A Birkhoff-Bruhat Atlas for partial flag varieties
aa r X i v : . [ m a t h . R T ] J u l A BIRKHOFF-BRUHAT ATLAS FOR PARTIAL FLAGVARIETIES
HUANCHEN BAO AND XUHUA HE
Abstract.
A partial flag variety P K of a Kac-Moody group G has a natu-ral stratification into projected Richardson varieties. When G is a connectedreductive group, a Bruhat atlas for P K was constructed in [7]: P K is locallymodeled with Schubert varieties in some Kac-Moody flag variety as stratifiedspaces. The existence of Bruaht atlases implies some nice combinatorial andgeometric properties on the partial flag varieties and the decomposition intoprojected Richardson varieties.A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac-Moody group due to combinatorial and geometric reasons. To overcome ob-structions, we introduce the notion of Birkhoff-Bruhat atlas. Instead of theSchubert varieties used in a Bruhat atlas, we use the J -Schubert varieties fora Birkhoff-Bruhat atlas. The notion of the J -Schubert varieties interpolatesBirkhoff decomposition and Bruhat decomposition of the full flag variety (of alarger Kac-Moody group). The main result of this paper is the construction ofa Birkhoff-Bruhat atlas for any partial flag variety P K of a Kac-Moody group.We also construct a combinatorial atlas for the index set Q K of the projectedRichardson varieties in P K . As a consequence, we show that Q K has some nicecombinatorial properties. This gives a new proof and generalizes the work ofWilliams [21] in the case where the group G is a connected reductive group. Introduction
The flag variety and its decomposition into Richardson varieties.
Let G be a connected reductive group and B be the full flag variety of G . Anopen Richardson variety is the intersection of a Bruhat cell with an oppositeBruhat cell. We then have the decomposition of B into the disjoint union of theopen Richardson varieties. This decomposition has many remarkable properties,including:(1) each stratum is smooth;(2) the closure of each stratum is a union of other strata;(3) the closure of each stratum is normal, Cohen-Macaulay, with rational singu-larities;(4) over positive characteristic, there exists a Frobenius splitting on B which com-patibly splits all the strata;(5) over complex numbers, there exists a Poisson structure on B for which the T -leaves are exactly the strata; Mathematics Subject Classification.
Key words and phrases.
Partial flag varieties, projected Richardson varieties, Kac-Moodygroups, Bruhat atlas. (6) over real numbers, the intersection of the totally nonnegative flag variety X > with each stratum gives a cellular decomposition of X > ;(7) the poset of the strata is thin and EL-shellable.Many of these remarkable properties remain valid for the full flag variety of anyKac-Moody group.1.2. Partial flag varieties.
Now we consider the partial flag variety P K = G/P K of a Kac-Moody group G . The variety P K has a natural stratification into theprojected Richardson varieties. The projected Richardson varieties in P K arethe image of certain Richardson varieties in the full flag B under the projectionmap π : B → P K . However, the combinatorial and geometric structures of theprojected Richardson varieties in P K are more complicated than the Richardsonvarieties in B .Let Q K be the index set of the projected Richardson varieties in P K and (cid:22) be thepartial order on Q K . Williams [21] showed that if G is a connected reductive group,then the partial order set Q K has remarkable combinatorial properties: thinness,shellability, etc. Such combinatorial properties are used later by Galashin, Karpand Lam [6] to prove the conjecture of Postnikov and Williams that the totallynonnegative part of P K is a regular CW complex.1.3. The Bruhat atlas of [7] . One of the motivations in the unpublished workof Knutson, Lu and the second-named author [7] is to use the Richardson va-rieties in the full flag variety ˜ B of a “much larger” Kac-Moody group ˜ G as themodel for the decomposition of P K into projected Richardson varieties, and manyother stratified spaces arising in Lie theory. Consequently, many remarkable com-binatorial properties and geometric properties on these stratified spaces may bededuced directly from those on the Bruhat order of the Weyl group ˜ W of ˜ G andthe Richardson varieties of ˜ B .By definition, a Bruhat atlas for a stratified space M = ⊔ y ˚ M y consists of a largeKac-Moody group ˜ G and an open covering M = ∪ U , such that • for each U , there exists an isomorphism of stratified spaces from U to a Schubertcell in the flag variety ˜ B of ˜ G ; • for each y and U , U ∩ ˚ M y is mapped isomorphically to an open Richardsonvariety of ˜ B .The group ˜ G is called the atlas group for this Bruhat atlas.The first example of a Bruhat atlas was constructed by Snider [18] for Grass-mannian with the positroid stratification. A Bruhat atlas for the full flag varietyof a connected reductive group G was constructed by Knutson, Woo, and Yong in[15]. A Bruhat atlas for any partial flag variety of a connected reductive group anda Bruhat atlas for the wonderful compactification of a semisimple adjoint groupwas then constructed in [7]. A different “atlas” for the partial flag varieties of aconnected reductive group was constructed recently by Galashin, Karp and Lamin [6] and by Huang in [10]. A Bruhat atlas for the wonderful compactification ofthe symmetric space P SO (2 n ) /SO ( n −
1) was recently given by Huang in [11].1.4.
The Birkhoff-Bruhat atlas.
In the works discussed above, the group G involved is a connected reductive group, i.e. a Kac-Moody group of finite type. A Bruhat atlas for P K does not exists when G is of infinite type, due to thefollowing reasons. First, the partial flag variety P K , in general, are infinite-dimensional. Thus one can not use Schubert cells (which is finite-dimensional)as an atlas for P K . This gives a geometric obstruction. There is also a combi-natorial obstruction arising from comparison of the partial orders. Note that thepartial order (cid:22) on Q K involved both the Bruhat order and the opposite Bruhatorder in the Weyl group W of G . By the definition of Bruhat atlas, one needsto embed Q K into the Weyl group ˜ W of an atlas group. It is only possible if theWeyl group W has the longest element, which interchanges the Bruhat order andthe opposite Bruhat order on W .The main purpose of this paper is to introduce a suitable “atlas model” for thepartial flag varieties of any Kac-Moody group. We use the open J -Schubert cellsin the full flag variety ˜ B of an atlas group ˜ G instead of the Schubert cells in ˜ B asin the original definition of Bruhat atlas.The decomposition of ˜ B into the J -Schubert cell was introduced by Billig andDyer in [1], which simultaneously generalizes both the Bruhat decomposition of B into the Schubert cells and the Birkhoff decomposition of B into the oppositeSchubert cells. A J -Schubert cell, in general, is neither finite dimensional norfinite codimensional. A Birkhoff-Bruhat atlas for P K consists of an atlas group ˜ G and an open covering P K = ∪ U such that • for each U , an embedding of U into the full flag variety ˜ B of ˜ G ; • the intersection of U with any projected Richardson variety is mapped isomor-phically to a J -Richardson variety in ˜ B .The main result of this paper is Theorem 1.1.
Any partial flag variety P K of a Kac-Moody group admits a Birkhoff-Bruhat atlas. It is also worth mentioning that even in the finite type case, the atlas groupsfrom the Birkhoff-Bruhat atlas we constructed and those from the Bruhat atlasin [7] are different. Thus our construction provides a new “atlas model” for thepartial flag varieties for connected reductive groups.We also construct a combinatorial “atlas model” for the poset Q K . This “atlasmodel” identifies the poset ( Q K , (cid:22) ) with a convex subset of the Weyl group ˜ W ofthe atlas group ˜ G with respect to a twisted Bruhat order I ♭ . This combinatorial“atlas model” is valid for Q K from an arbitrary Coxeter group. Theorem 1.2.
The partial order (cid:22) on Q K is thin and EL-shellable. We refer to section 4.9 for the definition of thinness and EL-Shellability. Thisresult generalizes the previous work of Williams [21].Galashin, Karp and Lam [6, Conjecture 10.2] conjectured that the totally non-negative part of P K is a regular CW complex for a Kac-Moody group G , and Q K isits face poset. Theorem 1.2 confirms the combinatorial aspect of their conjecture.1.5. Organization.
This paper is organized as follows. We recall preliminaries ofKac-Moody groups and J-Schubert cells in Section 2. We then define a Birkhoff-Bruhat atlas and construct such an atlas for the partial flag variety P K of arbitrarytype in Section 3. We discuss some combinatorial consequences in Section 4.9. We HUANCHEN BAO AND XUHUA HE construct another Birkhoff-Bruhat atlas for the partial flag variety P K when is K is of finite type in Section 5. We discuss examples in Section 6. Acknowledgement:
We thank Thomas Lam and Lauren Williams for helpfulcomments.HB is supported by a NUS startup grant. XH is partially supported by a start-up grant and by funds connected with Choh-Ming Chair at CUHK, and by HongKong RGC grant 14300220. 2.
Preliminary
Minimal Kac-Moody groups.
Let I be a finite set and A = ( a ij ) i,j ∈ I be asymmetrizable generalized Cartan matrix in the sense of [13, § Kac-Moodyroot datum associated to A is a quintuple D = ( I, A, X, Y, ( α i ) i ∈ I , ( α ∨ i ) i ∈ I ) , where X is a free Z -module of finite rank with Z -dual Y , and the elements α i of X and α ∨ i of Y such that h α ∨ j , α i i = a ij for i, j ∈ I . We denote by ω i ∈ X the elementthat h α ∨ j , ω i i = δ ij . We shall assume the root datum D is simply connected.We have natural actions of W on both X and Y . Let∆ re = { w ( ± α i ) ∈ X | i ∈ I, w ∈ W } ⊂ X be the set of real roots. Then ∆ re = ∆ re + ⊔ ∆ re − is the union of positive real rootsand negative real roots.Let k be an algebraically closed field. The minimal Kac-Moody group G as-sociated to the Kac-Moody root datum D is the group generated by the torus T = Y ⊗ Z k × and the root subgroup U α ∼ = k for each real root α , subject to theTits relations [20]. Let U + ⊂ G (resp. U − ⊂ G ) be the subgroup generated by U α for α ∈ ∆ re + (resp. α ∈ ∆ re − ). Let B ± ⊂ G be the Borel subgroup generatedby T and U ± . We fix a pinning of G consisting of ( T, B + , B − , x i , y i ; i ∈ I ) withone parameter subgroups x i : k → U α i and y i : k → U − α i analogous to [17]. Wehave an anti-involution Ψ of G analogous to [17, § x i ( a )) = y i ( a ),Ψ( y i ( a )) = x i ( a ) and Ψ( t ) = t for a ∈ k , t ∈ T .Let J ⊂ I (not necessarily of finite type). We denote by P + J the subgroup of G generated by B + and U − α j for j ∈ J . Let W J be the subgroup of W generatedby { s j } j ∈ J . Let W J be the set of minimal-length coset representatives of W/W J and J W be the set of minimal-length coset representatives of W J \ W . For i ∈ I ,we define ˙ s i = x i ( − y i (1) x i ( − ∈ G. For any w ∈ W with reduced expression w = s i · · · s i n , we define˙ w = ˙ s i · · · ˙ s i n ∈ G. It is known that ˙ w is well-defined and independent of the reduced expression.Let L J be the subgroup of P + J generated by T , U ± α j for j ∈ J . We denoteby ∆ reJ = { w ( ± α j ) ∈ X | j ∈ J, w ∈ W J } ⊂ ∆ re the set of real roots of L J . Wewrite ∆ reJ, ± = ∆ reJ ∩ ∆ re ± . We denote by U P + J the unipotent radical of P + J , which isgenerated by U α for α ∈ ∆ re + − ∆ reJ . We have following Levi decomposition of P + J [12, Theorem B.39] P + J = L J ⋉ U P + J . (2.1) We similarly define the subgroup P − J of G min as the subgroup generated by B − and U α j for j ∈ J , with the Levi decomposition P − J = L J ⋉ U P − J . (2.2)2.2. The full flag variety.
In this subsection, we recall several results on theKac-Moody flag varieties.We denote by B the (thin) full flag variety [16], equipped with the ind-varietystructure. Let v, w ∈ W . Define, respectively, the Schubert cell, the oppositeSchubert cell and the open Richardson variety by˚ X w = B + ˙ wB + /B + , ˚ X v = B − ˙ vB + /B + , ˚ R v,w = ˚ X w ∩ ˚ X v . We have the Bruhat decomposition B = ⊔ w ∈ W ˚ X w and the Birkhoff decomposi-tion B = ⊔ v ∈ W ˚ X v . It is known that ˚ R v,w = ∅ if and only if v w . In this case,˚ R v,w is irreducible of dimension ℓ ( w ) − ℓ ( v ). We also have the decomposition B = ⊔ v w ˚ R v,w . Let X w , X v , R v,w be the (Zariski) closure of ˚ X w , ˚ X v , ˚ R v,w respectively. By [16,Proposition 7.1.15&7.1.21], X w = G w ′ w ˚ X w ′ , X v = G v ′ > v ˚ X v ′ . As the Schubert varieties and opposite Schubert varieties intersect transversally,we also have (see e.g. [14]) R v,w = G v v ′ w ′ w ˚ R v ′ ,w ′ . The J -Schubert cells and J -Richardson varieties. Let J ⊂ I . Following[1, Closure patterns], we define the partial order J on W as follows. The partialorder J on W is generated by the relations s β w J < w for w ∈ W and β ∈ Ψ J with w − ( β ) ∈ ∆ re, − . Define a (non-standard) length function J ℓ on W by J ℓ ( w ) = ℓ ( w ) − ♯ (∆ re, + J ∩ w − (∆ re, − )) . Note that any element in W can be written in a unique way as xy for x ∈ W J and y ∈ J W . We have J ℓ ( xy ) = ℓ ( y ) − ℓ ( x ), where ℓ ( · ) denotes the usual lengthfunction. We defineΨ + J = ∆ re, − J ⊔ (∆ re, + − ∆ re, + J ) , Ψ − J = ∆ re, + J ⊔ (∆ re, − − ∆ re, − J ) . Remark 2.1.
The relation between our partial order J and the partial order Ψ J used in [1, Closure pattern] is the following v J w if and only if v − Ψ J w − . Note that v J w is not equivalent to v − J w − in general [1, Page 18]. Remark 2.2.
Let W J be a finite Weyl group. In this case, we denote by w J thelongest element of W J . Then for w, w ′ ∈ W , w ′ J w if and only if w J w ′ w J w .Moreover, we have J ℓ ( w ) = ℓ ( w J w ) − ℓ ( w J ). HUANCHEN BAO AND XUHUA HE
Let J B + be the subgroup of G generated by T and U α for α ∈ Ψ + J . Then J B + is the opposite Borel subgroup of the standard parabolic subgroup P + J . Similarly,let J B − be the subgroup of G generated by T and U α for α ∈ Ψ − J . In the casewhere J = ∅ , we have J B + = B + and J B − = B − . In the case where J = I , wehave J B + = B − and J B − = B + . Let B ± J = L J ∩ B ± and U ± J = L J ∩ U ± .Thanks to the Levi decompositions (2.1)&(2.2), we have J B + = B − J ⋉ U P + J , J B − = B + J ⋉ U P − J . Let v, w ∈ W . Define, respectively, the J -Schubert cell, the opposite J -Schubertcell and the open J -Richardson variety by J ˚ X w = J B + ˙ wB + /B + , J ˚ X v = J B − ˙ vB + /B + , J ˚ R v,w = J ˚ X w ∩ J ˚ X v . Lemma 2.3.
We have isomorphisms ( U − J ∩ ˙ wU − ˙ w − ) × ( U P + J ∩ ˙ wU − ˙ w − ) → J ˚ X w , ( x , x ) x x ˙ wB + /B + ;( U + J ∩ ˙ vU − ˙ v − ) × ( U P − J ∩ ˙ vU − ˙ v − ) −→ J ˚ X v , ( x , x ) x x ˙ vB + /B + . Proof.
We prove the first statement here. The second one is entirely similar.It follows from the Levi decompositions and [16, Theorem 5.2.3] that we haveisomorphisms ( U − J ∩ ˙ wU − ˙ w − ) × ( U − J ∩ ˙ wU + ˙ w − ) −→ U − J , ( U P + J ∩ ˙ wU − ˙ w − ) × ( U P + J ∩ ˙ wU + ˙ w − ) −→ U P + J . Therefore we have J B + ˙ wB + /B + = ( U − J ∩ ˙ wU − w − ) · ( U P + J ∩ ˙ wU − w − ) · ˙ wB + /B + . Now the lemma follows from the restriction of the isomorphism˙ wU − ˙ w − −→ ˙ wU − B + /B + , g g ˙ wB + /B + . (cid:3) By [1, Theorem 1], we have B = ⊔ w ∈ W J ˚ X w = ⊔ v ∈ W J ˚ X v . (2.3)Let J X w and J X v be the (Zariski) closure of J ˚ X w and J ˚ X v respectively. By [1,Theorem 4], we have J X w = J ˚ X w = G w ′ J w ˚ X w ′ , J X v = J ˚ X v = G v J v ′ ˚ X v ′ . (2.4) Proposition 2.4.
Let v, w ∈ W . Then the following conditions are equivalent:(1) J ˚ X w ∩ J ˚ X v = ∅ ;(2) J X w ∩ J X v = ∅ ;(3) v J w .Proof. It is obvious that (1) ⇒ (2).We show that (2) ⇒ (3). If J X w ∩ J X v = ∅ , then there exists z ∈ W , suchthat J X w ∩ J X v ∩ X z = ∅ . Since X z is finite dimensional, J X w ∩ J X v ∩ X z is stillfinite dimensional. It is projective and stable under the left action of T . By [16,Exercise 7.1.E.5], J X w ∩ J X v ∩ X z has a T -fixed point. Hence J X w ∩ J X v has a T -fixed point.Note that the T -fixed points in X are { ˙ wB + /B + ; w ∈ W } . Thus by (2.3),˙ wB + /B + is the only T -fixed point of J ˚ X w and of J ˚ X w . By (2.4), the T -fixed points in J X w are { ˙ w ′ B + /B + ; w ′ J w } and the T -fixed points in J X v are { ˙ v ′ B + /B + ; v J v ′ } .Since J X w and J X v have a common T -fixed point, we must have v J w .We then show that (3) ⇒ (1). Let U v = ˙ vU − B + /B + ⊂ B . Thanks to the Levidecomposition and [16, Theorem 5.2.3], we have the isomorphisms( U − J U P + J ∩ ˙ vU − ˙ v − ) × ( U + J U P − J ∩ ˙ vU − ˙ v − ) ˙ vU − ˙ v − U v , ( g , g ) g g g g ˙ vB + /B + . ∼ ∼ By Lemma 2.3, we have the isomorphism( U − J U P + J ∩ ˙ vU − ˙ v − ) × J ˚ X v ∼ −→ U v . Since J ˚ X w is U − J U P + J -stable, we have, via restriction,( U − J U P + J ∩ ˙ vU − ˙ v − ) × ( J ˚ X w ∩ J ˚ X v ) ∼ −→ J ˚ X w ∩ U v . It remains to prove J ˚ X w ∩ U v = ∅ when v J w . Since U v is open in B , we have J ˚ X w ∩ U v = ∅ if and only if J X w ∩ U v = ∅ . Thanks to (2.4), we have ˙ vB + /B + ∈ J X w if v J w . We clearly have ˙ vB + /B + ∈ U v . Therefore ˙ vB + /B + ∈ U v ∩ J X w .The claim follows. (cid:3) J-Schubert decompositions.
In this subsection, we study a more refinedversion of Lemma 2.3.For a group K and subsets K ′ , K , K , . . . , K n , we write K ′ = K ⊙ K ⊙ · · · ⊙ K n if any element k ′ ∈ K ′ can be written uniquely as k ′ = k k · · · k n with k i ∈ K i . Lemma 2.5.
Let v ∈ W J and w ∈ W J ∩ J W . We have U − J U P + J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ( U − J ∩ ˙ vU − J ˙ v − ) ⊙ ˙ v ( U + P + J ∩ ˙ wU − J ˙ w − ) ˙ v − . Proof.
Note that ℓ ( vw ) = ℓ ( v ) + ℓ ( w ). We recall the following decompositionsfrom [16, Theorem 5.2.3]:( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ( U − ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ) ⊙ ( U + ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ); ( ♦ U − ∩ ˙ vU − ˙ v − = ˙ v ( U − ∩ ˙ wU + ˙ w − ) ˙ v − ⊙ ( U − ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ); ( ♦ U + ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ˙ v ( U + ∩ ˙ wU − ˙ w − ) ˙ v − ⊙ ( U + ∩ ˙ vU − ˙ v − ) . ( ♦ P + J = L J ⋉ U P + J , the decompositions arecompatible with the restriction from U ± to U ± J as well as from U + to U P + J .It follows from ( ♦
1) that U − J U P + J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ( U − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ) ⊙ ( U P + J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ) . Since w ∈ J W , we have U − J ∩ ˙ wU + ˙ w − = { e } . Therefore it follows from ( ♦ U − J ∩ ˙ vU − J ˙ v − = U − J ∩ ˙ vU − ˙ v − = U − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − . HUANCHEN BAO AND XUHUA HE
Finally, since v ∈ W J , we have U P + J ∩ ˙ vU − ˙ v − = { e } . Since w ∈ W J , it followsfrom ( ♦
3) that U P + J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ˙ v ( U P + J ∩ ˙ wU − ˙ w − ) ˙ v − = ˙ v ( U P + J ∩ ˙ wU − J ˙ w − ) ˙ v − The lemma follows. (cid:3)
The following result follows easily from Lemma 2.3 and Lemma 2.5.
Corollary 2.6.
Let J , J ⊂ I . Let v ∈ W J and w ∈ W J ∩ J W . We have anisomorphism ( U − J ∩ ˙ vU − J ˙ v − ) × ˙ v ( U + P + J ∩ ˙ wU − J ˙ w − ) ˙ v − −→ J ˚ X vw , ( g , g ) g g ( ˙ v ˙ w ) B + . Lemma 2.7.
Let v ∈ W J and w ∈ W J ∩ J W . We have U + J U P − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − =( U + J ∩ ˙ vU − J ˙ v − ) ⊙ ˙ v ( U − J ∩ U P − J ∩ ˙ wU − J ˙ w − ) ˙ v − ⊙ ˙ v ( U P − J ∩ U P − J ∩ ˙ wU − ˙ w − ) ˙ v − . Proof.
Note that ℓ ( vw ) = ℓ ( v )+ ℓ ( w ). We recall again the following decompositionsfrom [16, Theorem 5.2.3]:( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ( U + ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ) ⊙ ( U − ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ); ( ♦ U + ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ˙ v ( U + ∩ ˙ wU − ˙ w − ) ˙ v − ⊙ ( U + ∩ ˙ vU − ˙ v − ) . ( ♦ P + J = L J ⋉ U P + J , the decompositions aboveare compatible with the restriction from U ± to U ± J as well as from U + to U P + J .It follows from ( ♦
1) that U + J U P − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ( U + J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ) ⊙ ( U P − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − ) . Since w ∈ W J ∩ J W and v ∈ W J , it follows from ( ♦
3) that U + J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = U + J ∩ ˙ vU − ˙ v − = U + J ∩ ˙ vU − J ˙ v − . Since v ∈ W J , we have U P − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ˙ v ( U P − J ∩ ˙ wU − ˙ w − ) ˙ v − . Thanks to the Levi decomposition of P − J , we further have U P − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − = ˙ v ( U P − J ∩ ˙ wU − ˙ w − ) ˙ v − = ˙ v ( U − J ∩ U P − J ∩ ˙ wU − ˙ w − ) ˙ v − ⊙ ˙ v ( U P − J ∩ U P − J ∩ ˙ wU − ˙ w − ) ˙ v − = ˙ v ( U − J ∩ U P − J ∩ ˙ wU − J ˙ w − ) ˙ v − ⊙ ˙ v ( U P − J ∩ U P − J ∩ ˙ wU − ˙ w − ) ˙ v − . The lemma is proved. (cid:3)
Product of Parabolic subgroups.
Let J , J ⊂ I . We study the decom-position of P + J P + J /B + with respect to the J -Schubert cells and the opposite J -Schubert cells.We first consider the decomposition into the J -Schubert cells. Proposition 2.8.
Let J , J ⊂ I . Then we have P + J P + J = L J L J B + = G ˜ w ∈ W J W J J B + ˙˜ wB + . Proof.
It follows from (2.3) that S ˜ w ∈ W J W J J B + ˙˜ wB + is a disjoint union. We have L J B + = G w ∈ W J ∩ J W ( L J ∩ P J ) ˙ wB + = G w ∈ W J ∩ J W ( L J ∩ L J )( L J ∩ U P + J ) ˙ wB + . Thus L J L J B + = [ w ∈ W J ∩ J W L J ( L J ∩ U P + J ) ˙ wB + = [ v ∈ W J ,w ∈ W J ∩ J W ( L J ∩ U − ) ˙ v ( L J ∩ U + )( L J ∩ U P + J ) ˙ wB + ⊂ [ v ∈ W J ,w ∈ W J ∩ J W ( L J ∩ U − ) ˙ vU P + J ( L J ∩ U + ) ˙ wB + = [ v ∈ W J ,w ∈ W J ∩ J W ( L J ∩ U − ) U P + J ˙ v ˙ w (cid:0) ˙ w − ( L J ∩ U + ) ˙ w ) B + = [ v ∈ W J ,w ∈ W J ∩ J W J B + ˙ v ˙ wB + . Now we proceed with the reverse inclusion. Note that any element in W J W J can be written in a unique way as vw for some v ∈ W J and w ∈ W J ∩ J W .By definition and the Levi decomposition (2.1), J B + = ( L J ∩ U − ) U P + J T . Since v ∈ W J , the conjugation action of ˙ v stabilizes U P + J T . Thus J B + ˙ v ˙ wB + = ( L J ∩ U − ) ˙ v ( U P + J ˙ wB + ) = ( L J ∩ U − ) ˙ v ( U P + J ∩ ˙ wB − ˙ w − ) ˙ wB + . Since w ∈ W J , we have U P + J ∩ ˙ wB − ˙ w − ⊂ U P + J ∩ L J . Thus J B + ˙ v ˙ wB + ⊂ ( L J ∩ U − ) ˙ v ( L J ∩ U + ) ˙ wB + ⊂ L J L J B + . The statement is proved. (cid:3)
We then consider the decomposition into opposite J -Schubert cells. Proposition 2.9.
We have P + J P + J = G ˜ w ∈ W J W J J B − ˙˜ wB + ∩ P + J P + J . Proof.
Note that P − J ∩ L J is an opposite parabolic subgroup of L J . By theBirkhoff decomposition of L J and the Levi decomposition of P + J , we have P + J = G w ∈ W J ∩ J W ( P − J ∩ L J ) ˙ wB + = G w ∈ W J ∩ J W ( L J ∩ L J )( U − J ∩ U P − J ) ˙ wB + . Hence P + J P + J = [ w ∈ W J ∩ J W L J ( U − J ∩ U P − J ) ˙ wB + = [ v ∈ W J ,w ∈ W J ∩ J W U + J ˙ vU + J ( U − J ∩ U P − J ) ˙ wB + = [ v ∈ W J ,w ∈ W J ∩ J W ( U + J ∩ ˙ vU − J ˙ v − ) ˙ vU + J ( U − J ∩ U P − J ) ˙ wB + . We show that(a) U + J ( U − J ∩ U P − J ) ˙ wB + = ( U − J ∩ U P − J ) ˙ wB + .We have the decomposition U + J = ( U + J ∩ U + J )( U + J ∩ U P + J ). Since ( U − J ∩ U P − J ) ˙ w ⊂ L J , we have U P + J ( U − J ∩ U P − J ) ˙ w = ( U − J ∩ U P − J ) ˙ wU P + J and( U + J ∩ U + J )( U + J ∩ U P + J )( U − J ∩ U P − J ) ˙ wB + = ( U + J ∩ U + J )( U − J ∩ U P − J ) ˙ wB + . Thanks to the Levi decomposition of L J ∩ P − J , we further have( U + J ∩ U + J )( U − J ∩ U P − J ) = ( U − J ∩ U P − J )( U + J ∩ U + J ) . Since w ∈ J W , we have ( U + J ∩ U + J ) ˙ wB + = ˙ wB + .Thus (a) is proved.Now we have P + J P + J = [ v ∈ W J ,w ∈ W J ∩ J W ( U + J ∩ ˙ vU − J v − ) ˙ v ( U − J ∩ U P − J ) ˙ wB + . (2.5)Since ˙ v ( U − J ∩ U P − J ) ⊂ U P − J ˙ v , we have( U + J ∩ ˙ vU − J ˙ v − ) ˙ v ( U − J ∩ U P − J ) ˙ wB + ⊂ J B − ˙ v ˙ wB + . Thanks to (2.3), we have P + J P + J ⊂ G v ∈ W J ,w ∈ W J ∩ J W J B − ˙ v ˙ wB + . Then we have that ( U + J ∩ ˙ vU − J ˙ v − ) ˙ v ( U − J ∩ U P − J ) ˙ wB + = J B − ˙ v ˙ wB + ∩ P + J P + J and P + J P + J = G v ∈ W J ,w ∈ W J ∩ J W ( J B − ˙ v ˙ wB + ∩ P + J P + J ) . Since W J W J = W J ( W J ∩ J W ), we have P + J P + J = F ˜ w ∈ W J W J J B − ˙˜ wB + ∩ P + J P + J and the proposition is proved. (cid:3) Combining Proposition 2.9 and Lemma 2.7, we have the following proposition. Proposition 2.10.
Let J , J ⊂ I and v ∈ W J , w ∈ W J ∩ J W . We have anisomorphism ( U + J ∩ ˙ vU − J ˙ v − ) × (cid:0) ˙ v ( U − J ∩ U P − J ∩ ˙ wU − J ˙ w − ) ˙ v − (cid:1) → J ˚ X vw ∩ P + J P + J /B + . Proof.
By Proposition 2.9, we have J ˚ X vw ∩ P + J P + J /B + = ( U + J ∩ ˙ vU − J ˙ v − ) ˙ v ( U − J ∩ U P − J ) ˙ wB + = ( U + J ∩ ˙ vU − J ˙ v − ) ˙ v ( U − J ∩ U P − J ∩ ˙ wU − J ˙ w − ) ˙ wB + . By Lemma 2.3, we have the isomorphism (cid:0) U + J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − (cid:1) × (cid:0) U P − J ∩ ( ˙ v ˙ w ) U − ( ˙ v ˙ w ) − (cid:1) ∼ −→ J ˚ X vw . Thanks to Lemma 2.7, the lemma follows from the restriction of the above iso-morphism to( U + J ∩ ˙ vU − J ˙ v − ) × (cid:0) ˙ v ( U − J ∩ U P − J ∩ ˙ wU − J ˙ w − ) ˙ v − (cid:1) ∼ −→ J ˚ X vw ∩ P + J P + J /B + . (cid:3) A Birkhoff-Bruhat atlas
Definitions.
Let M be an ind-variety over k . A stratification on M is afamily of locally closed, finite dimensional subvarieties { ˚ M y } y ∈Y indexed by aposet Y such that • M = ⊔ y ∈Y ˚ M y ; • For any y ∈ Y , the Zariski closure M y of ˚ M y equals ⊔ y ′ y ˚ M y ′ .Assume furthermore that the minimal strata in the stratification Y min of M arepoints. A Birkhoff-Bruhat atlas on ( M, Y ) is the following data:(1) an open covering for M consisting of open sets U f around the minimal strata f ∈ Y ;(2) a (minimal) Kac-Moody group ˜ G and a subset J of the set of simple roots of˜ G ;(3) for any minimal stratum f ∈ Y , an embedding c f from U f into the flag variety˜ B of ˜ G such that c f ( U f ∩ ˚ M y ) is an open J -Richardson variety of ˜ B for any y ∈ ˜ Y .3.2. Partial flag varieties.
Let K ⊂ I and P K = G/P + K be the partial flagvariety. Then we have the decomposition P K = ⊔ w ∈ W K B + ˙ wP + K /P + K = ⊔ v ∈ W K B − ˙ vP + K /P + K . Let Q K = { ( v, w ) ∈ W × W K | v w } . Define the partial order (cid:22) on Q K asfollows:( v ′ , w ′ ) (cid:22) ( v, w ) if there exists u ∈ W K such that v v ′ u w ′ u w. For any ( v, w ) ∈ Q K , set˚Π v,w = π K ( ˚ R v,w ) and Π v,w = π K ( R v,w ) , where π K : B → P K is the projection map. Then Π v,w is the (Zariski) closure of˚Π v,w in P K . We call ˚Π v,w an open projected Richardson variety and Π v,w a closedprojected Richardson variety . By [14, Proposition 3.6], we have P K = ⊔ ( v,w ) ∈ Q K ˚Π v,w and Π v,w = ⊔ ( v ′ ,w ′ ) ∈ Q K ;( v ′ ,w ′ ) (cid:22) ( v,w ) ˚Π v ′ ,w ′ . (3.1) Let K ⊂ I . The goal of the rest of this section is devoted to con-struct an Birkhoff-Bruhat atlas for the stratified space M = P K withthe stratification { ˚Π v,w } ( v,w ) ∈ Q K considered in § .3.3. The Kac-Moody group ˜ G . We construct the set of simple roots and theassociated generalized Cartan matrix of the Kac-Moody ˜ G from the original Kac-Moody group G . We list some examples of such construction in § I of simple roots is the union of two copies of I , glued along K . Moreprecisely, let I ♭ = { i ♭ | i ∈ I } and I ♯ = { i ♯ | i ∈ I } be the two copies of I . Then˜ I = I ♭ ∪ I ♯ with I ♭ ∩ I ♯ = { k ♭ = k ♯ | k ∈ K } . For any i ∈ I , we set ( i ♭ ) ♮ = ( i ♯ ) ♮ = i .The generalized Cartan matrix ˜ A = (˜ a ˜ i, ˜ i ′ ) ˜ i, ˜ i ′ ∈ ˜ I is defined as follows: • for ˜ i, ˜ i ′ ∈ I ♭ , ˜ a ˜ i, ˜ i ′ = a ˜ i ♮ , (˜ i ′ ) ♮ ; • for ˜ i, ˜ i ′ ∈ I ♯ , ˜ a ˜ i, ˜ i ′ = a ˜ i ♮ , (˜ i ′ ) ♮ ; • for ˜ i ∈ ˜ I − I ♭ and ˜ i ′ ∈ ˜ I − I ♯ , ˜ a ˜ i, ˜ i ′ = ˜ a ˜ i ′ , ˜ i = 0.Since I ♯ ∩ I ♭ = { k b = k ♯ | k ∈ K } , we have ˜ a k ♭ , ( k ′ ) ♭ = a k,k ′ = ˜ a k ♯ , ( k ′ ) ♯ and thegeneralized Cartan matrix ˜ A is well-defined. The generalized Cartan matrix ˜ A issymmetrizable.Let ˜ G min be the minimal Kac-Moody group of simply connected type associatedto ( ˜ I, ˜ A ) and ˜ W be its Weyl group. Let ˜ W I ♭ and ˜ W I ♯ be the parabolic subgroupof ˜ W generated by simple reflections in I ♭ and I ♯ respectively. We have naturalidentifications W → ˜ W I ♭ , w w ♭ and W → ˜ W I ♯ , w w ♯ . For w ∈ W K , w ♭ = w ♯ .For any w ∈ W , we set ( w ♭ ) ♮ = w and ( w ♯ ) ♮ = w .Similarly, we have natural embedding G min → ˜ L I ♭ , g g ♭ and G min → ˜ L I ♯ , g g ♯ . For g ∈ L K , g ♭ = g ♯ .We denote by ˜ X the flag variety of ˜ G min , and denote by I ♭ ˚˜ X − , I ♭ ˚˜ X − , I ♭ ˚˜ R − , − the I ♭ -Schubert cells, the opposite I ♭ -Schubert cells and the open I ♭ -Richardsonvariety respectively.3.4. The Kac-Moody group in a Birkhoff-Bruhat atlas.
Let r ∈ W . Thefollowing multiplication maps are isomorphisms of ind-varieties:( ˙ rU ˙ r − ∩ U + ) × ( ˙ rU ˙ r − ∩ U − ) −→ ˙ rU ˙ r − , ( g , g ) g g ; (3.2)( ˙ rU ˙ r − ∩ U − ) × ( ˙ rU ˙ r − ∩ U + ) −→ ˙ rU ˙ r − , ( h , h ) h h . (3.3)We define morphisms of ind-varieties σ r, − : ˙ rU ˙ r − → ˙ rU ˙ r − ∩ U − , g g g , and σ r, + : ˙ rU ˙ r − → ˙ rU ˙ r − ∩ U + , h h h . Lemma 3.1. [6, Proposition 8.2]
Let r ∈ W . The map σ r = ( σ r, + , σ r, − ) : ˙ rU − ˙ r − −→ ( U + ∩ ˙ rU − ˙ r − ) × ( U − ∩ ˙ rU − ˙ r − ) is an isomorphism of ind-varieties. Note that the isomorphism is compatible with Levi decompositions. The re-striction of σ r gives the isomorphism˙ rU P − K ˙ r − −→ ( U + ∩ ˙ rU P − K ˙ r − ) × ( U − ∩ ˙ rU P − K ˙ r − ) . Let r ∈ W K and U r = ˙ rB − P + K /P + K ⊂ P K . We have an isomorphism˙ rU P − K ˙ r −→ U r , g g ˙ rP + K /P + K . Finally, for g ∈ ˙ rU − ˙ r − , we write σ ♭/♯r, ± ( g ) for σ r, ± ( g ) ♭/♯ . Theorem 3.2.
For r ∈ W K , we define the map ˜ c r : U r −→ ˜ X,g ˙ rP + K /P + K σ ♭r, + ( g ) · ˙ r ♭ ( ˙ r − ) ♯ · σ ♯r, − ( g ) − · ˜ B + / ˜ B + for g ∈ ˙ rU P − K ˙ r − . Then (˜ c r ) r ∈ W K gives a Birkhoff-Bruhat atlas for P K .Proof. Since r ∈ W K , we have U + ∩ ˙ rU P − K ˙ r = U + ∩ ˙ rU − ˙ r .Therefore σ ♭r, + ( U P − K ) = ( U + ) ♭ ∩ ˙ r ♭ ( U − ) ♭ ( ˙ r − ) ♭ . On the other hand, we have˙ r ♭ ( ˙ r − ) ♯ · σ ♯r, − ( U P − K ) − · ˙ r ♯ ( ˙ r − ) ♭ = ˙ r ♭ (cid:16) ( ˙ r − ) ♯ ( U − ) ♯ ˙ r ♯ ∩ ( U P − K ) ♯ (cid:17) ( ˙ r − ) ♭ = ˙ r ♭ (cid:16) ( ˙ r − ) ♯ ( U − ) ♯ ˙ r ♯ ∩ ˜ U − I ♯ ∩ ˜ U − P − I♭ (cid:17) ( ˙ r − ) ♭ . By Proposition 2.10, ˜ c r is an embedding with image I ♭ ˚˜ X ˜ ν ( r,r ) ∩ ˜ L I ♭ ˜ L I ♯ ˜ B + / ˜ B + .We then check the stratifications. Let ( v, w ) ∈ Q K . Suppose that g ∈ ˙ rU P − K ˙ r − with g ˙ rP + K /P + K ∈ ˚Π v,w . Then there exists l ∈ L J such that g ˙ rl ∈ B + ˙ wB + ∩ B − ˙ vB + . By (3.2) and (3.3), we have σ r, + ( g ) ˙ rl ∈ B − g ˙ rl ⊂ B − ˙ vB + and σ r, − ( g ) ˙ rl ∈ B + g ˙ rl ⊂ B + ˙ wB + . Therefore σ ♭r, + ( g ) · ˙ r ♭ ( ˙ r − ) ♯ · σ ♯r, − ( g ) − = ( σ r, + ( g ) ˙ r ) ♭ · (cid:0) ( σ r, − ( g ) ˙ r ) − (cid:1) ♯ = ( σ r, + ( g ) ˙ rl ) ♭ · (cid:0) ( σ r, − ( g ) ˙ rl ) − (cid:1) ♯ ∈ ( U − ) ♭ · ˙ v ♭ · ( B + ) ♭ ( B + ) ♯ · ( ˙ w − ) ♯ · ( B + ) ♯ ⊂ ( U − ) ♭ ˙ v ♭ · ˜ U P + I♭ ∩ I♯ ˜ U + I ♭ ∩ I ♯ · ( ˙ w − ) ♯ ( B + ) ♯ ( ♥ ) ⊂ I ♭ B + ˙ v ♭ ( ˙ w − ) ♯ ( B + ) ♯ = I ♭ ˚˜ X ˜ ν ( v,w ) , where ( ♥ ) follows from˙ v ♭ ˜ U P + I♭ ∩ I♯ = ˜ U P + I♭ ∩ I♯ ˙ v ♭ and ˜ U + I ♭ ∩ I ♯ ( ˙ w − ) ♯ = ( ˙ w − ) ♯ ˜ U + I ♭ ∩ I ♯ . By Proposition 2.9,˜ c r ( U r ) = ∪ ( v,w ) ∈ Q J ˜ c r ( U r ∩ ˚Π v,w ) ⊂ ∪ ( v,w ) ∈ Q J I ♭ ˚˜ R r ♭ ( r − ) ♯ ,v ♭ ( w − ) ♯ ⊂ I ♭ ˚˜ X ˜ ν ( r,r ) ∩ ˜ L I ♭ ˜ L I ♯ ˜ B + / ˜ B + . Since ˜ c r ( U r ) = I ♭ ˚˜ X ˜ ν ( r,r ) ∩ ˜ L I ♭ ˜ L I ♯ ˜ B + / ˜ B + , all the inclusion above are actuallyequalities. In particular, for any ( v, w ) ∈ Q J , ˜ c r ( U r ∩ ˚Π v,w ) = I ♭ ˚˜ R r ♭ ( r − ) ♯ ,v ♭ ( w − ) ♯ .The theorem is proved. (cid:3) Combinatorial atlas
In this section, we assume W is an arbitrary Coxeter group and we discuss acombinatorial analog of the geometric “atlas model” in Section 3.4.1. Posets.
Let Q be a poset with partial order . For any x, y ∈ Q , let [ x, y ] = { z ∈ Q | x z y } be the interval from x to y . The covering relation is denote by ⋗ . In other words, y ⋗ x if [ x, y ] = { x, y } . For any x, y ∈ Q with x y , a maximalchain from x to y is a finite sequence of elements y = w ⋗ w ⋗ · · · ⋗ w n = x forsome n ∈ N and w , w , . . . , w n ∈ Q . The number n is called the length of thechain. Note that the maximal chain may not exists in general.We say that Q is pure if for any x, y ∈ Q with x y , the maximal chains from x to y always exist and have the same length. Such length is also called the lengthof the interval [ x, y ]. A pure poset Q is called thin if every interval of length 2 hasexactly 4 elements, i.e. has exactly two elements between x and y .A subset C ⊂ Q is called convex if for any x, y ∈ C , we have [ x, y ] ⊂ C .4.2. EL-Shellability.
Now we recall the notion of EL-shellability introduced byBjorner in [2].Suppose that the poset Q is pure. An edge labeling of Q is a map λ from theset of all covering relations in Q to a poset Λ. The labeling λ sends any maximalchain of an interval of Q to a tuple of Λ. A maximal chain is called increasing if the associated tuple of Λ is increasing. The edge labeling λ also allows one toorder the maximal chains of any interval of Q by ordering the corresponding tupleslexicographically.An edge labeling of Q is called EL-shellable if for every interval, there exists aunique increasing maximal chain, and all the other maximal chains of this intervalare less than this maximal chain (with respect to the lexicographical order).For a poset Q , we define the augmented poset ˆ Q = Q ⊔ { ˆ0 } , where ˆ0 is theminimal element of ˆ Q .Thinness and EL-shellability are important combinatorial properties of posets.For example, Bjorner proved in [3] that if a finite poset ˆ Q is thin and EL-shellable,then it is the face poset of some regular CW complex homeomorphic to a sphere.The main result of this section is the following. Theorem 4.1.
The poset ˆ Q K is thin and EL-shellable. The strategy of the proof is as follows: we first establish a combinatorial atlasmodel for the poset Q K ; then we prove ˆ Q K is thin in § Q K isEL-shellable in § W is a finite Weyl group was first established by Williams in [21,Theorem 1 & Theorem 2]. Another proof for Q K when W is a finite Weyl groupwas given in [8]. Our approach to handle Q K are different. As to the augmentedelement ˆ0, we follow a similar idea as in [21]. Combinatorial atlas.
Let K ⊂ I . Recall Q K = { ( v, w ) ∈ W × W K | v w } equipped with the partial order (cid:22) defined by( v ′ , w ′ ) (cid:22) ( v, w ) if there exists u ∈ W K such that v v ′ u w ′ u w. The construction of ˜ W in § W . Proposition 4.2.
Define the map ˜ ν : Q K −→ ˜ W , ( v, w ) ( v ) ♭ ( w − ) ♯ . Then(1) the map ˜ ν induces an isomorphism of the posets ( Q K , (cid:22) ) and (˜ ν ( Q K ) , I ♭ ) ;(2) ˜ ν ( Q K ) = { ˜ w ∈ ˜ W I ♭ ˜ W I ♯ | r ♭ ( r − ) ♯ I ♭ ˜ w for some r ∈ W K } ;(3) ˜ ν ( Q K ) is a convex subset of the poset ( ˜ W , I ♭ ) . We shall first give a geometric proof of the proposition in § W is aWeyl group of some Kac-Moody group G . We then give a combinatorial proof ofthe proposition in § W .4.4. Geometric proof of Proposition 4.2.
In this subsection, we assume that W is a Weyl group of some Kac-Moody group G . We deduce Proposition 4.2 forsuch W as a consequence of Theorem 3.2.Recall § Q K , (cid:22) ) is the index set of the stratification of P K into the unions of projected Richardson varieties. By (2.4), the poset ( ˜ W , I ♭ ) isthe index set of the stratification of ˜ X into the unions of the I ♭ -Schuber varieties.Proposition 4.2 (1) follows directly from Theorem 3.2.We show Proposition 4.2 (2). By Proposition 2.4, we have that c r ( U r ) = G ˜ w ∈ ˜ WI♭ ˜ WI♯ , r ♭ ( r − ) ♯ I♭ ˜ wI ♭ ˚˜ X ˜ ν ( r,r ) ∩ I ♭ ˚˜ X ˜ w and for any ˜ w ∈ ˜ W I ♭ ˜ W I ♯ with r ♭ ( r − ) ♯ I ♭ ˜ w , I ♭ ˚˜ X ˜ ν ( r,r ) ∩ I ♭ ˚˜ X ˜ w = ∅ . Hence˜ ν ( { ( v, w ) ∈ Q K | U r ∩ ˚Π v,w = ∅} ) = { ˜ w ∈ ˜ W I ♭ ˜ W I ♯ | r ♭ ( r − ) ♯ I ♭ ˜ w } . We then show Proposition 4.2 (3). Let x, y ∈ ˜ ν ( Q K ) with r ♭ ( r − ) ♯ I ♭ x forsome r ∈ W K . Since ˜ P + I ♭ ˜ P + I ♯ / ˜ B + is closed in ˜ G/ ˜ B + , we see that ˜ W I ♭ ˜ W I ♯ is closedin ˜ W under the partial order I ♭ thanks to Proposition 2.8. Then for any z ∈ [ x, y ], since z I ♭ y , we have z ∈ ˜ W I ♭ ˜ W I ♯ . We also have r ♭ ( r − ) ♯ I ♭ x I ♭ z , hence z ∈ ˜ ν ( Q K ).Proposition 4.2 then follows for such W .4.5. The partial order J . Let W be an arbitrary Coxeter group and J bea subset of the set of simple reflections in W . In this subsection, we give anequivalent description of the partial order J defined in § Lemma 4.3.
Let x ∈ W J and y ∈ W J . The partial order J is generated by(a) s β xy − J Recall Ψ J = ∆ re, − J ⊔ (∆ re, + − ∆ re, + J ). Let β ∈ Ψ J with s β xy − J xy − ,hence yx − ( β ) ∈ ∆ re, − .If β ∈ ∆ re, − J , then x − ( β ) ∈ ∆ reJ . Since y ∈ W J , yx − ( β ) ∈ ∆ re, − is equivalentto x − ( β ) ∈ ∆ re, − J . In this case, we equivalently have s β x > x .If β ∈ ∆ re, + − ∆ re, + J , then x − ( β ) ∈ ∆ re, + − ∆ re, + J . Set β ′ = x − ( β ). Then y ( β ′ ) ∈ ∆ re, − . In this case, s β ′ y − < y − . On the other hand, if s β ′ y − < y − forsome β ′ ∈ ∆ re, + , then we must have β ′ / ∈ ∆ re, + J since y ∈ W J . (cid:3) Corollary 4.4. Let x ∈ W J and y ∈ W J . The partial order J is generated by(a ′ ) x y − J Let x, x ′ , u ∈ W such that x x ′ u . Then thereexists u ′ u , such that x ( u ′ ) − x ′ .Proof. In the notations of [9, Lemma A.3], Lemma 4.5 is equivalent to x ⊳ u − x ′ if and only if x x ′ ∗ u. The lemma follows. (cid:3) Proposition 4.6. Let x, x ′ ∈ W J and y, y ′ ∈ W J . The following conditions areequivalent:(1) x ′ ( y ′ ) − J xy − ;(2) There exists u ∈ W J such that x x ′ u and y ′ u y .Proof. It suffices to replace condition (1) with the generating relations ( a ′ ) and( b ′ ) considered in Corollary 4.4.Note that Corollary 4.4 ( a ′ ) ⇒ (2) is trivial. For Corollary 4.4 ( b ′ ), we write y as y ′ u for y ′ ∈ W J and u ∈ W J . Then xy − = ( xu − )( y ′ ) − . And we have( xu − ) u = x and y ′ u y . So (1) ⇒ (2) is proved.Now we show that (2) ⇒ (1). Suppose that there exists u ∈ W J such that x x ′ u and y ′ u y . By Lemma 4.5, there exists u ′ u such that x ( u ′ ) − x ′ .Since y ′ ∈ W J , y ′ u ′ y ′ u y . By Corollary 4.4, we have x ′ ( y ′ ) − J x ( u ′ ) − ( y ′ ) − = x ( y ′ u ′ ) − J xy − . This finishes the proof. (cid:3) Combinatorial proof of Proposition 4.2. We show (1). For w ∈ W K ,we have w − ∈ K W and hence ( w − ) ♯ ∈ I ♭ ˜ W . It is easy to see that ˜ ν is injective.We prove the compatibility of the partial orders. Let( v, w ) , ( v ′ , w ′ ) ∈ Q K . By Proposition 4.6, ( v ′ ) ♭ (( w ′ ) − ) ♯ I ♭ ( v ) ♭ ( w − ) ♯ if and only if there exits z ∈ ˜ W I ♭ such that ( v ) ♭ ( v ′ ) ♭ z and z − (( w ′ ) − ) ♯ ( w − ) ♯ .Note that (( w ′ ) − ) ♯ , ( w − ) ♯ ∈ ˜ W I ♯ . Thus z − (( w ′ ) − ) ♯ ( w − ) ♯ implies that z ∈ ˜ W I ♯ ∩ ˜ W I ♭ . In this case, z ♮ ∈ W K . Hence v ♭ ( v ′ ) ♭ z ⇐⇒ v v ′ z ♮ ,z − (( w ′ ) − ) ♯ ( w − ) ♯ ⇐⇒ w ′ z ♮ w. So ( v ′ ) ♭ (( w ′ ) − ) ♯ I ♭ v ♭ ( w − ) ♯ if and only if ( v ′ , w ′ ) (cid:22) ( v, w ). We show (2). For any ( v, w ) ∈ Q K , we have ( w, w ) (cid:22) ( v, w ). Therefore we have w ♭ ( w − ) ♯ I ♭ ˜ ν (( v, w )) for w ∈ W K . Hence˜ ν ( Q K ) ⊂ { ˜ w ∈ ˜ W I ♭ ˜ W I ♯ | r ♭ ( r − ) ♯ I ♭ ˜ w for some r ∈ W K } . Let ˜ w ∈ ˜ W I ♭ ˜ W I ♯ with r ♭ ( r − ) ♯ I ♭ ˜ w for some r ∈ W K . We write ˜ w = xy − with x ∈ ˜ W I ♭ , y ∈ ˜ W I ♯ ∩ ˜ W I ♭ . By Proposition 4.6, there exists u ∈ W I ♭ such that x r ♭ u and r ♯ u y . Therefore we must have u ∈ ˜ W I ♭ ∩ ˜ W I ♯ , that is, u ♮ ∈ W K .So x ♮ ru ♮ y ♮ . Hence we have ( x ♮ , y ♮ ) ∈ Q K and ˜ ν (( x ♮ , y ♮ )) = ˜ w . This showsthat ˜ ν ( Q K ) ⊃ { ˜ w ∈ ˜ W I ♭ ˜ W I ♯ | r ♭ ( r − ) ♯ I ♭ ˜ w for some r ∈ W K } . We show (3). By (2), it suffices to show the set ˜ W I ♭ ˜ W I ♯ is closed in ( ˜ W , I ♭ ).Then the argument in the last paragraph of § w I ♭ xy − with x ∈ ˜ W I ♭ , y ∈ ˜ W I ♯ ∩ ˜ W I ♭ . We write ˜ w = x ′ ( y ′ ) − with x ′ ∈ ˜ W I ♭ , y ′ ∈ ˜ W I ♭ . ByProposition 4.6, there exists u ∈ W I ♭ such that x x ′ u and y ′ u y . Then y ′ y and hence y ′ ∈ ˜ W I ♯ . So ˜ w ∈ ˜ W I ♭ ˜ W I ♯ .4.7. Thinness. Dyer proved in [5, Proposition 2.5] that the poset ( ˜ W , I ♭ ) isthin. We have shown in Proposition 4.2 that the poset ( Q K , (cid:22) ) can be identifiedwith a convex subset of the poset ( ˜ W , I ♭ ). So Q K is thin.As to the rank 2 intervals involving ˆ0, one can follow the same proof (the lastparagraph) as in [21, Proof of Theorem 1.1]. We shall omit the details here.4.8. Reflection orders. In order to prove the EL-shellability, we first recall thereflection orders introduced by Dyer in [4, Definition 2.1]. Let T be the set ofreflections in ˜ W . A total order (cid:22) on T is called a reflection order if for any s, t ∈ T , either s ≺ tst ≺ tstst ≺ · · · or t ≺ sts ≺ ststs ≺ · · · .For any covering relation w ⋗ w ′ , we label this edge by the reflection w ( w ′ ) − ∈ T .Dyer proved in [5, Proposition 3.9] (taking I = ∅ and J = S in loc.cit. ) that anyreflection order on T gives an EL-labelling on ( ˜ W , I ♭ ). In particular, the poset( ˜ W , I ♭ ) is EL-shellable.Now we take the reflection order (cid:22) on T such that ˜ W I ♭ ∩ T is a final section,i.e., for any t, t ′ ∈ T with t ∈ ˜ W I ♭ and t ′ / ∈ ˜ W I ♭ , we have t ′ ≺ t . The existence ofsuch reflection order was established by Dyer in [4, Proposition 2.3]. We define theaugmented totally order set ˆ T = T ⊔ {⊥} where a ≺ ⊥ ≺ b for any a ∈ T − ˜ W I ♭ and b ∈ ˜ W I ♭ ∩ T .Recall we can identify Q K with a convex subset of ˜ W . The edge labelling on˜ W defined above induces an edge labelling on Q K . For any edge of ˆ Q K involvingˆ0, we label the edge by ⊥ ∈ ˆ T . This gives a labelling on all the edges in ˆ Q K .4.9. EL-shellability. In this subsection, we show that the edge labelling on ˆ Q K defined above is an EL-labelling. We have already seen that the edge labellingabove on ( ˜ W , I ♭ ) is an EL-labelling.Let [ x, y ] be an interval in ˆ Q K .We first consider the case x = ˆ0. By Proposition 4.2, the poset Q K can beidentified with a convex subset of ( ˜ W , I ♭ ). Then ˜ ν ([ x, y ]) is an interval in ˜ W .The claim follows in this case. Now we consider the interval [ˆ0 , y ]. Similar to the argument in the last twoparagraphs of [21, Proof of Theorem 1.2], for any ( v, w ) ∈ Q K , any lexicographi-cally minimal chain from ( v, w ) to ˆ0 does not involve edges labelled by ˜ W I ♭ ∩ T andmust have ( r, r ) ≻ ˆ0 as the last two terms, where r = min( vW J ). The claim for theinterval [ˆ0 , ( v, w )] in ˆ Q K now follows from the claim of the interval [( r, r ) , ( v, w )]in ( Q k , (cid:22) ) proved above.4.10. Comparison with the work [21] . It is worth pointing out that the edgeslabelled by T ∩ ˜ W I ♭ and ⊥ corresponds to the edges of type 2 and type 3 in thesense of [21, Corollary 6.4] respectively. Moreover, we may choose a reflectionorder on T so that its restriction to T ∩ ˜ W I ♭ matches with the order used for thetype 2 edges in [21]. However, it is not clear how to match T \ ˜ W I ♭ with the labelingon the type 1 edges in [21].5. A different Birkhoff-Bruhat atlas for W K finite case In this section we construction a different Birkhoff-Bruhat atlas for P K when K is of finite type. Let w K be the longest element of the finite Weyl group W K .We first construct the atlas group ˘ G . We list some examples of such constructionin § I be the union of two copies of I , glued along the ( − w K )-graph automor-phism of K . More precisely, let I ♭ and I ♯ be the two copies of I . Then ˘ I = I ♭ ∪ I ♯ with I ♭ ∩ I ♯ = { j ♭ ; j ∈ K } = { j ♯ ; j ∈ K } . For j, j ′ ∈ K , j ♭ = ( j ′ ) ♯ if and only if α j ′ = − w K ( α j ). Define ♮ : I ♭ → I, ( i ♭ ) ♮ = i and ♮ ′ : I ♯ → I, ( i ♯ ) ♮ ′ = i . Then for˘ i ∈ I ♭ ∩ I ♯ , ˘ i ♮ and ˘ i ♮ ′ are both contained in K and α ˘ i ♮ ′ = − w K ( α ˘ i ♮ ).The generalized Cartan matrix ˘ A = (˘ a ˘ i, ˘ i ′ ) ˘ i, ˘ i ′ ∈ ˘ I is defined as follows: • for ˘ i, ˘ i ′ ∈ I ♭ , ˘ a ˘ i, ˘ i ′ = a ˘ i ♮ , (˘ i ′ ) ♮ ; • for ˘ i, ˘ i ′ ∈ I ♯ , ˘ a ˘ i, ˘ i ′ = a ˘ i ♮ , (˘ i ′ ) ♮ ; • for ˘ i ∈ ˘ I − I ♭ and ˘ i ′ ∈ ˘ I − I ♯ , ˘ a ˘ i, ˘ i ′ = ˘ a ˘ i ′ , ˘ i = 0.Note that for j , j ′ , j , j ′ ∈ K with α j ′ = − w K ( α j ) and α j ′ = − w K ( α j ), wehave a j ,j = a j ′ ,j ′ . Thus for ˘ i, ˘ i ′ ∈ I ♭ ∩ I ♯ , we have a ˘ i ♮ , (˘ i ′ ) ♮ = a ˘ i ♮ ′ , (˘ i ′ ) ♮ ′ and thegeneralized Cartan matrix ˘ A is well-defined.Let ˘ G be the minimal Kac-Moody group of simply connected type associatedto ( ˘ I, ˘ A ) and ˘ W be its Weyl group. Let ˘ W I ♭ and ˘ W I ♯ be the parabolic subgroupof ˘ W generated by simple reflections in I ♭ and I ♯ respectively. We have naturalidentifications W → ˘ W I ♭ , w w ♭ and W → ˘ W I ♯ , w w ♯ . For w ∈ W J , w ♭ = w J w ♯ w − J . For any w ∈ W , we set ( w ♭ ) ♮ = w and ( w ♯ ) ♮ ′ = w .Similarly to Section 3, we have natural embeddings G → ˘ L I ♭ , g g ♭ and G → ˘ L I ♯ , g g ♯ . Lemma 5.1. Let L K, der be the derived group of L K . Then for any g ∈ L K, der , ( g ♭ ) − = Ψ( ˙ w K g ˙ w − K ) ♯ . Proof. Since g ∈ L K, der , it suffices to prove that for i ∈ K ,(( x α i ( a )) ♭ ) − = Ψ( ˙ w K x α i ( a ) ˙ w − K ) ♯ , (( y α i ( a )) ♭ ) − = Ψ( ˙ w K y α i ( a ) ˙ w − K ) ♯ . We prove the first identity. The second identity is proved in the same way.Let j ∈ K with w K ( α i ) = − α j . Then ( s − j w K )( α i ) = α j . It follows from [19,Proposition 9.3.5] and direct computations that˙ w K x α i ( a ) ˙ w − K = ˙ s j ( ˙ s − j ˙ w K ) x α i ( a )( ˙ s − j ˙ w K ) − ˙ s − j = ˙ s j x α j ( a ) ˙ s − j = y α j ( − a ) . The lemma is proved. (cid:3) We denote by ˘ B the flag variety of ˘ G , and denote by I ♭ ˚˘ X − , I ♭ ˚˘ X − , I ♭ ˚˘ R − , − the I ♭ -Schubert cells, the opposite I ♭ -Schubert cells and the open I ♭ -Richardson varietiesin ˘ B respectively. Theorem 5.2. Suppose that W K is finite. For any r ∈ W K , define the map ˘ c r : U r −→ ˘ X,g ˙ rP + K /P + K σ ♭r, − ( g ) · ˙ r ♭ ( ˙ w K ) ♭ ( ˙ r − ) ♯ · Ψ( σ ♯r, + ( g )) · ˘ B + / ˘ B + for g ∈ ˙ rU P − K ˙ r − . Then (˘ c r ) r ∈ W K gives a Birkhoff-Bruhat atlas for P K .Proof. We have σ ♭r, − ( ˙ rU P − K ˙ r − ) = ( U − ) ♭ ∩ ˙ r ♭ ( U P − K ) ♭ ( ˙ r − ) ♭ = ( U − ) ♭ ∩ ( ˙ r ˙ w K ) ♭ ( U − ) ♭ ( ˙ w K ˙ r − ) ♭ and ˙ r ♭ ( ˙ w K ) ♭ ( ˙ r − ) ♯ · ι (cid:16) σ ♯r, + ( ˙ rU P − K ˙ r − ) (cid:17) − · ˙ r ♯ ( ˙ w K ) ♭ ( ˙ r − ) ♭ = ˙ r ♭ ( ˙ w K ) ♭ ( ˙ r − ) ♯ · (cid:16) ( U − ) ♯ ∩ ( ˙ rU P + K ˙ r − ) ♯ (cid:17) · ˙ r ♯ ( ˙ w K ) ♭ ( ˙ r − ) ♭ = ˙ r ♭ ( ˙ w K ) ♭ · (cid:16) ( ˙ r − U − ˙ r ) ♯ ∩ ( U P + K ) ♯ (cid:17) · ( ˙ w K ) ♭ ( ˙ r − ) ♭ . Now it follows from Corollary 2.6 that ˘ c r is an embedding with image I ♭ ˚˘ X ˘ ν ( r,r ) .Let ( v, w ) ∈ Q K . Suppose that g ∈ ˙ rU P − K ˙ r − with g ˙ rP + K /P + K ∈ ˚Π v,w . Thenthere exists l ∈ L K, der such that g ˙ rl ∈ B + ˙ wB + ∩ B − ˙ vU P + K .By (3.2) and (3.3), we have σ r, + ( g ) ˙ rl ∈ B − g ˙ rl ⊂ B − ˙ vU P + K , σ r, − ( g ) ˙ rl ∈ B + g ˙ rl ⊂ B + ˙ wB + . Set l ′ = ˙ w − K l ˙ w K ∈ L K, der . Then (( l ′ ) ♭ ) − = Ψ( ˙ w K l ′ ˙ w − K ) = Ψ( l ) ♯ by Lemma 5.1.So σ ♭r, − ( g ) · ˙ r ♭ ( ˙ w K ) ♭ ( ˙ r − ) ♯ · Ψ( σ ♯r, + ( g )) · ˘ B + / ˘ B + = ( σ ♭r, − ( g ) ˙ r ˙ w K l ′ ) ♭ · (cid:0) Ψ( σ ♯r, + ( g ) ˙ rl ) (cid:1) ♯ · ˘ B + / ˘ B + = ( σ ♭r, − ( g ) ˙ rl ˙ w K ) ♭ · (cid:0) Ψ( σ ♯r, + ( g ) ˙ rl ) (cid:1) ♯ · ˘ B + / ˘ B + ∈ ( U + ) ♭ · ˙ w ♭ · ( U + ) ♭ · ( ˙ w K ) ♭ · ( U P − K ) ♯ · ( ˙ v − ) ♯ · ( U + ) ♯ · ˘ B + / ˘ B + = ( U + ) ♭ · ˙ w ♭ · ( U P + K ) ♭ · ( ˙ w K ) ♭ · ( U P − K ) ♯ · ( ˙ v − ) ♯ · ( U + ) ♯ · ˘ B + / ˘ B + ⊂ ( U + ) ♭ · ˙ w ♭ · U P − I♭ · ( ˙ w K ) ♭ · U P + I♯ · ( ˙ v − ) ♯ · ( U + ) ♯ · ˘ B + / ˘ B + = I ♭ B − · ˙ w ♭ · ( ˙ w K ) ♭ · ( ˙ v − ) ♯ · · ˘ B + / ˘ B + . By Proposition 2.8,˘ c r ( U r ) = ∪ ( v,w ) ∈ Q K ; v w ˘ c r ( U r ∩ ˚Π v,w ) ⊂ ∪ ( v,w ) ∈ Q K ; v wI ♭ ˚˘ R ( ww J ) ♭ ( v − ) ♯ , ( rw J ) ♭ ( r − ) ♯ ⊂ I ♭ ˚˘ X ˘ ν ( r,r ) . Since ˘ c r ( U r ) = I ♭ ˚˘ X ˘ ν ( r,r ) , all the inclusions above are actually equalities. Inparticular, we have˘ c r ( U r ∩ ˚Π v,w ) = I ♭ ˚˘ R ( ww J ) ♭ ( v − ) ♯ , ( rw J ) ♭ ( r − ) ♯ , for ( v, w ) ∈ Q K . The theorem is proved. (cid:3) In the case where W is finite, the multiplication by ˙ w I ♭ interchanges the oppositeSchubert cells with the I ♭ -Schubert cells, where w I ♭ is the longest element of W I ♭ .Thus for W finite case, the “atlas model” constructed here is essentially the sameas the Bruhat atlas constructed in [7]. They differ by the multiplication by ˙ w I ♭ .6. Some examples P K ˜ G ˘ G Bruhat atlas ◦ ◦ ▽ ▽▽ ▽ ▽ ▽▽ ▽ ▽ ▽▽ ▽ ◦ • ▽ (cid:7)▽ ▽ (cid:7) ▽ ▽ (cid:7) ▽ ◦ • ◦ ▽ ▽(cid:7)▽ ▽ ▽ ▽(cid:7)▽ ▽ ▽ ▽(cid:7)▽ ▽ • ◦ • ▽(cid:7) (cid:7)▽ ▽(cid:7) (cid:7)▽ ▽(cid:7) (cid:7)▽ ◦ • • ▽ (cid:7) (cid:7)▽ ▽ (cid:7) (cid:7) ▽ ▽ (cid:7) (cid:7) ▽ ◦ • • ∞ ▽ (cid:7) (cid:7)▽ ∞∞ ▽ (cid:7) (cid:7) ▽ ∞ ∞ N/A • • ◦ ∞ ▽(cid:7) (cid:7) ▽ ∞ N/A N/A In this section, we provide some examples of the Dynkin diagrams of the atlasgroups. Here ˜ G and ˘ G are the atlas groups constructed in § § G and ˘ G are the same, the atlases constructed in § § G . The subset K is the set of vertices filled with black color. In the other columns, the Dynkindiagrams consists of three types of vertices: △ , the vertices in ( I − J ) ♯ ; ▽ , thevertices in ( I − J ) ♭ ; (cid:7) , the vertices in I ♯ ∩ I ♭ = J ♯ = J ♭ .We also would like to point out that in certain cases, the atlas groups ˜ G and˘ G constructed in section 3 and section 5 coincide. However, the Birkhoff-Bruhatatlases constructed there are still quite different in these cases, as one may seefrom the maps ˜ c r and ˘ c r in Theorem 3.2 and Theorem 5.2 respectively. References [1] Y. Billig and M. J. Dyer, Decompositions of Bruhat type for the Kac-Moody groups , NovaJ. Algebra Geom. (1994), no. 1, 11–39.[2] A. Bjorner, Shellable and Cohen-Macaulay partially ordered sets , Trans. Amer. Math. Soc. (1980), 159– 183.[3] A. 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