 ) (cid:55)→ y := g ( J )1 · ˙ τ uλ ( g ( J )2 ) − ˙ τ − uλ .Let Θ := u Φ ( J ) − ∩ Φ + and Θ := u Φ ( J ) − ∩ Φ − , thus U ( J )1 = U (Θ ), U ( J )2 = U − (Θ ),and Θ (cid:116) Θ = u Φ ( J ) − . By the proof of (7.16), τ uλ Θ ⊂ ∆ +re \ Φ + and τ − uλ Θ ⊂ ∆ − re , thus Θ (cid:116) τ uλ Θ ⊂ Inv( τ − uλ ). Let Θ ⊂ ∆ +re be defined by Θ := Inv( τ − uλ ) \ (Θ (cid:116) τ uλ Θ ). ByLemma A.1, the multiplication map gives a biregular isomorphism(7.23) U (Θ ) × U ( τ uλ Θ ) × (cid:89) α ∈ Θ U α ∼ −→ U (Inv( τ − uλ )) = U ( τ uλ ) , where U (Θ) denotes the subgroup generated by {U α } α ∈ Θ . In particular, U (Θ ) · U ( τ uλ Θ ) isa closed subvariety of U ( τ uλ ) isomorphic to C | Θ | + | Θ | = C (cid:96) ( w J ) . Observe that U ( τ uλ Θ ) =˙ τ uλ U ( J )2 ˙ τ − uλ , hence ¯ ϕ ‡ u essentially coincides with the restriction of the map (7.23) to U (Θ ) ×U ( τ uλ Θ ) × { } . We have thus shown that ¯ ϕ ‡ u gives a biregular isomorphism between U ( J )1 × U ( J )2 and a closed (cid:96) ( w J )-dimensional subvariety of U ( τ uλ ). Therefore ¯ ϕ u gives a biregularisomorphism between C ( J ) u and a closed (cid:96) ( w J )-dimensional subvariety ¯ ϕ u ( C ( J ) u ) of ◦ X τ uλ . ByProposition A.2, X τ Jλ ∩ ◦ X τ uλ is a closed irreducible subvariety of ◦ X τ uλ , and by (7.8) andProposition A.2, it has dimension (cid:96) ( w J ). Since ¯ ϕ u ( C ( J ) u ) ⊂ X τ Jλ ∩ ◦ X τ uλ , it follows that¯ ϕ u ( C ( J ) u ) = X τ Jλ ∩ ◦ X τ uλ . We are done with the proof of Theorem 7.2(1). Remark 7.7. Alternatively, the proof of Theorem 7.2(1) could be deduced from Deodhar-type parametrizations [Had84, Had85, BD94] of ◦ R τ uλ vτ λ w − , by observing that any reduced wordfor τ uλ that is compatible with the length-additive factorization τ uλ = u · τ Jλ · w J u − in (7.8)contains a unique reduced subword for τ Jλ .7.7. Proof of Theorem 7.2(2). We use the notation and results from Section 6. Let x ∈ G be such that xP ∈ Π > v (cid:48) ,w (cid:48) . Since Π > v (cid:48) ,w (cid:48) = π J ( R > v (cid:48) ,w (cid:48) ), we may assume that xB ∈ R > v (cid:48) ,w (cid:48) . Then x ∈ ˙ uG ( J )0 by Lemma 6.9(ii), so ¯ ϕ u ( xP ) is defined. In addition, by Lemma 4.22(i) we mayassume that x ∈ ˙ uP − . By definition, ¯ ϕ u ( xP ) ∈ C vτ λ w − if and only if ˙ w ˙ τ − λ ˙ v − ¯ ϕ u ( xP ) ∈B − B / B . By (7.17), this is equivalent to(7.24) ˙ w ˙ τ − λ ˙ v − · κ x x · ˙ τ λ · x − ∈ B − B . By Theorem 6.4, x ∈ G ( J ) u,v , so ˙ v − κ x x ∈ G ( J )0 . Let us factorize y := ˙ v − κ x x as y =[ y ] ( J ) − [ y ] J [ y ] ( J )+ using Lemma 4.22(iii). By (7.13) and (7.15), we get˙ w ˙ τ − λ ˙ v − · κ x x · ˙ τ λ · x − = ˙ w · ˙ τ − λ [ y ] ( J ) − ˙ τ λ · ˙ τ − λ [ y ] J ˙ τ λ · ˙ τ − λ [ y ] ( J )+ ˙ τ λ · x − ∈ ˙ w · U ( I ) − · [ y ] J · U ( I ) · x − . Using (7.6), we can move U ( I ) − to the left and U ( I ) to the right, so we see that (7.24) isequivalent to ˙ w [ y ] J x − ∈ B − B . By Definition 6.1, we have [ y ] J = η ( x ), and by Lemma 6.3(ii),we have ζ ( J ) u,v ( x ) = xη ( x ) − = x [ y ] − J . By Theorem 6.4, ζ ( J ) u,v ( x ) ∈ BB − ˙ w , and after takinginverses, we obtain ˙ w [ y ] J x − ∈ B − B ⊂ B − B , finishing the proof. (cid:3) From Bruhat atlas to Fomin–Shapiro atlas We use Theorem 7.2 to prove Theorem 2.5.8.1. Affine Bruhat projections. We first define the affine flag variety version of the map¯ ν g from (2.1). We will need some results on Gaussian decomposition inside G , see Appen-dix A.5 for a proof. Lemma 8.1. Let G := B − · B . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 41 (i) The multiplication map gives a biregular isomorphism of ind-varieties: (8.1) U − × T × U ∼ −→ G . For x ∈ G , we denote by [ x ] − ∈ U − , [ x ] ∈ T , and [ x ] + ∈ U the unique elements suchthat x = [ x ] − [ x ] [ x ] + . (ii) For g ∈ ˜ W , the multiplication map gives biregular isomorphisms of ind-varieties: (8.2) µ : U ( g ) × U ( g ) ∼ −→ ˙ g U − ˙ g − , µ : U ( g ) × U ( g ) ∼ −→ ˙ g U − ˙ g − . The group ˙ g U − ˙ g − , as well as its subgroups U ( g ) and U ( g ), act on C g . The following result,which we state for the polynomial loop group G , holds in Kac–Moody generality. Proposition 8.2. Let g ∈ ˜ W . (i) For x ∈ G such that x B ∈ C g , there exist unique elements y ∈ U ( g ) and y ∈ U ( g ) such that y x B ∈ ◦ X g and y x B ∈ ◦ X g . (ii) The map ˜ ν g : C g ∼ −→ ◦ X g × ◦ X g sending x B (cid:55)→ ( y x B , y x B ) is a biregular isomorphismof ind-varieties. (iii) For all h, f ∈ ˜ W satisfying h ≤ g ≤ f , the map ˜ ν g restricts to a biregular isomorphism C g ∩ ◦ R fh ∼ −→ ◦ R fg × ◦ R gh of finite-dimensional varieties.Proof. Let us first prove an affine analog of Lemma 4.2. Let ν : ˙ g U − ˙ g − → U ( g ), ν :˙ g U − ˙ g − → U ( g ) denote the second component of µ − and µ − (cf. (8.2)), respectively, andlet ν := ( ν , ν ) : ˙ g U − ˙ g − → U ( g ) × U ( g ). We claim that ν is a biregular isomorphism.By Lemma 8.1(ii), ν is a regular morphism. Let us now compute the inverse of ν . Given x ∈ U ( g ) and x ∈ U ( g ), we claim that there exist unique y ∈ U ( g ) and y ∈ U ( g ) suchthat y x = y x . Indeed, this equation is equivalent to y − y = x x − , so we must have y = [ x x − ] − − and y = [ x x − ] + . Clearly, ν − ( x , x ) = y x = y x , and by Lemma 8.1(i),the map ν − is regular. Applying (7.5) finishes the proof of (i) and (ii).We now prove (iii). Observe that if x B ∈ C g ∩ ◦ R fh for some h ≤ f ∈ ˜ W then x ∈B − ˙ h B ∩ B ˙ f B . Let y , y be as in (ii). Then y ∈ U ( g ) ⊂ U , so y x ∈ B ˙ f B . Similarly, y ∈ U ( g ) ⊂ U − , so y x ∈ B − ˙ h B . It follows that if x B ∈ C g ∩ ◦ R fh then ˜ ν g ( x B ) ∈ ◦ R gh × ◦ R fg .In particular, we must have h ≤ g ≤ f , and we are done by (7.3). (cid:3) Torus action. Recall that T = C ∗ × T is the affine torus. The group C ∗ acts on G via loop rotation as follows. For t ∈ C ∗ , we have t · g ( z ) = g ( tz ). We form the semidirectproduct C ∗ (cid:110) G with multiplication given by ( t , x ( z )) · ( t , x ( z )) := ( t t , x ( z ) x ( t z )),for ( t , x ( z )) , ( t , x ( z )) ∈ C ∗ × G . Let Y ( T ) := Hom( C ∗ , T ) ∼ = Z d ⊕ Y ( T ). For λ ∈ Y ( T ), t ∈ C ∗ , t (cid:48) ∈ C , and α ∈ ∆ re , we have(8.3) λ ( t ) x α ( t (cid:48) ) λ ( t ) − = x α ( t (cid:104) λ,α (cid:105) t (cid:48) ) , where x α : C ∼ −→ U α is as in Section 7.1, and (cid:104)· , ·(cid:105) : Y ( T ) × X ( T ) → Z extends the pairingfrom Section 4.1 in such a way that (cid:104) d, δ (cid:105) = 1 and (cid:104) d, α i (cid:105) = (cid:104) α ∨ i , δ (cid:105) = 0 for i ∈ I .Let g ∈ ˜ W and denote N := (cid:96) ( g ). If Inv( g ) = { α (1) , . . . , α ( N ) } , then by Lemma A.1, themap x g : C N → U ( g ) given by(8.4) x g ( t , . . . , t N ) := x α (1) ( t ) · · · x α ( N ) ( t N ) is a biregular isomorphism. For t = ( t , . . . , t N ) ∈ C N , define (cid:107) t (cid:107) := ( | t | + · · · + | t N | ) ∈ R ≥ , and let (cid:107) · (cid:107) : U ( g ) → R ≥ be defined by (cid:107) y (cid:107) := (cid:107) x − g ( y ) (cid:107) . Identifying U ( g ) with ◦ X g via (7.5), we get a function (cid:107) · (cid:107) : ◦ X g → R ≥ .We say that ˜ ρ ∈ Y ( T ) is a regular dominant integral coweight if (cid:104) ˜ ρ, δ (cid:105) ∈ Z > and (cid:104) ˜ ρ, α i (cid:105) ∈ Z > for all i ∈ I . In this case, we have (cid:104) ˜ ρ, α (cid:105) ∈ Z > for any α ∈ ∆ +re . Let us choose such acoweight ˜ ρ , and define ϑ g : R > × G / B → G / B by ϑ g ( t, x B ) := ˜ ρ ( t ) x B .It follows from (8.3) that if g ∈ ˜ W and y ∈ U ( g ) is such that x − g ( y ) = ( t , . . . , t N ) thenthere exist k , . . . , k N ∈ Z > satisfying(8.5) (cid:107) ϑ g ( t, y ˙ g B ) (cid:107) = (cid:0) t k | t | + · · · + t k N | t N | (cid:1) for all t ∈ R > .8.3. Proof of Theorem 2.5. By Corollary 4.20, (( G/P ) R , ( G/P ) ≥ , Q J ) is a TNN spacein the sense of Definition 2.1. Thus it suffices to construct a Fomin–Shapiro atlas.Let ( u, u ) (cid:22) ( v, w ) ∈ Q J , and denote f := ( u, u ), g := ( v, w ). Thus we have ψ ( f ) = τ uλ and ψ ( g ) = vτ λ w − . Moreover, for the maximal element ˆ1 = (id , w J ) ∈ Q J , we have ψ (ˆ1) = τ Jλ . By Theorem 7.2(1), the map ¯ ϕ u gives an isomorphism C ( J ) u ∼ −→ X ψ (ˆ1) ∩ ◦ X ψ ( f ) .Let O C g ⊂ C ( J ) u be the preimage of C ψ ( g ) ∩ X ψ (ˆ1) ∩ ◦ X ψ ( f ) under ¯ ϕ u , and denote by O g := O C g ∩ ( G/P ) R . Since C ψ ( g ) is open in G / B , we see that O C g is open in C ( J ) u which is open in G/P , so O g is an open subset of ( G/P ) R . By Theorem 7.2(2), O g contains Star ≥ g , whichshows (FS5). Moreover, we claim that O g ⊂ Star g . Indeed, if h (cid:23) f but h (cid:54)(cid:23) g then ψ ( h ) (cid:54)≤ ψ ( g ). The map ¯ ϕ u sends ◦ Π h ∩ C ( J ) u to ◦ R ψ ( f ) ψ ( h ) , which does not intersect C ψ ( g ) by (A.3).We now define the smooth cone ( Z g , ϑ g ). Throughout, we identify ◦ X ψ ( g ) with C N g for N g := (cid:96) ( ψ ( g )) via (8.4). We set Z C g := X ψ (ˆ1) ∩ ◦ X ψ ( g ) and ◦ Z C g,h := ◦ R ψ ( g ) ψ ( h ) for g (cid:22) h ∈ Q J .We let Z g := Z C g ∩ R N g and ◦ Z g,h := ◦ Z C g,h ∩ R N g denote the corresponding sets of real points.Thus (FS1) follows. The action ϑ g restricts to R N g , and by (8.5), it satisfies (SC2). Aswe discussed in Section 8.2, the action of ϑ g also preserves Z g (showing (SC1)) and ◦ Z g,h (showing (FS2)).Finally, we define a map ¯ ν g : O C g → ( ◦ Π g ∩ O C g ) × C N g as follows. Let ˜ ν g = (˜ ν g, , ˜ ν g, ) : C g ∼ −→ ◦ X g × ◦ X g be the map from Proposition 8.2. We let ¯ ν g, := ˜ ν g, ◦ ¯ ϕ u , so it sends O C g → C ψ ( g ) → ◦ X ψ ( g ) ∼ = C N g . By Proposition 8.2(iii), the image of ¯ ν g, is precisely Z C g . Wealso let ¯ ν g, := ¯ ϕ − u ◦ ˜ ν g, ◦ ¯ ϕ u , thus it sends O C g ∼ −→ C ψ ( g ) ∩ X ψ (ˆ1) ∩ ◦ X ψ ( f ) → ◦ R ψ ( f ) ψ ( g ) ∼ −→ ◦ Π g ∩ O C g . It follows from Theorem 7.2(1) and Proposition 8.2 that ¯ ν g := (¯ ν g, , ¯ ν g, ) gives a biregularisomorphism O C g ∼ −→ ( ◦ Π g ∩ O C g ) × Z C g . All maps in the definition of Z C g are defined over R ,thus ¯ ν g gives a smooth embedding O g → ( ◦ Π R g ∩ O g ) × R N g with image ( ◦ Π R g ∩ O g ) × Z g . ByLemma 3.3, we find that Z g is an embedded submanifold of R N g , so we get a diffeomorphism¯ ν g : O g ∼ −→ ( ◦ Π R g ∩ O g ) × Z g . By Theorem 7.2(1) and Proposition 8.2(iii), we find that for h (cid:23) g , ¯ ν g sends ◦ Π h ∩ O g to( ◦ Π g ∩ O g ) × ◦ Z g,h , showing (FS3). When xP ∈ ◦ Π g ∩ O g , we have ¯ ϕ u ( xP ) ∈ ◦ R ψ ( f ) ψ ( g ) , so EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 43 ˜ ν g, ( ¯ ϕ u ( xP )) = ¯ ϕ u ( xP ) and ˜ ν g, ( ¯ ϕ u ( xP )) ∈ ◦ R ψ ( g ) ψ ( g ) . Thus ¯ ν g, ( xP ) = x and ¯ ν g, ( xP ) = 0,showing (FS4). We have checked all the requirements of Definitions 2.1, 2.2, and 2.3. (cid:3) The case G = SL n In this section, we illustrate our construction in type A . We mostly focus on the case when G/P is the Grassmannian Gr( k, n ) so that ( G/P ) ≥ is the totally nonnegative Grassmannian Gr ≥ ( k, n ) of Postnikov [Pos07]. Throughout, we assume K = C .9.1. Preliminaries. Fix an integer n ≥ n ] := { , , . . . , n } . For 0 ≤ k ≤ n ,let (cid:0) [ n ] k (cid:1) denote the set of all k -element subsets of [ n ].Let G = SL n be the group of n × n matrices over C of determinant 1. We have subgroups B, B − , T, U, U − ⊂ G consisting of upper triangular, lower triangular, diagonal, upper unitri-angular, and lower unitriangular matrices of determinant 1, respectively. The Weyl group W is the group S n of permutations of [ n ], and for i ∈ I = [ n − s i ∈ W is the simpletransposition of elements i and i + 1. If w ∈ W is written as a product w = s i . . . s i l , thenthe action of w on [ n ] is given by w ( j ) = s i ( . . . ( s i l ( j )) . . . ) for j ∈ [ n ]. For S ⊂ [ n ], weset wS := { w ( j ) | j ∈ S } . For example, if n = 3 and w = s s then w (1) = 3, w (2) = 1, w (3) = 2, and w { , } = { , } .For i ∈ [ n − φ i : SL → G just sends a matrix A ∈ SL to the n × n matrix φ i ( A ) ∈ SL n which has a 2 × A in rows and columns i, i + 1. Thusif n = 3 then ˙ s = (cid:104) − (cid:105) , ˙ s = (cid:104) − 10 1 0 (cid:105) , and if w = s s then ˙ w = (cid:104) − − 11 0 0 (cid:105) . In general,given w ∈ S n , ˙ w contains a ± w ( j ) and column j for each j ∈ [ n ], and the sign ofthis entry is − ± w ( j ) , j )-th entry of ˙ w equals ( − { i The group B acts on G = SL n by right multiplication, and G/B is the complete flag variety in C n . It consists of flags { } = V ⊂ V ⊂ · · · ⊂ V n = C n in C n suchthat dim V i = i for i ∈ [ n ]. For a matrix x ∈ SL n , the element xB ∈ G/B gives rise to a flag V ⊂ V ⊂ · · · ⊂ V n such that V i is the span of columns 1 , . . . , i of x . For k ∈ [ n ], S ∈ (cid:0) [ n ] k (cid:1) ,and x ∈ SL n , we denote by ∆ flag S the determinant of the k × k submatrix of x with row set S and column set [ k ]. Thus for each k ∈ [ n ], we have a map ∆ flag k : G/B → CP ( nk ) − sending xB to (∆ flag S ( x )) S ∈ ( [ n ] k ). Here (cid:0) [ n ] k (cid:1) is identified with the set W ω k from Lemma 4.21(iv).9.3. Partial flag variety. For J ⊂ [ n ], we have a parabolic subgroup P ⊂ G , and thepartial flag variety G/P consists of partial flags { } = V ⊂ V j ⊂ · · · ⊂ V j l ⊂ V n = C n ,where { j < · · · < j l } := [ n − \ J and dim V j i = j i for i ∈ [ l ]. The projection π J : G/B → G/P sends a flag ( V , V , . . . , V n ) to ( V , V j , . . . , V j l , V n ). When J = ∅ , we have P = B and G/P = G/B . We will focus on the “opposite” special case:Unless otherwise stated, we assume that J = [ n − \ { k } for some fixed k ∈ [ n − In this case, G/P is the (complex) Grassmannian Gr( k, n ), which we will identify with thespace of n × k full rank matrices modulo column operations. Let us write matrices in SL n in block form (cid:20) A BC D (cid:21) , where A is of size k × k and D is of size ( n − k ) × ( n − k ). For amatrix x = (cid:20) A BC D (cid:21) ∈ SL n , we denote by [ x | := (cid:20) AC (cid:21) the n × k submatrix consisting of thefirst k columns of x . Thus every x ∈ SL n gives rise to an element xP of G/P which is a k -dimensional subspace V k ⊂ C n equal to the column span of [ x | . The map ∆ flag k in this caseis the classical Pl¨ucker embedding ∆ flag k : Gr( k, n ) (cid:44) → CP ( nk ) − .The set W J from Section 4.6 consists of Grassmannian permutations : we have w ∈ W J if and only if w = id or every reduced word for w ends with s k . Equivalently, w ∈ W J ifand only if w (1) < · · · < w ( k ) and w ( k + 1) < · · · < w ( n ), so the map w (cid:55)→ w [ k ] givesa bijection W J → (cid:0) [ n ] k (cid:1) . The parabolic subgroup W J (generated by { s j } j ∈ J ) consists ofpermutations w ∈ S n such that w [ k ] = [ k ], and the longest element w J ∈ W J is given by( w J (1) , . . . , w J ( n )) = ( k, . . . , , n, . . . , k + 1). The maximal element w J of W J is given by( w J (1) , . . . , w J ( n )) = ( n − k + 1 , . . . , n, , . . . , n − k ). We have U J = (cid:110)(cid:104) U k U n − k (cid:105)(cid:111) , U ( J ) − = (cid:110)(cid:104) I k C I n − k (cid:105)(cid:111) , L J = (cid:110)(cid:104) A D (cid:105)(cid:111) , P = (cid:110)(cid:104) A B D (cid:105)(cid:111) , where U r is an r × r upper unitriangular matrix, I r is an r × r identity matrix, A ∈ SL k , D ∈ SL n − k , and B , C are arbitrary k × ( n − k ) and ( n − k ) × k matrices, respectively.9.4. Affine charts. We have G ( J )0 := { x ∈ G | ∆ flag[ k ] ( x ) (cid:54) = 0 } , and for x = (cid:20) A BC D (cid:21) ∈ G ( J )0 (such that det A = ∆ flag[ k ] ( x ) (cid:54) = 0), the factorization x = [ x ] ( J ) − [ x ] ( J )0 [ x ] ( J )+ from Lemma 4.22(iii)is given by(9.1) (cid:104) A BC D (cid:105) = (cid:104) I k CA − I n − k (cid:105) · (cid:104) A D − CA − B (cid:105) · (cid:104) I k A − B I n − k (cid:105) . The matrix D − CA − B is called the Schur complement of A in x .For u ∈ W J , the set C ( J ) u ⊂ G/P from Section 4.12 consists of elements xP such that∆ flag u [ k ] ( x ) (cid:54) = 0. The (inverse of the) isomorphism (4.31) essentially amounts to computing thereduced column echelon form of an n × k matrix: if x ∈ G is such that xP ∈ C ( J ) u is sent to g ( J ) ∈ ˙ uU ( J ) − ˙ u − via (4.31), then the n × k matrices [ x | and (cid:2) g ( J ) ˙ u (cid:12)(cid:12) have the same columnspan, and the submatrix of (cid:2) g ( J ) ˙ u (cid:12)(cid:12) with row set u [ k ] is the k × k identity matrix. Let ussay that an n × k matrix M is in u [ k ] -echelon form if its submatrix with row set u [ k ] is the k × k identity matrix.The matrices g ( J )1 ˙ u and g ( J )2 ˙ u from Definition 4.23 are obtained from g ( J ) ˙ u simply byreplacing some entries with 0. Explicitly, let ( M i,j ) := (cid:2) g ( J ) ˙ u (cid:12)(cid:12) , ( M (cid:48) i,j ) := (cid:104) g ( J )1 ˙ u (cid:12)(cid:12)(cid:12) , and( M (cid:48)(cid:48) i,j ) := (cid:104) g ( J )2 ˙ u (cid:12)(cid:12)(cid:12) be the corresponding n × k matrices. Thus M i,j = δ i,u ( j ) for all i ∈ u [ k ]and j ∈ [ k ], and we have M (cid:48) i,j = (cid:40) M i,j , if i ≤ u ( j ),0 , otherwise, M (cid:48)(cid:48) i,j = (cid:40) M i,j , if i ≥ u ( j ),0 , otherwise, for all i ∈ [ n ] and j ∈ [ k ].The operation M (cid:55)→ M (cid:48) , which we call u -truncation , will play an important role. EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 45 Example 9.1. Let G/P = Gr(2 , 4) and u = s s ∈ W J , so u [ k ] = { , } . We have x = g ( J ) ˙ u = (cid:20) x x − x x − (cid:21) , [ x | = (cid:20) x x x x (cid:21) , (cid:104) g ( J )1 ˙ u (cid:12)(cid:12)(cid:12) = (cid:20) x x (cid:21) , (cid:104) g ( J )2 ˙ u (cid:12)(cid:12)(cid:12) = (cid:20) x x (cid:21) , where blank entries correspond to zeros.9.5. Positroid varieties. We review the background on positroid varieties inside Gr( k, n ),which were introduced in [KLS13], building on the work of Postnikov [Pos07]. Let ˜ S n bethe group of affine permutations , i.e., bijections f : Z → Z such that f ( i + n ) = f ( i ) + n for all i ∈ Z . We have a function av : ˜ S n → Z sending f to av( f ) := n (cid:80) ni =1 ( f ( i ) − i ),which is an integer for all f ∈ ˜ S n . For j ∈ Z , denote ˜ S j,n := { f ∈ ˜ S n | av( f ) = j } . Every f ∈ ˜ S n is determined by the sequence f (1) , . . . , f ( n ), and we write f in window notation as f = [ f (1) , . . . , f ( n )]. For λ ∈ Z n , define τ λ ∈ ˜ S n by τ λ := [ d , . . . , d n ], where d i = i + nλ i for all i ∈ [ n ]. Let Bound( k, n ) ⊂ ˜ S k,n be the set of bounded affine permutations , whichconsists of all f ∈ ˜ S n satisfying av( f ) = k and i ≤ f ( i ) ≤ i + n for all i ∈ Z . The subset˜ S ,n is a Coxeter group with generators s , . . . , s n − , s n = s , where for i ∈ [ n ], s i : Z → Z sends i (cid:55)→ i + 1, i + 1 (cid:55)→ i , and j (cid:55)→ j for all j (cid:54)≡ i, i + 1 (mod n ). We let ≤ denote theBruhat order on ˜ S ,n , and (cid:96) : ˜ S ,n → Z ≥ denote the length function. We have a bijection˜ S ,n → ˜ S k,n sending ( i (cid:55)→ f ( i )) to ( i (cid:55)→ f ( i ) + k ), which induces a poset structure and alength function on ˜ S k,n . When f ≤ g , we write g ≤ op f , and we will be interested in theposet (Bound( k, n ) , ≤ op ), which has a unique maximal element τ k := [1 + k, k, . . . , n + k ].It is known that Bound( k, n ) is a lower order ideal of ( ˜ S k,n , ≤ op ). We fix λ = 1 k n − k :=(1 , . . . , , , . . . , ∈ Z n (with k τ λ = [1 + n, . . . , k + n, k + 1 , . . . , n ] is one of the (cid:0) [ n ] k (cid:1) minimal elements of (Bound( k, n ) , ≤ op ). The group S n is naturally a subset of ˜ S ,n , andwe have τ k = τ λ ( w J ) − = τ Jλ , where τ Jλ was introduced in Section 7.2.Given an n × k matrix M and i ∈ [ n ], we let M i denote the i th row of M . We extendthis to all i ∈ Z in such a way that M i + n = ( − k − M i for all i ∈ Z . Thus we view M as a periodic Z × k matrix. (The sign ( − k − is chosen so that if M ∈ Gr ≥ ( k, n ), thenthe matrix with rows M i , . . . , M i + n − belongs to Gr ≥ ( k, n ) for all i ∈ Z , see Section 9.11.)Every n × k matrix M of rank k gives rise to a map f M : Z → Z sending i ∈ Z to theminimal j ≥ i such that M i belongs to the linear span of M i +1 , . . . , M j . It is easy to see that f M ∈ Bound( k, n ) and f M depends only on the column span of M . For h ∈ Bound( k, n ),the (open) positroid variety ◦ Π h ⊂ Gr( k, n ) is the subset ◦ Π h := { M ∈ Gr( k, n ) | f M = h } .Its Zariski closure inside Gr( k, n ) is Π h = (cid:70) g ≤ op h ◦ Π g , see [KLS13, Theorem 5.10].For h ∈ Bound( k, n ), define the Grassmann necklace I h = ( I a ) a ∈ Z of h by(9.2) I a := { h ( i ) | i < a, h ( i ) ≥ a } for a ∈ Z .Then I a is a k -element subset of [ a, a + n ), where for a ≤ b ∈ Z we set [ a, b ) := { a, a +1 , . . . , b − } . For a ≤ b ∈ Z and M ∈ Gr( k, n ), define rank( M ; a, b ) to be the rank of the submatrixof M with row set [ a, b ). For a, b ∈ Z and h ∈ ˜ S n , define r a,b ( h ) := { i < a | h ( i ) ≥ b } . Wedescribe two well known characterizations of open positroid varieties, see [KLS13, § Proposition 9.2. Let h ∈ Bound( k, n ) and let I h = ( I a ) a ∈ Z be its Grassmann necklace. (i) The set ◦ Π h consists of all M ∈ Gr( k, n ) such that for each a ∈ Z , I a is the lexicograph-ically minimal k -element subset S of [ a, a + n ) such that the rows ( M i ) i ∈ S are linearlyindependent. (ii) For M ∈ Gr( k, n ) , we have M ∈ ◦ Π h if and only if (9.3) k − rank( M ; a, b ) = r a,b ( h ) for all a ≤ b ∈ Z . We use window notation for Grassmann necklaces as well, i.e., we write I h = [ I , . . . , I n ].Recall that we have fixed λ = 1 k n − k ∈ Z n . For ( v, w ) ∈ Q J , define f v,w ∈ ˜ S n by(9.4) f v,w := vτ λ w − . Theorem 9.3 ([KLS13, Propositions 3.15 and 5.4]) . The map ( v, w ) (cid:55)→ f v,w gives a posetisomorphism ( Q J , (cid:22) ) ∼ −→ (Bound( k, n ) , ≤ op ) . For ( v, w ) ∈ Q J , we have ◦ Π v,w = ◦ Π f v,w and Π v,w = Π f v,w as subsets of G/P = Gr( k, n ) . Example 9.4. There are n positroid varieties of codimension 1, each given by the condition∆ flag { i − k +1 ,...,i } = 0 for some i ∈ [ n ]. Indeed, the top element (id , w J ) ∈ Q J covers n elements,namely ( s i , w J ) for i ∈ [ n − , s n − k w J ). In the former case we have f s i ,w J = s i τ Jλ , which corresponds to the variety ∆ flag { i − k +1 ,...,i } = 0. In the latter case we have f id ,s n − k w J = τ Jλ s n − k , which corresponds to the variety ∆ flag { n − k +1 ,...,n } = 0. Example 9.5. One can check directly from (9.4) and (9.2) that the first element of theGrassmann necklace of f v,w is I = v [ k ]. Similarly, w [ k ] = { i ∈ [ n ] | f v,w ( i ) > n } . Example 9.6. Elements of Bound( k, n ) and Q J are in bijection with L -diagrams of [Pos07].The bijection between Q J and the set of L-diagrams is described in [Pos07, § v, w ) ∈ Q J gives rise to a L-diagram whose shape is a Young diagram inside a k × ( n − k )rectangle, corresponding to the set w [ k ]. The squares of the L-diagram correspond to theterms in a reduced expression for w , as shown in Figure 2 (top left): the box with coordinates( i, j ) in matrix notation is labeled by s k + j − i , and we form the expression by reading boxesfrom right to left, bottom to top. The terms in the positive subexpression for v inside w correspond to the squares of the L-diagram that are not filled with dots, see Figure 2 (bottomleft). Thus the bijection of Theorem 9.3 can be pictorially represented as in Figure 2 (right).We refer to [Pos07, § 19] or [Wil07, Appendix A] for the precise description. For the examplein Figure 2, we have v = s , w = s s s s s , and f v,w = [3 , , , , 6] in window notation,which is obtained by following the strands in Figure 2 (right) from top to bottom.9.6. Polynomial loop group. We explain how the construction in Section 7 applies to thecase G/P = Gr( k, n ). Recall that A := C [ z, z − ]. Let GL n ( A ) denote the polynomial loopgroup of GL n , consisting of n × n matrices with entries in A whose determinant is a nonzeroLaurent monomial in z , i.e., an invertible element of A . (We use GL n ( A ) instead of SL n ( A )as the constructions are combinatorially more elegant.) We have a group homomorphismval : GL n ( A ) → Z sending x ∈ GL n ( A ) to j ∈ Z such that det x = cz − j for some c ∈ C ∗ ,and we let GL ( j ) n ( A ) := { x ∈ GL n ( A ) | val x = j } . The subgroups GL n ( A + ) and GL n ( A − )are contained inside the group GL (0) n ( A ) of matrices whose determinant belongs to C ∗ .We have subgroups U ( A + ) := ¯ev − ( U ), U − ( A − ) := ¯ev − ∞ ( U − ), B ( A + ) := ¯ev − ( B ) and B − ( A − ) := ¯ev − ∞ ( B − ) of GL (0) n ( A ). Thus in the notation of Section 7 for G = SL n , we have G = SL n ( A ) (cid:40) GL (0) n ( A ), B = SL n ( A ) ∩ B ( A + ) (cid:40) B ( A + ), U = U ( A + ), and U − = U − ( A − ).To each matrix x ∈ GL n ( A ), we associate a Z × Z matrix ˜ x = (˜ x i,j ) i,j ∈ Z that is uniquelydefined by the conditions EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 47 s s s s s 11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 − − − − − − − − w − τ λ v Figure 2. A L-diagram (bottom left), the labeling of its squares by simpletranspositions (top left), and the result of applying the bijection of Theo-rem 9.3 (right). See Example 9.6 for details.(1) ˜ x i,j = ˜ x i + n,j + n for all i, j ∈ Z , and(2) the entry x i,j ( z ) equals the finite sum (cid:80) d ∈ Z ˜ x i,j + dn z d for all i, j ∈ [ n ].One can check that if x = x x , then ˜ x = ˜ x ˜ x . With this identification, the subgroups U , U − , B ( A + ), and B − ( A − ) have a very natural meaning. For example, x ∈ GL n ( A ) belongsto U if and only if ˜ x i,j = 0 for i > j and ˜ x i,i = 1 for all i ∈ Z . Similarly, B ( A + ) consists ofall elements x ∈ GL n ( A ) such that ˜ x i,j = 0 for i > j and ˜ x i,i (cid:54) = 0 for all i ∈ Z .To each affine permutation f ∈ ˜ S k,n , we associate an element ˙ f ∈ GL n ( A ) so that thecorresponding Z × Z matrix ˜ f satisfies ˜ f i,j = 1 if i = f ( j ) and ˜ f i,j = 0 otherwise, for all i, j ∈ Z . In other words, if for i, j ∈ [ n ] there exists d ∈ Z such that f ( j ) = i + dn then ˙ f i,j ( z ) := z − d , and otherwise ˙ f i,j ( z ) := 0. Observe that val ˙ f = k for all f ∈ ˜ S k,n , thus ˙ f ∈ GL ( k ) n ( A ).Recall that we have fixed λ = 1 k n − k ∈ Z n . We obtain ˙ τ λ = diag (cid:0) z , . . . , z , , . . . , (cid:1) with k entries equal to z , and for u ∈ W J , we therefore get ˙ τ uλ = diag( c , . . . , c n ), where c i = z for i ∈ u [ k ] and c i = 1 for i / ∈ u [ k ].9.7. Affine flag variety. The quotient GL ( k ) n ( A ) /B ( A + ) is isomorphic to the affine flagvariety G / B of Section 7 for the case G = SL n . Indeed, GL (0) n ( A ) acts simply transitively onGL ( k ) n ( A ) and we clearly have GL (0) n ( A ) /B ( A + ) ∼ = G / B . For f ≤ op h ∈ ˜ S k,n and g ∈ ˜ S k,n ,we have subsets ◦ X f , ◦ X h , ◦ R fh , C g ⊂ GL ( k ) n ( A ) /B ( A + ) defined by ◦ X f := B ( A + ) · ˙ f · B ( A + ) /B ( A + ) , ◦ X h := B − ( A − ) · ˙ h · B ( A + ) /B ( A + ) , ◦ R fh := ◦ X h ∩ ◦ X f , C g := ˙ g · B − ( A − ) · B ( A + ) /B ( A + ) . Let us now calculate the map ϕ u from (7.11). Recall that it sends xP ∈ C ( J ) u to g ( J )1 · ˙ τ uλ · ( g ( J )2 ) − . Assuming as before that x = g ( J ) ˙ u ∈ ˙ uU ( J ) − , consider the corresponding n × k matrix ( M i,j ) := [ x | in u [ k ]-echelon form. Proposition 9.7. The matrix y := ϕ u ( xP ) ∈ GL ( k ) n ( A ) is given for all i, j ∈ [ n ] by (9.5) y i,j ( z ) = δ i,j , if j / ∈ u [ k ] , − M i,s , if i > j and j = u ( s ) for some s ∈ [ k ] , M i,s z , if i ≤ j and j = u ( s ) for some s ∈ [ k ] .Proof. This follows by directly computing the product g ( J )1 · ˙ τ uλ · ( g ( J )2 ) − . (cid:3) Example 9.8. In the notation of Example 9.1, we have(9.6) y = g ( J )1 · ˙ τ uλ · ( g ( J )2 ) − = (cid:20) x x (cid:21) · (cid:34) z z (cid:35) · (cid:20) − x − x (cid:21) = z − x x z − x x z z . Remark 9.9. The map ¯ ϕ u : xP (cid:55)→ g ( J )1 · ˙ τ uλ · ( g ( J )2 ) − · B ( A + ) is a slight variation of asimilar embedding of [Sni10] which we denote ¯ ϕ (cid:48) u . We have ¯ ϕ (cid:48) u ( xP ) = g ( J )1 · ˙ τ uλ · g ( J )2 · B ( A + ),and the corresponding matrix y (cid:48) = ϕ (cid:48) u ( xP ) := g ( J )1 · ˙ τ uλ · g ( J )2 is given by (9.5) except that − M i,s should be replaced by M i,s . Thus y (cid:48) is obtained from y by substituting z (cid:55)→ − z andthen changing the signs of all columns in u [ k ]. In particular, y (cid:48) and y are related by anelement of the affine torus from Section 8.2. Proposition 9.14 below is due to Snider [Sni10].Theorem 7.2(1) generalizes Snider’s result to arbitrary G/P .We give a standard convenient characterization of ◦ X h using lattices . For each x ∈ GL n ( A )and column a ∈ Z , we introduce a Laurent polynomial x a ( t ) ∈ C [ t, t − ] defined by x a ( t ) := (cid:80) i ∈ Z ˜ x i,a t i , and an infinite-dimensional linear subspace L a ( x ) ⊂ C [ t, t − ] given by L a ( x ) :=Span { x j ( t ) | j < a } , where Span denotes the space of all finite linear combinations. For b ∈ Z , define another linear subspace E b ⊂ C [ t, t − ] by E b := Span { t i | i ≥ b } . Finally,for a, b ∈ Z , define r a,b ( x ) ∈ Z to be the dimension of L a ( x ) ∩ E b . In other words, r a,b ( x )is the dimension of the space of Z × b − , b − , . . . andcan be obtained as finite linear combinations of columns a − , a − , . . . of ˜ x . Recall fromSection 9.5 that for a, b ∈ Z and h ∈ ˜ S n , we denote r a,b ( h ) := { i < a | h ( i ) ≥ b } . Lemma 9.10. Let x ∈ GL ( d ) n ( A ) and h ∈ ˜ S d,n for some d ∈ Z . Then (9.7) x · B ( A + ) ∈ ◦ X h if and only if r a,b ( x ) = r a,b ( h ) for all a, b ∈ Z .Proof. It is clear that r a,b ( x ) = r a,b ( h ) when x = ˙ h . One can check that r a,b ( y − xy + ) = r a,b ( x )for all x ∈ GL ( d ) n ( A ), y − ∈ B − ( A − ), y + ∈ B ( A + ), and a, b ∈ Z . This proves (9.7) sinceGL ( d ) n ( A ) /B ( A + ) = (cid:70) h ∈ ˜ S d,n ◦ X h by (A.2). (cid:3) Remark 9.11. A lattice L is usually defined (see e.g. [Kum02, § C [[ z ]]-submodule of C (( t )) ∼ = C (( z )) n (where z = t n ) satisfying L ⊗ C [[ z ]] C (( z )) ∼ = C (( z )) n . The C [[ z ]]-submodule generated by our L a ( x ) gives a lattice L a ( x ) in the usual sense. Definition 9.12. Suppose we are given an n × k matrix M in u [ k ]-echelon form. Recall thatwe have defined the row M a for all a ∈ Z in such a way that M a + n = ( − k − M a . For a ∈ Z and j ∈ [ k ], denote by θ ua,j ∈ [ a, a + n ) the unique integer that is equal to u ( j ) modulo n .Define the u -truncation M tr au of M to be the [ a, a + n ) × k matrix M tr au = ( M tr au i,j ) such thatfor i ∈ [ a, a + n ) and j ∈ [ k ], the entry M tr au i,j is equal to M i,j if i ≤ θ ua,j and to 0 otherwise, EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 49 see Example 9.18. Thus M tr au is obtained from the matrix with rows M a , . . . , M a + n − bysetting an entry to 0 if it is below the corresponding ± a, . . . , a + n − , . . . , n . For example, if x = g ( J ) ˙ u and M = [ x | then M tr u = (cid:104) g ( J )1 ˙ u (cid:12)(cid:12)(cid:12) , cf. Example 9.1. Lemma 9.13. Let x = g ( J ) ˙ u ∈ ˙ uU ( J ) − , M := [ x | , and y := ϕ u ( xP ) . Then for all a ∈ Z , thespace L a ( y ) has a basis (9.8) { t i | i < a } (cid:116) { P ( t ) , . . . , P k ( t ) } , where P s ( t ) := a + n − (cid:88) i = a M tr au i,s t i for s ∈ [ k ] . Proof. For a subset S ⊂ Z , define S + n Z := { j + in | j ∈ S, i ∈ Z } . The space L a ( y ) is thespan of y j ( t ) for all j < a . If j / ∈ u [ k ] + n Z then y j ( t ) = t j by definition. If j ∈ u [ k ] + n Z then y j − n ( t ) = t j + (cid:80) j − n
G/P = Gr( k, n ). Proposition 9.14. For h ∈ Bound( k, n ) such that τ uλ ≤ op h , the map ¯ ϕ u gives isomor-phisms ¯ ϕ u : C ( J ) u ∼ −→ ◦ X τ uλ , ¯ ϕ u : C ( J ) u ∩ ◦ Π h ∼ −→ ◦ R τ uλ h . Proof. It is clear from (9.5) that we have a biregular isomorphism U ( J )1 × U ( J )2 ∼ −→ U ( τ uλ )sending ( g ( J )1 , g ( J )2 ) (cid:55)→ g ( J )1 · ˙ τ uλ ( g ( J )2 ) − ˙ τ − uλ . Thus the map ( g ( J )1 , g ( J )2 ) (cid:55)→ g ( J )1 · ˙ τ uλ · ( g ( J )2 ) − · B ( A + ) gives a parametrization of ◦ X τ uλ , see (7.5). Since C ( J ) u = (cid:70) h ∈ Bound( k,n ) ( C ( J ) u ∩ ◦ Π h ),let us fix h ∈ Bound( k, n ) and x = g ( J ) ˙ u ∈ ˙ uU ( J ) − . Denote M := [ x | and y := ϕ u ( xP ).By (9.3), we have M ∈ ◦ Π h if and only if k − rank( M ; a, b ) = r a,b ( h ) for all a ≤ b ∈ Z .By (9.7), we have y · B ( A + ) ∈ ◦ X h if and only if r a,b ( y ) = r a,b ( h ) for all a, b ∈ Z . If a > b then r a,b ( y ) = r a,b +1 ( y ) + 1 by (9.8) and r a,b ( h ) = r a,b +1 ( h ) + 1 since h ∈ Bound( k, n ) satisfies h − ( b ) ≤ b , so h − ( b ) < a . We have shown that y · B ( A + ) ∈ ◦ X h if and only if r a,b ( y ) = r a,b ( h )for all a ≤ b ∈ Z . Thus it suffices to show(9.9) r a,b ( y ) + rank( M ; a, b ) = k for all a ≤ b ∈ Z .By (9.8), r a,b ( y ) is the dimension of Span { P ( t ) , . . . , P k ( t ) }∩ E b . By the rank-nullity theorem, k − r a,b ( y ) is the rank of the submatrix of M tr au with row set [ a, b ), which is obtained bydownward row operations from the submatrix of M with row set [ a, b ). This shows (9.9). (cid:3) Remark 9.15. By Theorem 7.2(1), the image of ¯ ϕ u is X τ Jλ ∩ ◦ X τ uλ , where τ Jλ = τ λ ( w J ) − .But recall from Section 9.5 that τ λ ( w J ) − = τ k , and since ◦ X τ k is dense in GL ( k ) n ( A ) /B ( A + ),we find that X τ Jλ ∩ ◦ X τ uλ = ◦ X τ uλ . Example 9.16. Suppose that x = g ( J ) ˙ u is given in Example 9.1, then y = ϕ u ( xP ) isthe matrix from Example 9.8. It is clear that y ∈ B ( A + ) · ˙ τ uλ regardless of the values of x , x , x , x , and therefore y · B ( A + ) belongs to ◦ X τ uλ . We can try to factorize y as anelement of B − ( A − ) · ˙ τ k · B ( A + ): y = x x x − x x z x x z x x x x x x − x x · (cid:34) z z (cid:35) · x x − x x x − x x x x x − x x − x x x − x x x − x z z x . This factorization makes sense only when all denominators on the right-hand side arenonzero, which shows that y · B ( A + ) ∈ ◦ R τ uλ τ k whenever the minors ∆ flag12 ( x ) = x , ∆ flag23 = x x − x x , and ∆ flag34 = x are nonzero. Observe also that ∆ flag14 ( x ) = 1. Thus y · B ( A + ) ∈ ◦ R τ uλ τ k precisely when xP ∈ ◦ Π τ k , where τ k = [3 , , , 6] in window notation. If x = 0 then xP ∈ ◦ Π h for h = [2 , , , h = (cid:34) z z (cid:35) , y | x =0 = (cid:34) − x z x x z − x x x x (cid:35) · (cid:34) z z (cid:35) · − x x x x − x x − x z z x . Therefore y | x =0 belongs to ◦ R τ uλ h whenever x , x , x (cid:54) = 0. Observe that the Grassmannnecklace of h is given by I h = [ { , } , { , } , { , } , { , } ] in window notation, and thecorresponding flag minors of x | x =0 are given by ∆ flag13 = x , ∆ flag23 = x x , ∆ flag34 = x ,∆ flag14 = 1, in agreement with Proposition 9.14.9.8. Preimage of C g . For this section, we fix τ uλ ≤ op g ∈ Bound( k, n ). We would like tounderstand the preimage of (cid:16) ◦ X τ uλ ∩ C g (cid:17) ⊂ GL ( k ) n ( A ) /B ( A + ) under the map ¯ ϕ u . For a set S ⊂ [ a, a + n ) of size k , define ∆ tr au S ( M ) to be the determinant of the k × k submatrix of M tr au with row set S . Let I g = ( I a ) a ∈ Z be the Grassmann necklace of g . Proposition 9.17. Suppose that xP ∈ C ( J ) u and let M := (cid:2) g ( J ) ˙ u (cid:12)(cid:12) . Then ¯ ϕ u ( xP ) ∈ C g ifand only if ∆ tr au I a ( M ) (cid:54) = 0 for all a ∈ [ n ] .Proof. Let h ∈ ˜ S n be the unique element such that ˙ g − ¯ ϕ u ( xP ) belongs to ◦ X h , thus ¯ ϕ u ( xP ) ∈C g if and only if h = id. Since val ϕ u ( xP ) = k and val ˙ g − = − k , we get h ∈ ˜ S ,n . Hence h = id if and only if r a,a ( h ) = 0 for all a ∈ Z . Let y := ϕ u ( xP ) and y (cid:48) := ˙ g − y . Thenfor a ∈ Z , we get L a ( y (cid:48) ) = g − L a ( y ), where g − acts on C [ t, t − ] by a linear map sending t j (cid:55)→ t g − ( j ) . In particular, L a ( y (cid:48) ) ∩ E a = ( g − L a ( y )) ∩ E a has the same dimension as L a ( y ) ∩ gE a . Let us denote H a := { t i | i ≥ a } , so E a = Span( H a ) and gE a = Span( gH a ).Since g ( i ) ≥ i for all i ∈ Z , it follows from (9.2) that gH a = H a \ { t j } j ∈ I a . Thereforeby (9.8), L a ( y ) ∩ gE a = { } if and only if Span { P j ( t ) } j ∈ [ k ] ∩ Span ( H a \ { t j } j ∈ I a ) = { } ,which happens precisely when the submatrix of M tr au with row set I a is nonsingular, i.e.,∆ tr au I a ( M ) (cid:54) = 0. (cid:3) Example 9.18. Suppose that x is the matrix from Example 9.1, then y := ϕ u ( xP ) is givenin Example 9.8. We have M = (cid:20) x x x x (cid:21) , M tr u = (cid:20) x x (cid:21) , M tr u = (cid:20) x x x x − (cid:21) , M tr u = (cid:20) x x − (cid:21) , M tr u = (cid:20) − (cid:21) . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 51 Suppose that g = [2 , , , 7] as in Example 9.16, then its Grassmann necklace is I g =[ { , } , { , } , { , } , { , } ] in window notation. This gives(9.10) ∆ tr u ( M ) = x , ∆ tr u ( M ) = x x − x x , ∆ tr u ( M ) = x , ∆ tr u ( M ) = 1 . On the other hand, recall from Example 9.16 that ˙ g = (cid:34) z z (cid:35) . Since y ∈ C g if and onlyif ˙ g − y ∈ B − ( A − ) · B ( A + ), we can factorize it as(9.11) ˙ g − y = (cid:34) − x x z z − x z z x (cid:35) = x x z − x x x − x x x z − x x x − x x x x · − x x − x x x − x x x x x − x x − x x x − x x x − x z z x . Again, this is valid only when the denominators in the right-hand side are nonzero. We thussee that ˙ g − y belongs to B − ( A − ) · B ( A + ) precisely when all minors in (9.10) are nonzero,in agreement with Proposition 9.17.9.9. Fomin–Shapiro atlas. The computation in (9.11) can now be used to find the maps¯ ν g and ϑ g . As in Section 8.3, denote by O g ⊂ C ( J ) u the preimage of C g ∩ ◦ X τ uλ under ¯ ϕ u . Thusfor our running example, O g is the subset of C ( J ) u where all minors in (9.10) are nonzero.We are interested in the map ¯ ν g = (¯ ν g, , ¯ ν g, ) : O g → ( ◦ Π g ∩ O g ) × Z g from (2.1), defined inSection 8.3. The first component is ¯ ν g, = ¯ ϕ − u ◦ ˜ ν g, ◦ ¯ ϕ u , where ˜ ν g : C g ∩ ◦ X τ uλ ∼ −→ ◦ R τ uλ g × ◦ X g is the map from Proposition 8.2(ii). In order to compute it, we consider the factorization˙ g − y = y − · y + ∈ U − · B ( A + ) from (9.11). The group U ( g ) is 1-dimensional since (cid:96) ( g ) = 1,and the corresponding element y ∈ U ( g ) from Proposition 8.2(ii) can be computed byfactorizing ˙ gy − ˙ g − as an element of U ( g ) · U ( g ):˙ gy − ˙ g − = − x x x − x x z x x z x x − x x x − x x x = (cid:34) x x (cid:35) · (cid:34) − x x x − x x z x x z − x x x − x x x (cid:35) , y = (cid:34) − x x (cid:35) . Therefore the map ˜ ν g, sends y · B ( A + ) from (9.6) to y y · B ( A + ) = z − x x − x x x − x x − x x z z · B ( A + ) = z − x x − x x x − x x z z · B ( A + ) . Applying ¯ ϕ − u to the right-hand side, we see that the map ¯ ν g, is given by¯ ν g, : O g → ◦ Π g ∩ O g , (cid:20) x x x x (cid:21) (cid:55)→ (cid:34) x x − x x x x x (cid:35) . Similarly, factorizing ˙ gy − ˙ g − as an element of U ( g ) · U ( g ), we find that˜ ν g, ( y · B ( A + )) = y y · B ( A + ) = (cid:34) x x (cid:35) · ˙ g · B ( A + ) . We have N g = (cid:96) ( g ) = 1, and the map ¯ ν g, : O g → Z g = R sends (cid:20) x x x x (cid:21) (cid:55)→ x x . Torus action. We compute the maps from Section 8.2. Let ˜ ρ ∈ Y ( T ) denote the grouphomomorphism ˜ ρ : C ∗ → C ∗ × T sending t (cid:55)→ ˜ ρ ( t ) := ( t n , diag( t n − , . . . , t, x ∈ GL n ( A )is represented by a Z × Z matrix (˜ x i,j ) then the element y := ˜ ρ ( t ) x ˜ ρ ( t ) − ∈ GL n ( A ) satisfies˜ y i,j = t j − i ˜ x i,j for all i, j ∈ Z . Example 9.19. Continuing the above example, we find that˜ ρ ( t ) · y y · ˜ ρ ( t ) − · B ( A + ) = (cid:34) tx x (cid:35) · ˙ g · B ( A + ) , and (cid:107) y y · B ( A + ) (cid:107) = | x || x | . Thus the action of ϑ g on Z g is given by ϑ g (cid:16) t, x x (cid:17) = tx x . The pullback of this actionto O g ⊂ C ( J ) u via ¯ ν − g preserves x , x , and x x − x x (since it preserves ¯ ν g, ( x )), butmultiplies x x by t . Therefore it is given by¯ ν − g ◦ (id × ϑ g ( t, · )) ◦ ¯ ν g : O g → O g , (cid:20) x x x x (cid:21) (cid:55)→ (cid:20) x +( t − x x x tx x x (cid:21) . The maps κ and ζ ( J ) u,v . The subset ˙ uG ( J )0 consists of matrices x ∈ G such that∆ flag u [ k ] ( x ) (cid:54) = 0. Suppose that x = g ( J ) ˙ u ∈ ˙ uU ( J ) − . Then the elements g ( J )1 ˙ u and g ( J )2 ˙ u areobtained from x by setting some entries to zero, see Section 9.4. The map x (cid:55)→ κ x x fromDefinition 4.23 sends x = g ( J ) ˙ u to g ( J )1 ˙ u , e.g., if [ x | = (cid:20) x x x x (cid:21) then [ κ x x | = (cid:20) x x (cid:21) as inExample 9.1. Comparing this to Section 9.8, we see that if M = [ x | is in u [ k ]-echelon formthen [ κ x x | is the u -truncation M tr u .Let now ( v, w ) ∈ Q (cid:23) ( u,u ) J , thus τ uλ ≤ op g := f v,w , and denote I g := ( I a ) a ∈ Z . The set G ( J ) u,v from (6.1) consists of x ∈ G such that ∆ flag u [ k ] ( x ) (cid:54) = 0 and ∆ flag v [ k ] ( κ x x ) (cid:54) = 0. But recall fromExample 9.5 that v [ k ] = I . Thus(9.12) G ( J ) u,v = (cid:110) x ∈ G | ∆ flag u [ k ] ( x ) (cid:54) = 0 and ∆ tr u I ( M ) (cid:54) = 0 (cid:111) , where M := (cid:2) g ( J ) ˙ u (cid:12)(cid:12) . Example 9.20. We compute the maps κ and ζ ( J ) u,v for our running example. Suppose that x = g ( J ) ˙ u is given in Example 9.1, and let g = [2 , , , 7] as in Example 9.18. Then g = s τ k ,so under the correspondence (9.4), we have g = f v,w for v = s and w = w J = s s s s ,see also Example 9.4. Since v [ k ] = I = { , } , we see that x ∈ G ( J ) u,v whenever x (cid:54) = 0. Wehave just computed that [ κ x x | = (cid:20) x x (cid:21) , thus ˙ v − κ x x = (cid:20) x − − x (cid:21) . Factorizing it as anelement of U ( J ) − · L J · U ( J ) via (9.1), we get˙ v − κ x x = (cid:20) x − − x (cid:21) = (cid:34) − x x x (cid:35) · x − x x x · (cid:34) − x (cid:35) , [ ˙ v − κ x x ] J = x − x x x . Thus we have computed η ( x ) = [ ˙ v − κ x x ] J from Definition 6.1. Since x ∈ ˙ uU ( J ) − , we useLemma 6.3(ii) to find ζ ( J ) u,v ( x ) = xη ( x ) − = x x x − − x x − x x , thus ζ ( J ) u,v ( x ) ˙ w − = − − x x x x − x x x . EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 53 Therefore the bottom-right principal minors of ζ ( J ) u,v ( x ) ˙ w − are(9.13) ∆ ± ( ζ ( J ) u,v ( x ) ˙ w − ) = 1 x , ∆ ± ( ζ ( J ) u,v ( x ) ˙ w − ) = x x , ∆ ± ( ζ ( J ) u,v ( x ) ˙ w − ) = x x − x x x . By Proposition 9.17, the preimage of C g under ¯ ϕ u is described by ∆ tr au I a ( M ) (cid:54) = 0 for all a ∈ [ n ]. Alternatively, as we showed in Section 7.7, the preimage of C g under ¯ ϕ u is describedby ∆ ± i ( ζ ( J ) u,v ( x ) ˙ w − ) (cid:54) = 0 for all i ∈ [ n − n ≤ k ∈ [ n ], and ( u, u ) (cid:22) ( v, w ) ∈ Q J : Conjecture 9.21. Let ( u, u ) (cid:22) ( v, w ) ∈ Q J . Denote g := f v,w , and let I g := ( I a ) a ∈ Z be theGrassmann necklace of g . Suppose that x = g ( J ) ˙ u ∈ G ( J ) u,v and let M := [ x | . Then(9.14) ∆ ± n +1 − i ( ζ ( J ) u,v ( x ) ˙ w − ) = ∆ tr iu I i ( M )∆ tr u I ( M ) for all i ∈ [ n ].For example, compare (9.13) with (9.10). Recall also that when i = 1, ∆ ± n ( ζ ( J ) u,v ( x ) ˙ w − ) := 1,so in this case (9.14) holds trivially.9.11. Total positivity. We recall some background on the totally nonnegative Grassman-nian Gr ≥ ( k, n ) of [Pos07]. By a result of Whitney [Whi52], G ≥ is the set of matricesin SL n ( R ) all of whose minors (of arbitrary sizes) are nonnegative. We have the followingcharacterizations: ( G/B ) ≥ = (cid:110) xB ∈ ( G/B ) R | ∆ flag S ( x ) ≥ S ⊂ [ n ] (cid:111) , (9.15) Gr ≥ ( k, n ) = ( G/P ) ≥ = (cid:110) xP ∈ ( G/P ) R | ∆ flag S ( x ) ≥ S ∈ (cid:0) [ n ] k (cid:1)(cid:111) . (9.16)The equality (9.16) is due to Rietsch, see [Lam16, Remark 3.8] for a proof. The equal-ity (9.15) can be proved using arguments from [Whi52] (cf. the proof of Lemma 4.17). Wecaution the reader that the analogous statement can fail to hold for other choices of J . Forinstance, when G = SL and J = { } , ( G/P ) ≥ does not contain all xP ∈ ( G/P ) R suchthat ∆ flag S ( x ) ≥ S ∈ (cid:0) [ n ]1 (cid:1) ∪ (cid:0) [ n ]3 (cid:1) , see [Che11, § f ∈ Bound( k, n ), we let Π > f := ◦ Π f ∩ Gr ≥ ( k, n ) and Π ≥ f := Π f ∩ Gr ≥ ( k, n ). Thusfor ( v, w ) ∈ Q J , we have Π > f v,w = Π > v,w and Π ≥ f v,w = Π ≥ v,w by Theorem 9.3. Proposition 9.22. Let τ uλ ≤ op g ≤ op h ∈ Bound( k, n ) , and denote by I g := ( I a ) a ∈ Z theGrassmann necklace of g . Suppose that a matrix M in u [ k ] -echelon form belongs to Π > h .Then (9.17) M tr au ∈ Gr ≥ ( k, n ) and ∆ tr au I a ( M ) > for all a ∈ Z . Proof. Applying Theorem 9.3, we have ( u, u ) (cid:22) ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) ∈ Q J , where g = f v,w and h = f v (cid:48) ,w (cid:48) . By (4.22), we get v (cid:48) ≤ vr (cid:48) ≤ ur ≤ wr (cid:48) ≤ w (cid:48) for some r, r (cid:48) ∈ W J .Suppose first that a = 1. Let x ∈ G be such that M = (cid:2) g ( J ) ˙ u (cid:12)(cid:12) and xP ∈ Π > h , and denote M (cid:48) := M tr u . We may assume that xB ∈ R > v (cid:48) ,w (cid:48) . By Corollary 6.10, we find that κ x xP ∈ Π > v (cid:48) ,u ,where ¯ v (cid:48) := v (cid:48) (cid:47) r − w for some r w ∈ W J satisfying r w ≥ r , see Lemma 6.9(ii). This showsthat M (cid:48) ∈ Gr ≥ ( k, n ). Since ur ≤ ur w , we find that ur (cid:47) r − w ≤ u by Lemma 4.6(iii),therefore ur (cid:47) r − w = u . Applying (cid:47)r − w to v (cid:48) ≤ vr (cid:48) ≤ ur via Lemma 4.6(iii), we see that¯ v (cid:48) ≤ ( vr (cid:48) (cid:47) r − w ) ≤ u . Let v = v v for v ∈ W J and v ∈ W J be the parabolic factorization of v . Then vr (cid:48) (cid:47) r − w ∈ v W J , thus ( v , v ) (cid:22) (¯ v (cid:48) , u ) ∈ Q J , which is equivalent to ∆ flag v [ k ] ( κ x x ) > v [ k ] = I , and v [ k ] = v [ k ] since v ∈ v W J , so ∆ tr u I ( M ) =∆ flag I ( κ x x ) > 0. We have shown (9.17) for a = 1. Applying the cyclic shift χ : Gr ≥ ( k, n ) → Gr ≥ ( k, n ) (which takes M to the matrix with rows ( M a +1 ) a ∈ [ n ] ), we obtain (9.17) for all a ∈ Z . (cid:3) Note that our proof of Proposition 9.22 involves a lifting from G/P to G/B , so it does notstay completely inside Gr( k, n ). Problem 9.23. Give a self-contained proof of Proposition 9.22. Example 9.24. We now consider an example for the case G/P = Gr(2 , u := s ∈ W J , so u [ k ] = { , } . Consider ( v (cid:48) , w (cid:48) ) ∈ Q J given by v (cid:48) = s , w (cid:48) = s s s s s as in Figure 2,thus h := f v (cid:48) ,w (cid:48) = [3 , , , , from Section 4.9.1to compute x ∈ G such that xB ∈ R > v (cid:48) ,w (cid:48) and xP ∈ Π > h : x := y ( t ) ˙ s y ( t ) y ( t ) y ( t ) = (cid:34) − t t t t t t t t t t t (cid:35) , M := (cid:2) g ( J ) ˙ u (cid:12)(cid:12) = t t t − t t − t t t , where t = ( t , t , t , t ) ∈ R > . Observe that xB ∈ ( G/B ) ≥ since all flag minors of x arenonnegative. (For instance, the first column of x consists of nonnegative entries.) In fact,flag minors of x are subtraction-free rational expressions in t , cf. (5.19). The n × k matrix[ x | is not in u [ k ]-echelon form, but the matrix M := (cid:2) g ( J ) ˙ u (cid:12)(cid:12) is. Up to a common scalar, the2 × M are the same as the corresponding flag minors of x , however, other(i.e., 1 × 1) flag minors of M are not necessarily nonnegative. The Grassmann necklace of h is I h = [ { , } , { , } , { , } , { , } , { , } ]. Using Proposition 9.2(i), we check that indeed xP ∈ Π > h .Let us choose ( v, w ) ∈ Q J for v = s s , w = s s s s s , so that g := f v,w = [2 , , , , u, u ) (cid:22) ( v, w ) (cid:22) ( v (cid:48) , w (cid:48) ) and τ uλ ≤ op g ≤ op h . Wecompute the elements κ x = h ( J )2 ∈ U ( J )2 , π ˙ uP − ( x ), η ( x ), and ζ ( J ) u,v ( x ) = π ˙ uP − ( x ) · η ( x ) − fromDefinition 6.1: g ( J ) ˙ u = t t t − − t t − t t t , κ x = − t t t t t t t , κ x x = − t t t t t t t t ,π ˙ uP − ( x ) = − − t t t t t t t t t t t t , η ( x ) = t t t t t t t , ζ ( J ) u,v ( x ) = − t − t t − t t t t t . We see that all flag minors of κ x x are nonnegative, cf. Lemma 6.9(ii). Observe that κ g ( J ) ˙ u = κ x by Lemma 6.2(iii), so by Lemma 6.3(ii), we could alternatively compute ζ ( J ) u,v ( x ) as the For the Grassmannian case, Marsh–Rietsch parametrizations are closely related to BCFW bridgeparametrizations , see [BCFW05, AHBC + 16, Kar16]. EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 55 product g ( J ) ˙ u · η ( g ( J ) ˙ u ) − : η ( g ( J ) ˙ u ) = (cid:34) − (cid:35) , ζ ( J ) u,v ( x ) = g ( J ) ˙ u · η ( g ( J ) ˙ u ) − = t t t − − t t − t t t · (cid:34) − 11 1 1 1 (cid:35) . Finally, we compute the bottom-right i × i principal minors of ζ ( J ) u,v ( x ) ˙ w − and observe thatthey are all nonzero subtraction-free expressions in t , agreeing with Theorems 6.4 and 6.14: ζ ( J ) u,v ( x ) ˙ w − = − − t − t t t t − t t t , ∆ ± ( ζ ( J ) u,v ( x ) ˙ w − ) = t t t , ∆ ± ( ζ ( J ) u,v ( x ) ˙ w − ) = t t ,∆ ± ( ζ ( J ) u,v ( x ) ˙ w − ) = t t , ∆ ± ( ζ ( J ) u,v ( x ) ˙ w − ) = t t .Let us check that this agrees with Conjecture 9.21. The Grassmann necklace of g is I g =[ { , } , { , } , { , } , { , } , { , } ] in window notation. We see that the corresponding u -truncated minors of M = (cid:2) g ( J ) ˙ u (cid:12)(cid:12) are indeed given by∆ tr u ( M ) = 1 , ∆ tr u ( M ) = t t , ∆ tr u ( M ) = t t , ∆ tr u ( M ) = t t , ∆ tr u ( M ) = t t t . Further directions In addition to Theorem 1.1 and [Her14], we expect the regularity theorem to hold formany other spaces occurring in total positivity. The most natural immediate direction istotal positivity for Kac–Moody flag varieties.Let G min be a minimal Kac–Moody group, U min , U min − , B min , B min − be unipotent and Borelsubgroups, and ˜ W be the Weyl group as in Appendix A. Furthermore, let P min ⊃ B min denote a standard parabolic subgroup of G min (a group of the form G min ∩ P Y in the notationof [Kum02]). Definition 10.1. Define the totally nonnegative part U −≥ of U min − to be the subsemigroupgenerated by { x α i ( t ) | t ∈ R > , ≤ i ≤ r } . Define the totally nonnegative part of the flagvariety G min / P min to be the closure ( G min / P min ) ≥ := U −≥ P min / P min .We remark that our notion of U −≥ coincides with the one studied recently by Lusztig [Lus18,Lus19] in the simply laced case.When G min is an affine Kac–Moody group of type A , Definition 10.1 agrees with thedefinition of Lam and Pylyavskyy (cf. [LP12, Theorem 2.6]) for the polynomial loop group. Conjecture 10.2 (Regularity conjecture for Kac–Moody groups and flag varieties) . (1) The intersection of U −≥ with the Bruhat stratification {B min ˙ w B min | w ∈ ˜ W } of G min endows U −≥ with an (infinite) cell decomposition with closure partial order equal tothe Bruhat order of ˜ W . Furthermore, the link of the identity in any (closed) cell isa regular CW complex homeomorphic to a closed ball.(2) The intersection of ( G min / B min ) ≥ with the open Richardson stratification ◦ R vu of G min / B min endows ( G min / B min ) ≥ with the structure of a regular CW complex. Theclosure partial order is the interval order of the Bruhat order of ˜ W , and after addinga minimum, every interval of the closure partial order is thin and shellable. (3) The intersection of ( G min / P min ) ≥ with the open projected Richardson stratificationΠ ◦ v,w of G min / P min endows ( G min / P min ) ≥ with the structure of a regular CW complex.The closure partial order is the natural partial order on P -Bruhat intervals of ˜ W ,and after adding a minimum, every interval of the closure partial order is thin andshellable.Note that every interval in the Bruhat order of ˜ W is known to be thin and shellable [BW82].The stratification Π ◦ v,w and the P -Bruhat order can be defined analogously to [KLS14].We include a list of some other spaces occurring in total positivity which we expect tohave a natural regular CW complex structure.(1) The totally nonnegative part of double Bruhat cells [FZ99]. It has been expected thata link of a double Bruhat cell inside another double Bruhat cell is a regular CWcomplex homeomorphic to a closed ball. Our Theorem 3.12 confirms this in type A ,since double Bruhat cells for GL n embed in the Grassmannian Gr( n, n ), see [Pos07,Remark 3.11].(2) The compactified space of planar electrical networks [Lam18] and the space of bound-ary correlations of planar Ising models [GP18, Conjecture 9.1]. These spaces areknown to be homeomorphic to closed balls [GKL17, GP18], and have cell decompo-sitions [Lam18, GP18] whose face poset is graded, thin, and shellable [HK18].(3) Amplituhedra [AHT14] and, more generally, Grassmann polytopes [Lam16]. Grass-mann polytopes generalize convex polytopes into the Grassmannian Gr( k, n ). Theformer are well known to be regular CW complexes homeomorphic to closed balls.Some amplituhedra and Grassmann polytopes have been shown to be homeomor-phic to closed balls in [KW19, GKL17, BGPZ19], though we caution that not allGrassmann polytopes are balls.(4) The totally nonnegative part of the wonderful compactification [He07]. A cell decom-position of this space was constructed in [He07].We expect that most spaces in this list are TNN spaces that admit a Fomin–Shapiro atlas. Appendix A. Kac–Moody flag varieties We recall some background on Kac–Moody groups, and refer to [Kum02] for all missingdefinitions. We start by introducing the minimal Kac–Moody group G min and its flag variety G min / B min , and then explain how they relate to the polynomial loop group G and its flagvariety G / B from Section 7.A.1. Kac–Moody Lie algebras. Suppose that ˜ A is a generalized Cartan matrix [Kum02,Definition 1.1.1]. Thus ˜ A is an r × r integer matrix for some r ≥ 1. We assume ˜ A is symmetrizable , that is, there exists a diagonal matrix D ∈ GL r ( Q ) such that D ˜ A is asymmetric matrix. As in [Kum02, § g the Kac–Moody Lie algebra associatedwith ˜ A , and let h ⊂ g be its Cartan subalgebra , whose dual is denoted by h ∗ . Thus h and h ∗ are vector spaces over C of dimension ˜ r := 2 r − rank( ˜ A ), and we let (cid:104)· , ·(cid:105) : h × h ∗ → C denote the natural pairing.We let ∆ ⊂ h ∗ denote the root system of g , as defined in [Kum02, § { α i } ri =1 ⊂ h ∗ be the simple roots and { α ∨ i } ri =1 ∈ h be the simple coroots .Let ∆ re ⊂ ∆ denote the set of real roots and ∆ im ⊂ ∆ denote the set of imaginary roots ,so ∆ = ∆ re (cid:116) ∆ im . Also let ∆ = ∆ + (cid:116) ∆ − denote the decomposition of ∆ into positive and EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 57 negative roots, and denote ∆ +re := ∆ + ∩ ∆ re and ∆ − re := ∆ re ∩ ∆ − . Denote by ˜ W the Weylgroup associated with ˜ A as in [Kum02, § W acts on ∆, and preserves the subset∆ re . Moreover, ˜ W is generated by simple reflections s , . . . , s r ∈ ˜ W , and ( ˜ W , { s i } ri =1 ) is aCoxeter group by [Kum02, Proposition 1.3.21]. We let ( ˜ W , ≤ ) denote the Bruhat order on˜ W and (cid:96) : ˜ W → Z ≥ denote the length function.A.2. Kac–Moody groups. Let G min be the minimal Kac–Moody group associated to ˜ A byKac and Peterson [PK83, KP83], see [Kum02, § α ∈ ∆ re , there is aone-parameter subgroup U α ⊂ G min by [Kum02, Definition 6.2.7]. For each α ∈ ∆ re , we fixan isomorphism x α : C ∼ −→ U α of algebraic groups. Similarly to the subgroups U, U − , T, B, B − of G , we have subgroups U min , U min − , T min , B min , B min − of G min . The subgroup U min is generatedby {U α } α ∈ ∆ +re , and U min − is generated by {U α } α ∈ ∆ − re . Next, T min is an ˜ r -dimensional algebraictorus defined in [Kum02, § B min = T min (cid:110) U min is the standard positive Borel subgroup and B min − = T min (cid:110) U min − is the standard negative Borel subgroup .We define a bracket closed subset Θ ⊂ ∆ re in the same way as in Section 4.2, and for abracket closed subset Θ ⊂ ∆ +re (respectively, Θ ⊂ ∆ − re ), we have a subgroup U (Θ) ⊂ U min (respectively, U − (Θ) ⊂ U min − ), generated by U α for α ∈ Θ, see [Kum02, 6.1.1(6) and § w ∈ ˜ W , Inv( w ) := ∆ + ∩ w − ∆ − ⊂ ∆ +re is a bracket closed subset of size (cid:96) ( w ), cf. [Kum02,Example 6.1.5(b)]. We state the Kac–Moody analog of Lemma 4.1(i). Lemma A.1 ([Kum02, Lemma 6.1.4]) . Suppose that Θ = (cid:70) ni =1 Θ i and Θ , Θ , . . . , Θ n ⊂ ∆ +re are finite bracket closed subsets. Then U (Θ) , U (Θ ) , . . . , U (Θ n ) are finite-dimensionalunipotent algebraic groups, and the multiplication map gives a biregular isomorphism (A.1) U (Θ ) × · · · × U (Θ n ) ∼ −→ U (Θ) . A.3. Kac–Moody flag varieties. The Weyl group ˜ W equals N G min ( T min ) / T min , where N G min ( T min ) is the normalizer of T min in G min , cf. [Kum02, Lemma 7.4.2]. For f ∈ ˜ W , wedenote by ˙ f ∈ G min an arbitrary representative of f in N G min ( T min ).By [Kum02, Lemma 7.4.2, Exercise 7.4.E(9), and Theorem 5.2.3(g)], we have Bruhat andBirkhoff decompositions of G min :(A.2) G min = (cid:71) f ∈ ˜ W B min ˙ f B min , G min = (cid:71) h ∈ ˜ W B min − ˙ h B min . We let G min / B min denote the Kac–Moody flag variety of G min . For each h, f ∈ ˜ W , we haveSchubert cells ◦ X f := B min ˙ f B min / B min and opposite Schubert cells ◦ X h := B min − ˙ h B min / B min .If h (cid:54)≤ f ∈ ˜ W then by [Kum02, Lemma 7.1.22(b)], ◦ X h ∩ ◦ X f = ∅ . For h ≤ f , we denote ◦ R fh := ◦ X h ∩ ◦ X f . Therefore (7.3) follows from (A.2). The flag variety G min / B min is a projectiveind-variety by [Kum02, § ◦ X f and Schubert variety X f are finite-dimensional subvarieties, while the opposite Schubert cell ◦ X h and opposite Schubert variety X h are ind-subvarieties. Proposition A.2. Let h ≤ f ∈ ˜ W . Then X h ∩ X f is a closed irreducible ( (cid:96) ( f ) − (cid:96) ( h )) -dimensional subvariety of X f , and ◦ R fh is an open dense subset of X h ∩ X f . The results in [Kum02] are usually stated for the maximal Kac–Moody group which he denotes by G .However, these results apply to G min as well, see Remark A.3. Proof. By (7.5), ◦ X f is (cid:96) ( f )-dimensional, and by [Kum02, Lemma 7.3.10], ◦ X h ∩ X f hascodimension (cid:96) ( h ) in X f . The rest follows by [Kum17, Proposition 6.6]. (cid:3) For g ∈ ˜ W , let C g := ˙ g B min − B min / B min . We have(A.3) G min / B min = (cid:71) h ≤ f ◦ R fh and C g = (cid:71) h ≤ g ≤ f ( C g ∩ ◦ R fh ) , where the unions are taken over h, f ∈ ˜ W . The first part of (A.3) follows from (A.2), andfor the second part, see the proof of Proposition 8.2(iii). Remark A.3. Let ˆ G ⊃ G min be the “maximal” Kac–Moody group (denoted G in [Kum02])associated to ˜ A , and let ˆ B ⊃ B min be its standard positive Borel subgroup. Then thestandard negative Borel subgroup of ˆ G is still B min − . By [Kum02, 7.4.5(2)], we may identify G min / B min ∼ −→ ˆ G / ˆ B . By [Kum02, 7.4.2(3)], ◦ X f coincides with the variety ˆ B f ˆ B / ˆ B in [Kum02,Definition 7.1.13] for f ∈ ˜ W . Similarly, for h ∈ ˜ W , ◦ X h = B min − · ˙ h B min / B min coincides withthe variety B h ∅ := B min − h ˆ B / ˆ B defined in the last paragraph of [Kum02, § Affine Kac–Moody groups and polynomial loop groups. Suppose that ˜ A is theaffine Cartan matrix associated to a simple and simply-connected algebraic group G . Thuswe have r = | I | + 1, ˜ r = | I | + 2, and ˜ A is defined by [Kum02, 13.1.1(7)]. Let G denote thepolynomial loop group from Section 7. Our goal is to explain that the flag varieties G / B and G min / B min are isomorphic.Let C ⊂ T ⊂ G be the center of G , and let ˜ C ⊂ T min ⊂ G min be the center of G min ,see [Kum02, Lemma 6.2.9(c)]. By [Kum02, Corollary 13.2.9], there exists a surjective grouphomomorphism ψ : G min → ( C ∗ (cid:110) G ) /C with kernel ˜ C , where C ∗ acts on G as in Sec-tion 8.2, see also [Kum02, Definition 13.2.1]. The groups U , U − ⊂ G are identified with thegroups U min , U min − ⊂ G min , and we have T /C ∼ = T min / ˜ C . Thus ψ induces an isomorphism G min / B min ∼ −→ G / B between the affine Kac–Moody flag variety and the affine flag variety. TheWeyl groups ˜ W of G and G min are isomorphic by [Kum02, Proposition 13.1.7], and the rootsystems ∆ coincide by [Kum02, Corollary 13.1.4]. Therefore the subsets ◦ X f , ◦ X h , ◦ R fh , C g of G / B get sent by ψ to the corresponding subsets of G min / B min . As explained in the last para-graph of [Kum02, § G can be viewed as a subset of G min as well, and the restrictionof ψ to G is the quotient map G → G/C .We justify some of the other statements that we used in Sections 7.1 and 8.2. For (7.2),see [Kum02, § § Y ( T ) from Sec-tion 8.2, see [Kum02, § (cid:104)· , ·(cid:105) : Y ( T ) × X ( T ) → Z inthe same section, see [Kum02, § Gaussian decomposition and affine charts. By [Kum02, Theorem 7.4.14], G min isan affine ind-group . Similarly, U min , U min − , T , B min , B min − are affine ind-groups, see e.g. [Kum02, § G min0 := B min − B min and g ∈ ˜ W . Recall the subgroups U ( g ) and U ( g ) from (7.4). Then U ( g ) is a closed (cid:96) ( g )-dimensional subgroup of U min ∼ = U , and U ( g ) is a closed ind-subgroupof U min − ∼ = U − . Proof of Lemma 8.1. For (i), see [Kum02, Proposition 7.4.11]. For (ii), we use an argu-ment given in [Wil13, Proposition 2.5]: both maps are bijective morphisms by [Kum02, EGULARITY THEOREM FOR TOTALLY NONNEGATIVE FLAG VARIETIES 59 Lemma 6.1.3]. In particular, it follows that ˙ g U min − ˙ g − ⊂ G min0 and for x ∈ ˙ g U min − ˙ g − , we have[ x ] = 1. The inverse maps are given by µ − ( x ) = ([ x ] − , [ x ] + ), µ − ( x ) = ([ x − ] − , [ x − ] − − ).They are regular morphisms by (i), which proves (ii). (cid:3) Proof of (7.5) . The map ˙ g U min − ˙ g − ∼ −→ C g is a biregular isomorphism for g = id by [Kum02,Lemma 7.4.10]. Since ˜ W acts on G min / B min by left multiplication, the case of general g ∈ ˜ W follows as well. Since U ( g ), U ( g ) are closed ind-subvarieties of ˙ g U min − ˙ g − and ◦ X g , ◦ X g areclosed ind-subvarieties of C g , it suffices to show that the image of U ( g ) equals ◦ X g while theimage of U ( g ) equals ◦ X g . By [Kum02, Exercise 7.4.E(9) and 5.2.3(11)], we have U min = ( U min ∩ ˙ g U min − ˙ g − ) · ( U min ∩ ˙ g U min ˙ g − ) = U ( g ) · ( U min ∩ ˙ g U min ˙ g − ) , U min − = ( U min − ∩ ˙ g U min − ˙ g − ) · ( U min − ∩ ˙ g U min ˙ g − ) = U ( g ) · ( U min − ∩ ˙ g U min ˙ g − ) . Thus B min ˙ g B min = U ( g ) · ( U min ∩ ˙ g U min ˙ g − ) · ˙ g B min = U ( g ) · ˙ g · B min , B min − ˙ g B min = U ( g ) · ( U min − ∩ ˙ g U min ˙ g − ) · ˙ g B min = U ( g ) · ˙ g · B min . (cid:3) References [AHBC + 16] Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Post-nikov, and Jaroslav Trnka. Grassmannian Geometry of Scattering Amplitudes . Cambridge Uni-versity Press, Cambridge, 2016.[AHT14] Nima Arkani-Hamed and Jaroslav Trnka. The amplituhedron. J. High Energy Phys. , (10):33,2014.[BB05] Anders Bj¨orner and Francesco Brenti. Combinatorics of Coxeter groups , volume 231 of GraduateTexts in Mathematics . 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