Billey-Postnikov decompositions and the fibre bundle structure of Schubert varieties
aa r X i v : . [ m a t h . AG ] F e b BILLEY-POSTNIKOV DECOMPOSITIONS AND THE FIBRE BUNDLESTRUCTURE OF SCHUBERT VARIETIES
EDWARD RICHMOND AND WILLIAM SLOFSTRA
Abstract.
A theorem of Ryan and Wolper states that a type A Schubert variety issmooth if and only if it is an iterated fibre bundle of Grassmannians. We extend thistheorem to arbitrary finite type, showing that a Schubert variety in a generalized flagvariety is rationally smooth if and only if it is an iterated fibre bundle of rationallysmooth Grassmannian Schubert varieties. The proof depends on deep combinatorialresults of Billey-Postnikov on Weyl groups. We determine all smooth and rationallysmooth Grassmannian Schubert varieties, and give a new proof of Peterson’s theoremthat all simply-laced rationally smooth Schubert varieties are smooth. Taken together,our results give a fairly complete geometric description of smooth and rationally smoothSchubert varieties using primarily combinatorial methods. Introduction
Let G be a connected semisimple algebraic group over an algebraically closed field k ,and let X be a partial flag variety of type G . The variety X is stratified by Schubertvarieties X ( w ), which are indexed by minimal length coset representatives w in the Weylgroup W of G . When k = C and G = SL n ( C ) is the semisimple group of type A n − , thepartial flag varieties parametrize flags of subspaces in C n . When X is the complete flagvariety in type A , Ryan proved that smooth Schubert varieties are iterated fibre bundlesof Grassmannians [Rya87]. This result was extended to the partial flag varieties of type A over any algebraically closed field k of characteristic zero by Wolper [Wol89]. The mainresult of this paper is an extension of Ryan and Wolper’s theorems to flag varieties of allfinite types. We consider both the class of smooth Schubert varieties, and the larger classof rationally smooth Schubert varieties, proving the following result: a finite type Schubertvariety is (rationally) smooth if and only if it is an iterated fibre bundle of (rationally)smooth Grassmannian Schubert varieties.In type A , every smooth Grassmannian Schubert variety is a sub-Grassmannian [BL00,Theorem 9.3.1], and thus we recover Ryan and Wolper’s theorem from our results. However,in other types there are (rationally) smooth Grassmannian Schubert varieties which are notsub-Grassmannians. Fortunately, the singular locus of these Schubert varieties has beenextensively studied [LW90] [BP99]; a summary can be found in [BL00]. More recently,smooth Grassmannian Schubert varieties have been studied in the context of homologicalrigidity [Rob14] [HM13]. From this work, the list of smooth Schubert varieties is knownfor many generalized Grassmannians. As part of our results, we finish this line of inquiryby giving a complete list of both smooth and rationally smooth Grassmannian Schubertvarieties in all finite types. When combined with our extension of Ryan and Wolper’stheorem, this gives a fairly complete geometric description of (rationally) smooth Schubertvarieties in any finite type.There are several well known characterizations of (rationally) smooth Schubert varieties.If P w ( t ) is the Poincar´e polynomial of X ( w ), then a theorem of Carrell-Peterson states that X ( w ) is rationally smooth if and only if P w ( t ) is a palindromic polynomial [Car94]. Anothercharacterization says that X ( w ) is rationally smooth if and only if the Weyl group element w has trivial Kazhdan-Lusztig polynomials. For type A Schubert varieties, the Lakshmibai-Sandya theorem [LS90] states that a Schubert variety X ( w ) is smooth if and only if w , as apermutation, avoids 3412 and 4231. This pattern-avoidance criterion has been extended toclassical types by Billey [Bil98] and to all finite types using root-system pattern avoidanceby Billey-Postnikov [BP05]. These pattern avoidance criteria provide an efficient way totest if a given Schubert variety X ( w ) is smooth or rationally smooth. In contrast, ourresults provide information about the geometric structure of (rationally) smooth Schubertvarieties. For instance, Peterson’s theorem, proved in full generality by Carrell-Kuttler,states that if G is simply-laced then X ( w ) is rationally smooth if and only if it is smooth[CK03]. Other proofs of Peterson’s theorem have been given in [Dye] and [JW14]. Using ourresults, we give a new proof of Peterson’s theorem; compared to the proofs listed above, ourproof is more combinatorial. Using the list of (rationally) smooth Grassmannian Schubertvarieties, we can also efficiently generate the list of (rationally) smooth Schubert varietiesin any partial flag variety. This method can be extended to enumerate (rationally) smoothSchubert varieties in complete flag varieties; since this requires a new combinatorial datastructure, we leave this to another paper.The central idea behind our results is the notion of a Billey-Postnikov decomposition.To explain this, let X J denote the partial flag variety associated to a subset J of the simplegenerators S of W . If J ⊆ K ⊆ S , then there is a natural projection π : X J → X K . If X J ( w ) is a Schubert variety of X J and w = vu is the parabolic decomposition of w withrespect to K , then the restriction of π to X J ( w ) gives a projection(1) π : X J ( w ) → X K ( v )with generic fibre X J ( u ). While π : X J → X K is a fibre bundle, the restriction in equation(1) is not a fibre bundle in general (see, for instance, example 3.4). If this projection isa fibre bundle, then the Leray-Hirsch theorem states that the singular cohomology ring H ∗ ( X J ( w )) is a free H ∗ ( X K ( v ))-module over H ∗ ( X J ( u )). As a consequence, if P Jw ( t )denotes the Poincar´e polynomial of X J ( w ), then(2) P Jw ( t ) = P Kv ( t ) · P Ju ( t ) . We prove that equation (2) is also a sufficient condition for the projection in equation (1)to be a fibre bundle. This result is stated in Theorem 3.3, and holds for Schubert varietiesof any Kac-Moody group.Factorizations of P Jw ( t ) have been studied by a number of authors, most notably byGasharov [Gas98], Billey [Bil98], and Billey-Postnikov [BP05]. Billey and Postnikov showthat if X ∅ ( w ) is a rationally smooth Schubert variety of finite type, then there exists asubset K of simple generators for which either w or w − has a parabolic decomposition vu with respect to K such that the Poincar´e polynomial factors as P ∅ w ( t ) = P Kv ( t ) · P ∅ u ( t )[BP05]. Moreover, K can be choosen such that S \ K = { s } for some leaf s of theDynkin diagram of G . In [OY10], Oh and Yoo call such a parabolic decomposition aBilley-Postnikov decomposition. In this paper, we say that a parabolic decomposition w = vu with respect to K is Billey-Postnikov (BP) if the polynomial P Jw ( t ) factors as inequation (2) (dropping the condition that S \ K = { s } for some leaf s ). When J = ∅ , thisagrees with the definition used by the authors in [RS14]. We then prove that if X J ( w )is rationally smooth and of finite type, then there exists a nontrivial K containing J forwhich w has a BP decomposition. This result is stated in Theorem 3.6. The proof uses ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 3 the existence theorem in [BP05] combined with an inductive argument. Note that wecannot apply the existence theorem for BP decompositions stated in [BP05] to constructfibre bundle structures on Schubert varieties directly. While the Bruhat intervals [ e, w ]and [ e, w − ] are order isomorphic, the Schubert varieties X ∅ ( w ) and X ∅ ( w − ) are notnecessarily isomorphic. Hence a fibre bundle structure on X ∅ ( w − ) may not yield a fibrebundle structure on X ∅ ( w ).1.1. Acknowledgements.
We would like to thank Dave Anderson, Sara Billey, Jim Car-rell, and Alex Woo for helpful discussions. We thank the anonymous referee for usefulsuggestions on the manuscript. The first author was partially supported by the NaturalSciences and Engineering Research Council of Canada.2.
Background and terminology
We use the notation from the introduction throughout the paper. In particular, wework over a fixed algebraically closed field k of arbitrary characteristic. Let G denote asemisimple algebraic group over k or, as long as k = C , a Kac-Moody group. When workingwith Kac-Moody groups, we take G to be the minimum Kac-Moody group G min as definedin [Kum02]. Fix a choice of maximal torus and Borel T ⊂ B ⊂ G . Let W = N ( T ) /T denotethe Weyl group of G and fix a simple generating set S for W . We choose a representative inthe normalizer N ( T ) of T for each element w ∈ W . If S = { s , s , . . . , s n } is the generatingset of a specific Weyl group of finite type, we use the Bourbaki labeling of Dynkin diagramsin [Bou68] to describe the Coxeter relations.For basic facts about Schubert varieties, we refer again to [Kum02], and in particularto the facts about Tits systems in [Kum02, section 5.1]. As in loc. cit., a (standard)parabolic subgroup of W is a subgroup W J generated by a subset J ⊂ S . If J ⊆ S , we let P J := BW J B be the parabolic subgroup of G corresponding to J . The partial flag varietyassociated to J is then X J := G/P J . Let W J ≃ W/W J denote the set of minimal lengthcoset representatives. If w ∈ W J , then the Schubert variety associated to w is the closure X J ( w ) := BwP J /P J . The dimension of X J ( w ) is ℓ ( w ), the Coxeter length of w withrespect to S . Note that for Kac-Moody groups G , the flag variety X J can be an infinite-dimensional ind-variety, but the Schubert varieties X J ( w ) are always finite-dimensional.If J ⊆ K ⊆ S , then every element w ∈ W J can be written uniquely as w = vu , where v ∈ W K and u ∈ W K ∩ W J . This is called the (right) parabolic decomposition of w withrespect to K . We say w = w · · · w k is a reduced decomposition if ℓ ( w ) = P i ℓ ( w i ). Notethat all parabolic decompositions are reduced.If ≤ denotes Bruhat order on W , then X J ( w ) = a x ∈ [ e,w ] ∩ W J BxP J /P J where [ e, w ] denotes the interval in Bruhat order between e and w . If W J is finite and w ∈ W J , then the preimage of X J ( w ) in G/B is X ∅ ( w ′ ), where w ′ is the maximal lengthrepresentative of the coset wW J . Alternatively, w ′ = wu , where u is the longest elementof W J . If w and w are in W J , then w ≤ w if and only if w ′ ≤ w ′ , where w ′ i is the longestelement in the coset w i W J . Geometrically, w ≤ w if and only if X J ( w ) ⊆ X J ( w ).The Poincar´e polynomial P Jw ( t ) of an element w ∈ W J is defined as P Jw ( t ) := X x ∈ [ e,w ] ∩ W J t ℓ ( x ) . EDWARD RICHMOND AND WILLIAM SLOFSTRA
It is well-known that P Jw ( t ) = P dim H i ( X J ( w )) t i , the Poincar´e polynomial for the co-homology of X J ( w ). The Poincar´e polynomial is palindromic if P Jw ( t ) = t ℓ ( w ) P Jw ( t − ), orin other words if the coefficients form a palindrome when ordered by degree.The support of an element w ∈ W is defined as S ( w ) := { s ∈ S | s ≤ w } . Equivalently, S ( w ) is the set of simple reflections appearing in any reduced decompositionof w . The left and right descents set of w are D L ( w ) := { s ∈ S | ℓ ( sw ) ≤ ℓ ( w ) } D R ( w ) := { s ∈ S | ℓ ( ws ) ≤ ℓ ( w ) } . A generalized Grassmannian is a flag variety X J where | S \ J | = 1. In other words, P J isa maximal parabolic subgroup of G where X J = G/P J . A Schubert variety X J ( w ) of ageneralized Grassmannian is called a Grassmannian Schubert variety . An element w ∈ W is called a Grassmannian element if w ∈ W J for some generalized Grassmannian X J .Equivalently, w ∈ W is Grassmannian if and only if w has a unique right descent. By a Grassmannian parabolic decomposition , we mean a parabolic decomposition w = vu withrespect to a set K such that | K ∩ S ( w ) | = | S ( w ) | − Main results
As stated in the introduction, we define Billey-Postnikov decompositions as follows:
Definition 3.1.
Let w ∈ W J , and let w = vu be a parabolic decomposition with respectto K , where J ⊆ K ⊆ S . We say that w = vu is a Billey-Postnikov (BP) decompositionwith respect to (
J, K ) if P Jw ( t ) = P Kv ( t ) · P Ju ( t ) . When J = ∅ , we simply say that w = vu is a BP decomposition with respect to K . Some elementary equivalent definitions of BP decompositions are given in Proposition4.2. In particular, checking whether or not a given parabolic decomposition is a BP de-composition is computationally easy, and does not require working with Bruhat order.
Example 3.2.
Let G = SL ( k ) , with Weyl group W generated by S = { s , s , s } . If J = { s , s } , then the parabolic decomposition w = vu = ( s s s )( s s ) is a BP decompositionwith respect to J since P ∅ w ( t ) = P Jv ( t ) · P ∅ u ( t ) = ( t + 2 t + t + 1)( t + 2 t + 1)= t + 4 t + 6 t + 5 t + 3 t + 1 . The parabolic decomposition w = vu = ( s s s )( s ) is not a BP decomposition with respectto J since P ∅ w ( t ) = t + 3 t + 4 t + 3 t + 1 and P Jv ( t ) · P ∅ u ( t ) = ( t + 2 t + t + 1)( t + 1) = t + 3 t + 5 t + 2 t + 1 . Our first main theorem is a geometric characterization of BP decompositions. Note thatthis theorem holds when G is a general Kac-Moody group. ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 5
Theorem 3.3.
Let w ∈ W J and w = vu be a parabolic decomposition with respect to K .Then the following are equivalent:(a) The decomposition w = vu is a BP decomposition with respect to ( J, K ) .(b) The projection π : X J ( w ) → X K ( v ) is Zariski-locally trivial with fibre X J ( u ) .Consequently, if w = vu is a BP decomposition then:(1) X J ( w ) is (rationally) smooth if and only if X J ( u ) and X K ( v ) are (rationally)smooth.(2) The projection π : X J ( w ) → X K ( v ) is smooth if and only if X J ( u ) is smooth. Example 3.4.
Let G = SL ( k ) . Geometrically, we have G/B = { V • = ( V ⊂ V ⊂ V ⊂ k ) | dim V i = i } . Let E • denote the flag corresponding to eB . If w = s s s s s , then X ∅ ( w ) = { V • | dim( V ∩ E ) ≥ } . If J = { s , s } , then π ( V • ) = V . From Example 3.2, w = vu = ( s s s )( s s ) is a BPdecomposition with respect to J . In particular, the Schubert variety X J ( v ) = { V | dim( V ∩ E ) ≥ } and the fibre over V in the projection π : X ∅ ( w ) → X J ( v ) is π − ( V ) = { ( V , V ) | V ⊂ V ⊂ V } ∼ = X ∅ ( u ) ∼ = P × P . In this case, the uniform fibre X ∅ ( u ) is smooth. However since X J ( v ) is singular, we havethat X ∅ ( w ) is singular. If J = { s , s } , then π ( V • ) = V and w = vu = ( s s s )( s s ) isnot a BP decomposition. The fiber over V is given by π − ( V ) = { ( V , V ) | V ⊂ V ⊂ V and dim( V ∩ E ) ≥ }∼ = ( X ∅ ( s s ) if dim( V ∩ E ) = 1 X ∅ ( s s s ) if E ⊂ V Note that the fibres are not equidimensional. If Y is a variety over k , let H ∗ ( Y ) denote either etale cohomology, or, when k = C ,singular cohomology. For etale cohomology, we take coefficients in Q l , where l is a primenot equal to the characteristic of k , while for singular cohomology we take coefficients in C . Corollary 3.5.
Let w = vu be a parabolic decomposition with respect to K . Then thefollowing are equivalent:(a) The decomposition w = vu is a BP decomposition with respect to ( J, K ) .(b) There is an isomorphism H ∗ ( X J ( w )) ∼ = H ∗ ( X K ( v )) ⊗ H ∗ ( X J ( u )) as H ∗ ( X K ( v )) -modules. It is well known that the Poincar´e polynomial P w ( t ) = P H i ( X J ( w )) t i , so the ( b ) ⇒ ( a )direction of Corollary 3.5 follows immediately from the definition. The ( a ) ⇒ ( b ) directionof Corollary 3.5 and Theorem 3.3 will be proved in Section 4.2.Our second main theorem concerns the existence of BP decompositions when G issemisimple, or equivalently when W is finite. EDWARD RICHMOND AND WILLIAM SLOFSTRA
Theorem 3.6.
Let w ∈ W J , where W is finite, and suppose | S ( w ) \ J | ≥ . If X J ( w ) is rationally smooth, then w has a Grassmannian BP decomposition with respect to ( J, K ) for some maximal proper K containing J. As an application of our main theorems, we get the following extension of the Ryan-Wolper theorem to arbitrary finite type.
Corollary 3.7.
Let w ∈ W J , where W is finite, and set m = | S ( v ) \ J | . Then X J ( w ) isrationally smooth if and only if there is a sequence (3) X J ( w ) = X → X → · · · → X m − → X m = Spec k, where each morphism is a Zariski locally-trivial fibre bundle, and the fibres are rationallysmooth Grassmannian Schubert varieties.Similarly, X J ( w ) is smooth if and only if there is a sequence as in (3) where all thefibres, or equivalently, all the morphisms, are smooth. Each projection X i → X i +1 in Corollary 3.7 corresponds to a BP decomposition. How-ever, these BP decompositions are not usually Grassmannian, since the fibre of the pro-jection is Grassmannian rather than the base. To deduce Corollary 3.7 from Theorem3.6, we start with the morphisms X i → X m − (which do correspond to Grassmannian BPdecompositions), and then apply a certain associativity property (stated in Lemma 4.3)for BP decompositions. Theorem 3.6 and Corollary 3.7 are proved in Section 6.To complete the description of rationally smooth Schubert varieties, we list all rationallysmooth Grassmannian Schubert varieties of finite type. Theorem 3.8.
Let W be a finite Weyl group. Suppose w ∈ W J for some J = S \ { s } ,and that S ( w ) = S . Then X J ( w ) is rationally smooth if and only if either(1) w is the maximal element of W J , in which case X J ( w ) is smooth.(2) w is one of the following elements: W s w index set X J ( w ) smooth? B n s s k s k +1 · · · s n s n − · · · s < k ≤ n no B n s k u n,k +1 s · · · s k < k < n no B n s n s . . . s n n ≥ yes C n s s k s k +1 · · · s n s n − · · · s < k ≤ n yes C n s k u n,k +1 s · · · s k < k < n yes C n s n s . . . s n n ≥ no F s s s s s n/a no F s s s s s s s s s s n/a no F s s s s s s s s s s n/a yes F s s s s s n/a yes G s s s , s s s , s s s s n/a no G s s s n/a yes G s s s s , s s s s n/a noThe simple generators { s i } are the simple reflections corresponding to the labelledDynkin diagrams in [Bou68] . When W has type B n or C n , we let u n,k be themaximal element in W S \{ s ,s k } ∩ W S \{ s } . In each case, the set J = S \ { s } , where s is listed in the table. ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 7
Remark 3.9.
All the elements listed in part (2) of Theorem 3.8 satisfy a Coxeter-theoreticproperty which we term almost maximality . This property is defined in Definition 5.2, andplays an important role in the proof of Theorem 3.6.
Remark 3.10.
The assumption in Theorem 3.8 that S ( w ) = S does not weaken the char-acterization. If S ( w ) is a strict subset of S , then X J ( w ) is isomorphic to the Schubertvariety indexed by w in the smaller flag variety G S ( w ) /P S ( w ) ∩ J corresponding to the alge-braic subgroup G S ( w ) ⊂ G with Weyl group W S ( w ) . For a precise statement, see Lemma4.8. As mentioned in the introduction, the list of smooth Schubert varieties in a generalizedGrassmannian is known in many cases [LW90] [BP99] [BL00] [Rob14] [HM13]. In par-ticular, Hong and Mok show that if X J ( w ) is a smooth Schubert variety in a generalizedGrassmannian corresponding to a long root, then w must be the maximal element of W JS ( w ) (in the cominuscule case this also follows from the earlier work of Brion-Polo). The smoothSchubert varieties in C n with s = s k with 1 < k < n arise as “odd symplectic manifolds”,and have been studied by Mihai [Mih07]. To the best of the authors’ knowledge, the cases F , s = s and s = s are not covered by previous work, and the completeness of the abovelist has not been addressed outside of the cases mentioned above.It is well known that a Schubert variety X J ( w ) is rationally smooth if and only if the cor-responding Kazhdan-Lusztig polynomials are trivial. While there are a number of explicitformulas for Kazhdan-Lusztig polynomials of the Schubert varieties of minuscule and comi-nuscule generalized Grassmannians (see sections 9.1 and 9.2 of [BL00] for a summary), theauthors’ are not aware of any complete list of rationally smooth Grassmannian Schubertvarieties of finite type in previous work.Combining Theorem 3.8 with our main results, we get a new proof of Peterson’s theorem: Corollary 3.11.
Suppose W is simply-laced. Then X J ( w ) is rationally smooth if and onlyif it is smooth.Proof. If W is simply-laced, then all rationally smooth Grassmannian Schubert varietiesare smooth by Theorem 3.8, part (1). So if X J ( w ) is rationally smooth, then it is smoothby Corollary 3.7. (cid:3) Corollary 3.11 is a consequence of a more general theorem proved by Peterson, whichstates that if W is simply-laced then the rationally smooth and smooth locus of any Schu-bert variety coincide [CK03]. Peterson’s proof is more algebro-geometric, while our proof ismore combinatorial. Unfortunately our methods do not seem to apply to Peterson’s moregeneral theorem.4. Characterization of Billey-Postnikov decompositions
The goal of this section is to prove Theorem 3.3 and Corollary 3.5. We first give severalequivalent combinatorial characterizations of BP decompositions, and then apply thesecharacterizations to the geometry of Schubert varieties.4.1.
Combinatorial characterizations.
In this section, we can assume W is an arbitraryCoxeter group with simple generating set S . Note that Definition 3.1 still makes sense forarbitrary Coxeter groups. We restrict to the finite or crystallographic case only whennecessary. We start by proving some important facts about BP decompositions based onknown facts from the case J = ∅ . For notational simplicity, define W JK := W J ∩ W K EDWARD RICHMOND AND WILLIAM SLOFSTRA for any J ⊆ K ⊆ S . Lemma 4.1.
For every element w ∈ W J and subset K ⊆ S containing J , there is a uniquemaximal element ¯ u in [ e, w ] ∩ W JK with respect to Bruhat order. If w = vu is the parabolicdecomposition of w with respect to K , then ¯ u has a reduced decomposition ¯ u = ¯ vu , where ¯ v ∈ [ e, v ] ∩ W K .Proof. When J = ∅ , the existence of ¯ u is proved in [vdH74, Lemma 7]. Let u ′ denote themaximal element of [ e, w ] ∩ W K and let u ′ = ¯ uu ′′ be the parabolic decomposition of u ′ withrespect to J , so ¯ u ∈ W JK and u ′′ ∈ W J . If u ∈ [ e, w ] ∩ W JK , then u ≤ u and hence u ≤ ¯ u .For the second part of the lemma, if we take J = ∅ then it is easy to prove by inductionon ℓ ( u ) that u ′ above has a reduced decomposition u ′ = v ′ u , where v ′ ∈ [ e, v ] ∩ W K .For arbitrary J , observe that if u ∈ W JK and s ∈ K such that ℓ ( su ) = ℓ ( u ) + 1, theneither su ∈ W JK , or su = ut for some t ∈ J . Indeed, su = u t where u ∈ W J and t ∈ W J . Since u ≤ su , we get that u ≤ u , and if t = e then we must have u = u and ℓ ( t ) = 1. Taking a reduced decomposition s · · · s k for v ′ and considering the products s k u , s k − s k u , . . . , we eventually conclude that ¯ u has reduced decomposition s i · · · s i m u , where1 ≤ i < . . . < i m ≤ k . (cid:3) Recall that Bruhat order on W J induces a relative Bruhat order ≤ J on the coset space W/W J . By definition, w W J ≤ J w W J if and only if ¯ w ≤ ¯ w in the usual Bruhat order, where ¯ w i is the minimal length cosetrepresentative of w i W J . Note that if w ≤ w in Bruhat order, then w W J ≤ w W J evenif w , w W J . Define the descent set relative to J to be D JL ( w ) := { s ∈ S | swW J ≤ J wW J } Proposition 4.2.
Let w = vu ∈ W J be a parabolic decomposition with respect to K , so v ∈ W K , u ∈ W JK . The following are equivalent:(a) w = vu is a BP decomposition with respect to ( J, K ) .(b) The multiplication map (cid:0) [ e, v ] ∩ W K (cid:1) × (cid:0) [ e, u ] ∩ W JK (cid:1) → [ e, w ] ∩ W J is surjective.(c) The element u is the maximal element of [ e, w ] ∩ W JK .(d) S ( v ) ∩ K ⊆ D JL ( u ) .Furthermore, if W J is a finite Coxeter group and u ′ the maximal element of coset uW J ,then the following are equivalent to parts (a)-(d).(e) S ( v ) ∩ K ⊆ D L ( u ′ ) .(f ) The element u ′ has reduced decomposition u u , where u is the maximal elementof W S ( v ) ∩ K .(g) w = vu ′ is a BP decomposition with respect to K . Since the descent and support sets can be calculated efficiently, part (d) gives a practicalcriterion for checking whether a parabolic decomposition is a BP decomposition.
Proof.
Note that the multiplication map in part (b) is always injective. Hence part (b)is equivalent to part (c). The multiplication map is also length preserving, so part (b) isequivalent to part (a).
ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 9
To show that parts (c) and (d) are equivalent, suppose u is the maximal element of[ e, w ] ∩ W JK . If s ∈ S ( v ) ∩ K then su ∈ W K and hence suW J ≤ J uW J . For the converse,suppose ¯ u is the maximal element of [ e, w ] ∩ W JK , so u ≤ ¯ u ≤ w . By Lemma 4.1, if u < ¯ u then there is a simple reflection s ∈ S ( v ) ∩ K such that u < su ≤ ¯ u and su ∈ W J . Hence S ( v ) ∩ K is not a subset of D JL ( u ).The equivalence of parts (e) and (f) is immediate. Let w ′ be the maximal element in thecoset wW J . Then u is maximal in [ e, w ] ∩ W JK if and only if u ′ is maximal in [ e, w ′ ] ∩ W K .This implies part (f) is equivalent to part (c).Finally, u ′ is the maximal element of [ e, w ′ ] ∩ W K if and only if su ′ ≤ u ′ for all s ∈ S ( v ) ∩ K .Hence part (g) is equivalent to part (e). This completes the proof. (cid:3) In the case when J = ∅ , the equivalence of Proposition 4.2 parts (a)-(c) is proved in[BP05, Theorem 6.4], (a) ⇒ (d) in [OY10, Lemma 10], and (a) ⇐ (d) in [RS14, Lemma2.2]. Using Proposition 4.2, it is easy to show that BP decompositions are associative, inthe same way that parabolic decompositions are associative: Lemma 4.3.
Let I ⊆ J ⊆ K ⊆ S and w ∈ W I . Write w = xyz where x ∈ W K , y ∈ W JK ,and z ∈ W IJ . Then the following are equivalent:(a) x ( yz ) is a BP decomposition with respect to ( I, K ) and yz is a BP decompositionwith respect to ( I, J ) .(b) ( xy ) z is a BP decomposition with respect to ( I, J ) and xy is a BP decompositionwith respect to ( J, K ) . The last combinatorial property concerns Poincar´e polynomials.
Lemma 4.4.
Let W be a crystallographic Coxeter group and w ∈ W. Let w = vu ∈ W J bea parabolic decomposition with respect to K . If w = vu is a BP decomposition with respectto ( J, K ) , then P Jw ( t ) is palindromic if and only if P Kv ( t ) and P Ju ( t ) are palindromic.Proof. Let P ( t ) and P ( t ) be polynomials of degree d and d respectively. Suppose P j ( t ) = P i c ji t i has the property that c ji ≤ c jd j − i for all i ≤ ⌊ d j / ⌋ and j = 1 ,
2. Then itis easy to check that P · P is palindromic if and only if P and P are palindromic. By[BE09, Theorem A], the relative Poincar´e polynomials P Jw of elements in crystallographicCoxeter groups have this property. (cid:3) Remark 4.5.
For arbitrary Coxeter groups, Lemma 4.4 holds in the case that J = ∅ . Indeed, it suffices to prove that if P Jw ( t ) is palindromic, then P Ju ( t ) is palindromic. Thisfollows from [BP05, Lemma 6.6] and [Car94, Theorem B] along with the recent result in [EW14] that the Kazhdan-Lusztig polynomials of arbitrary Coxeter groups have nonnegativecoefficients. Geometric characterizations.
In this section we give some geometric properties ofBP decompositions, finishing with the proof of Theorem 3.3. We return to the assumptionthat W is the Weyl group of some Kac-Moody group G , and hence is crystallographic.For the remainder of the section, we fix J ⊆ K ⊆ S and the corresponding parabolicsubgroups P J ⊆ P K ⊆ G . For any g ∈ G , let [ g ] ∈ G/P K denote the image of g under theprojection G → G/P K . Lemma 4.6.
Let w = vu ∈ W J be a parabolic decomposition with respect to K and recallthe projection π : X J ( w ) → X K ( v ) . Let [ b v ] be a point of X K ( v ) , where b ∈ B and v ∈ [ e, v ] ∩ W K . Then π − ([ b v ]) = b v [ Bu ′ P J /P J where the union is over u ′ ∈ W JK such that v u ′ ≤ w .Proof. Using the parabolic decomposition of W K , we get that P K = BW K B = [ u ′ ∈ W JK Bu ′ P J . So the fibre of G → G/P K over [ b v ] is b v P K = [ u ′ ∈ W JK b v Bu ′ P J , Since ℓ ( v u ′ ) = ℓ ( v ) + ℓ ( u ′ ), we have that b v Bu ′ P J is a subset of Bv u ′ P J . The imageof the set Bv u ′ P J under the projection G → G/P J lies outside of X J ( w ) unless v u ′ ≤ w ,in which case the image is contained in X J ( w ). (cid:3) Lemma 4.6 yields the following intermediate criterion for BP decompositions.
Proposition 4.7.
Let w = vu ∈ W J be a parabolic decomposition with respect to K .Then w = vu is a BP decomposition with respect to ( J, K ) if and only if the fibres of theprojection π : X J ( w ) → X K ( v ) are equidimensional.Proof. Suppose w = vu is a BP decomposition and take b ∈ B , v ∈ [ e, v ] ∩ W K . Thenthe fibre π − ([ b v ]) = b v X J ( u ), since if u ′ ∈ [ e, w ] ∩ W JK then u ′ ≤ u by Proposition 4.2part (c).Conversely, the fibre π − ([ e ]) = X J (¯ u ), where ¯ u is the maximal element of [ e, w ] ∩ W JK ,while the fibre π − ([ v ]) = vX J ( u ). Hence if π : X J ( w ) → X K ( v ) is equidimensional, thenwe must have ℓ ( u ) = ℓ (¯ u ). But u ≤ ¯ u , so this implies u = ¯ u . (cid:3) To finish the proof of Theorem 3.3, we need two standard lemmas.
Lemma 4.8.
Let v ∈ W K and I = S ( v ) . Let G I be the reductive subgroup of P I , andlet P I,I ∩ K := G I ∩ P K be the parabolic subgroup of G I generated by I ∩ K . Finally,let X I ∩ KI ( v ) ⊆ G I /P I,I ∩ K be the Schubert variety indexed by v ∈ W KI . Then the map G I /P I,I ∩ K ֒ → G/P K induces an isomorphism X I ∩ KI ( v ) → X K ( v ) .Proof. It suffices to show that the induced map X I ∩ KI ( v ) ֒ → X K ( v ) is surjective. Write P I = G I N I where N I is the unipotent subgroup of P I and let B I = G I ∩ B denote theBorel of G I . If v ′ ∈ W I , then N I is stable under conjugation by v ′− . Write B = B I N I .Then for any v ′ ∈ W I , the Schubert cell Bv ′ P K /P K = B I N I v ′ P K /P K = B I ( v ′ v ′− ) N I v ′ P K /P K = B I v ′ P K /P K . Since v ∈ W KI ⊆ W I , we have that X I ∩ KI ( v ) ֒ → X K ( v ) is surjective. (cid:3) Lemma 4.9. If u ∈ W J , then X J ( u ) is closed under the action of P D JL ( u ) , the parabolicsubgroup generated by the left descent set D JL ( u ) of u relative to J .Proof. Let Z be the inverse image of X J ( u ) under the projection G → G/P J . Then Z = [ u ′ ≤ u Bu ′ BW J B = [ u ′ ≤ u Bu ′ W J B. If s ∈ D JL ( u ) and u ′ W J ≤ J uW J , then su ′ W J ≤ J uW J . Thus sBu ′ W J B ⊆ Bsu ′ W J B ∪ Bu ′ W J B ⊆ Z. So Z is closed under D JL ( u ), and therefore X J ( u ) is closed under P D JL ( u ) . (cid:3) ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 11
Proof of Theorem 3.3. If π : X J ( w ) → X K ( v ) is locally-trivial then the fibres of π areequidimensional. Thus w = vu is a BP decomposition with respect to ( J, K ) by Proposition4.7.Conversely, suppose that w = vu is a BP decomposition with respect to ( J, K ), andlet I = S ( v ) as in Lemma 4.8. Recall from the proof of Proposition 4.7 that if b ∈ B , v ∈ [ e, v ] ∩ W K , then the fibre π − ([ b v ]) = b v X J ( u ). Suppose g ∈ G I maps to [ b v ]in G/P K . Then we can write g = b v p , where b ∈ B I and p ∈ P I,I ∩ K . By Proposition4.2 part (d), I ∩ K ⊆ D JL ( u ). Hence by Lemma 4.9, pX J ( u ) = X J ( u ), and we concludethat gX J ( u ) = b v X J ( u ) is the fibre of π over [ g ] = [ b v ].By [Kum02, Corollary 7.4.15 and Exercise 7.4.5] the projection G I → G I /P I,I ∩ K islocally trivial, and thus has local sections. Given x ∈ X K ( v ), there is a Zariski openneighbourhood U x ⊆ X K ( v ) of x with a local section s : U x → G I ⊆ G of the projection G → G/P K . Let m : U x × X J ( u ) → G/P J denote the multiplication map ( u, y ) s ( u ) · y .The image of m is contained in X J ( w ), and thus we get a commuting square U x × X J ( u ) m / / (cid:15) (cid:15) X J ( w ) π (cid:15) (cid:15) U x (cid:31) (cid:127) / / X K ( v )in which the fibres of projection U x × X J ( u ) → U x are mapped bijectively onto the fibres of π . If z ∈ π − ( U x ) and g = s ( π ( z )), then z ∈ gX J ( u ). So m maps bijectively onto π − ( U x ),and we can define an inverse π − ( U x ) → U x × X J ( u ) by z ( π ( z ) , g − z ) where g = s ( π ( z )).We conclude that m is an isomorphism, and ultimately that π : X J ( w ) → X K ( v ) is locallytrivial.Now the projection U x × X J ( u ) → U x is smooth if and only if X J ( u ) is smooth, andthus the projection π : X J ( w ) → X K ( v ) is smooth if and only if X J ( u ) is smooth. Inparticular, if X J ( u ) and X K ( v ) are both smooth, then X J ( w ) is smooth. Conversely, if X J ( w ) is smooth then the product U x × X J ( u ) must be smooth whenever the projection G I → X K ( v ) has a local section over U . Looking at Zariski tangent spaces, we concludethat dim T x X K ( v ) + dim T y X J ( u ) ≤ ℓ ( w ) for all x ∈ X K ( v ), y ∈ X J ( u ). Since ℓ ( w ) = ℓ ( u ) + ℓ ( v ), both X K ( v ) and X J ( u ) must be smooth.The Schubert variety X J ( w ) is rationally smooth if and only if X J ( u ) and X K ( v ) arerationally smooth by Lemma 4.4. (cid:3) We finish the section by proving Corollary 3.5.
Proof of Corollary 3.5.
For singular cohomology, the proof follows easily from the Leray-Hirsch theorem. For etale cohomology, we use the Leray-Serre spectral sequence E ∗ , ∗ = H ret (cid:0) X K ( v ) , R s π ∗ Q l (cid:1) = ⇒ H r + set (cid:0) X J ( w ) , Q l (cid:1) for the projection π : X J ( w ) → X K ( v ) (see, e.g. [Tam94]). Since π is locally trivial,the sheaf R ∗ π ∗ Q l is locally constant, and by the proper base change theorem we see thatit is in fact isomorphic to H ∗ et ( X J ( u ) , Q l ). Since H ∗ et ( X K ( v ) , Q l ) and H ∗ et ( X J ( u ) , Q l ) areconcentrated in even dimensions, the Leray-Serre spectral sequence collapses at the E -term, and the spectral sequence converges to H ∗ et ( X K ( v )) ⊗ H ∗ et ( X J ( u )) as an algebra.Using the action of H ∗ et ( X K ( v )) on H ∗ et ( X J ( w )), we can solve the lifting problem to get anisomorphism H ∗ et (cid:0) X K ( v ) , Q l (cid:1) ⊗ H ∗ et (cid:0) X J ( u ) , Q l (cid:1) → H ∗ et (cid:0) X J ( w ) , Q l (cid:1) of H ∗ et ( X K ( v ))-modules. (cid:3) Rationally smooth Grassmannian Schubert varieties
In this section we define almost maximal elements of a Weyl group and prove Theorem3.8. We take G to a be simple Lie group of finite type, and hence the Weyl group W is a finite Coxeter group. It is well known that simple Lie groups are classified into fourclassical families A n , B n , C n , D n and exceptional types E , E , E , F and G . We beginwith the following theorem. Theorem 5.1 ([Bil98], [Gas98], [Las98], [BP05], [OY10]) . Let X ∅ ( w ) be rationally smoothSchubert variety with | S ( w ) | ≥ . Then there is a leaf s ∈ S ( w ) of the Dynkin diagram of W S ( w ) such that either w or w − has a BP decomposition vu with respect to J = S \ { s } .Furthermore, s can be chosen so that v is either the maximal length element in W J , orone of the following holds:(a) W S ( v ) is of type B n or C n , with either(1) J = S \ { s } , and v = s k s k +2 · · · s n s n − · · · s , for some < k ≤ n .(2) J = S \ { s n } with n ≥ , and v = s · · · s n .(b) W S ( v ) is of type F , with either(1) J = S \ { s } and v = s s s s .(2) J = S \ { s } and v = s s s s .(c) W S ( v ) is of type G , and v is one of the elements s s , s s s , s s s s , s s , s s s , s s s s . Note that the elements listed in parts (a)-(c) of Theorem 5.1 correspond to the elementslisted in part (2) of Theorem 3.8 for which s is a leaf of the Dynkin diagram. The resultthat w or w − has a BP decomposition with respect to a leaf is due to Billey [Bil98,Theorem 3.3 and Proposition 6.3] in the classical types and Billey-Postnikov [BP05] in theexceptional types. The type A case was also proved by Gasharov [Gas98] and Lascoux[Las98]. For the second part of Theorem 5.1 on the presentation of v , the proof for theclassical types is again due to Billey [Bil98]. The type E case is due to Oh-Yoo in [OY10] and types F and G can be easily verified by computer using Kumar’s criteria for rationalsmoothness [Kum96]. We remark that computer calculation plays an important role in theproof of Theorem 5.1. In particular, the results of [BP05] and [OY10] on the exceptionaltypes both require exhaustive computer verification.Note that the condition that either w or w − has a BP decomposition in Theorem 5.1can be rephrased as w having “left” or “right” sided BP decompositions. For any J ⊆ S, let J W ≃ W J \ W denote the set of minimal length left sided coset representatives. Any w ∈ W has unique left sided parabolic decomposition w = uv with respect to J where u ∈ W J and v ∈ J W. We say a left sided parabolic decomposition w = uv is a left sidedBP decomposition with respect to J if P w ( t ) = P u ( t ) · J P v ( t )where J P v ( t ) := X x ∈ [ e,v ] ∩ J W t ℓ ( x ) . This also follows from the geometric results of [HM13] together with Peterson’s theorem that all ratio-nally smooth elements in type E are smooth. ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 13
By a right sided parabolic or BP decomposition w = vu with respect to J, we simply meana usual parabolic or BP decomposition where v ∈ W J and u ∈ W J . With this terminology, w = vu is a right sided BP decomposition if and only if w − = u − v − is a left sidedBP decomposition. The combinatorial characterizations given in Proposition 4.2 have leftsided BP decomposition analogues. In particular, for Proposition 4.2, part (e) we let u ′ bethe maximal element of the coset ( W J ) u and replace the left descents D L ( u ′ ) with rightdescents D R ( u ′ ).The elements listed in Theorem 5.1 parts (a)-(c) share an important property we term“almost maximality”. Recall that an element w ∈ W is the maximal element in W S ( w ) ifand only if D L ( w ) = S ( w ), or equivalently if D R ( w ) = S ( w ). An element w ∈ W J is themaximal element in W S ( w ) ∩ JS ( w ) if and only if the longest element of wW S ( w ) ∩ J is in fact thelongest element of W S ( w ) . Based on these properties of maximal elements, we make thefollowing definition. Definition 5.2.
Given w ∈ W J , let w ′ be the longest element in wW S ( w ) ∩ J . We say that w is almost-maximal in W S ( w ) ∩ JS ( w ) if all of the following are true.(a) There are elements s, t ∈ S ( w ) (not necessarily distinct) such that D R ( w ′ ) = S ( w ′ ) \ { s } and D L ( w ′ ) = S ( w ′ ) \ { t } . (b) If w ′ = vu is the right sided parabolic decomposition with respect to D R ( w ′ ) , then S ( v ) = S ( w ′ ) .(c) If w ′ = uv is the left sided parabolic decomposition with respect to D L ( w ′ ) , then S ( v ) = S ( w ′ ) .Similarly, if w ∈ J W , we say w is almost-maximal in S ( w ) ∩ J W S ( w ) if parts (a)-(c) are truewith w ′ the longest element in the coset ( W S ( w ) ∩ J ) w . Note that an almost-maximal element is not maximal in W S ( w ) ∩ JS ( w ) by definition. If W S ( w ) ∩ JS ( w ) is clear from context, we will omit it. By the following lemma, the most interestingcase is when J = S \ { s } for some s ∈ S . Lemma 5.3.
Let w ∈ W J and assume J ⊆ S = S ( w ) . If w is almost maximal, then thereexists s / ∈ J and parabolic decomposition w = vu with respect to K = S \ { s } such that v is an almost-maximal element of W K and u is the maximal element of W JK .Proof. Assume that w ∈ W J is almost-maximal. Let w ′ be the longest element of wW J . Let s be the unique element of S \ D R ( w ′ ) and consider the parabolic decomposition w ′ = vu ′ with respect to K = S \ { s } = D R ( w ′ ). Note that S ( v ) = S ( w ′ ) = S and u ′ is maximal in W K since J ⊆ K . Write u ′ = uu where u and u are maximal in W JK and W J respectively.Then w ′ = wu = v ( uu ) | {z } u ′ which implies that w = vu . Since w is almost-maximal and w ′ is the maximal element of vW K , we have that v is almost maximal. (cid:3) Proof of Theorem 3.8.
We begin with the following proposition.
Proposition 5.4.
Let J = S \ { s } for some s and suppose v ∈ W J such that S ( v ) = S and X J ( v ) is rationally smooth. Then the following are equivalent.(i) v is not maximal in W J (ii) v is almost-maximal in W J (iii) v appears on the list in Theorem 3.8 part (2). Remark 5.5. If v ∈ W J is almost maximal, then X J ( v ) is not necessarily rationallysmooth. For example, let W be of type D with J = S \ { s } . Then v = s s s s ∈ W J isalmost maximal but X J ( v ) is not rationally smooth. Most of Theorem 3.8 follows from Proposition 5.4, leaving only the determination ofwhich Schubert varieties listed in Theorem 3.8 part (2) are smooth. Before we proveProposition 5.4, we first analyze the elements arising in parts (a) and (b) of Theorem 5.1.
Lemma 5.6.
Let W = B n or C n and J = { s , . . . , s n } . Let v = s k · · · s n − s n s n − · · · s where < k ≤ n and w ′ be the longest element of vW J . Then the following are true:(1) D L ( w ′ ) = S \ { s k − } .(2) The minimal length representative of W D L ( w ′ ) w ′ is s k − · · · s u − n,k , where u n,k is themaximal length element of W J \{ s k } J .Proof. Partition S into S = { s , . . . , s k − } and S = { s k , . . . , s n } . If w denotes themaximal element of W J then w ′ = vw . Since the elements of W S and W S \{ s k } commute,we can write w = u u u − n,k , where u is maximal in W S ∩ J and u is maximal in W S \{ s k } . Then w ′ = ( s k · · · s n · · · s )( u u u − n,k )= ( s k · · · s n · · · s k u )( s k − · · · s u ) u − n,k . Since ( s k · · · s n · · · s k ) is maximal in W S \{ s k } S , we have that ( s k · · · s n · · · s k u ) is a maximalelement in W S . In particular, S ⊆ D L ( w ′ ) . Similarly ( s k − · · · s ) is maximal in W S \{ s } S ,and hence ( s k − · · · s u ) is maximal in W S . Consequently s k − · · · s u = u s k − · · · s , where u is the maximal element in W S \{ s k − } . Now we have w ′ = ( s k · · · s n · · · s k u ) · ( u s k − · · · s u − n,k )= ( u s k · · · s n · · · s k u ) · ( s k − · · · s u − n,k ) . Thus S \ { s k − } ⊆ D L ( w ′ ) and hence S \ { s k − } ⊆ D L ( w ′ ). Since w ′ is not maximal in W , the element s k − / ∈ D L ( w ′ ) . This proves part (1), and part (2) follows from the factthat ( u s k · · · s n · · · s k u ) is maximal in W D L ( w ′ ) . (cid:3) Note that if k = 2 in Lemma 5.6, then D L ( w ′ ) = J , and the minimal length representa-tive of W J w ′ is s u − n,k = w − . Lemma 5.7.
Let W = B n or C n and J = { s , . . . , s n − } where n ≥ . Let v = s · · · s n and w ′ be the longest element of vW J . Then the following are true:(1) D L ( w ′ ) = J (2) The minimal length representative of W D L ( w ′ ) w ′ is s n · · · s . ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 15
Proof. If w denotes the longest element of W J , then w ′ = vw , and we can write w = u s n − · · · s , where u is the longest element of W J \{ s n − } . Hence w ′ = ( s · · · s n − s n )( u s n − · · · s ) = ( s · · · s n − u )( s n s n − · · · s ) . The lemma now follows from the fact that s · · · s n − u = w . (cid:3) Lemma 5.8.
Let W = F and J = { s , s , s } . Let v = s s s s and w ′ be the longestelement of wW J . Then the following are true:(1) D L ( w ′ ) = { s , s , s } (2) The minimal length representative of W D L ( w ′ ) w ′ is s s s s s s s s s .Proof. Computation in F shows that w ′ = ( s s s s )( s s s s s s s s s )= ( s s s s )( s s s s s s s s s ) . (cid:3) Note that Lemma 5.8 also applies to v = s s s s in F , since F has an automorphismsending the simple generator s k s − k for k ≤
4. (This automorphism is not a diagramautomorphism, and hence is not defined on the root system, but it is defined for the Coxetergroup).
Lemma 5.9.
Let v be almost-maximal in W S ( v ) ∩ JS ( v ) and let w ′ be the longest element in vW S ( v ) ∩ J . Let w ′ = u v be the left sided parabolic decomposition of w ′ with respect to J ′ := D L ( w ′ ) . Then the following are true:(1) v − is almost-maximal in W J ′ S ( v ) (2) X J ( v ) is rationally smooth if and only X J ′ ( v − ) is rationally smooth.Proof. Part (1) of the lemma is immediate from Definition 5.2 of almost-maximal. For thesecond part, let u be the maximal element of W S ( v ) ∩ J , so w ′ = vu . By Proposition 4.2, w ′ = vu is a BP decomposition and hence by Lemma 4.4, we have X J ( v ) is rationallysmooth if and only if X ∅ ( w ′ ) is rationally smooth. But P w ′ ( t ) = P ( w ′ ) − ( t ) , and so X ∅ ( w ′ ) is rationally smooth if and only if X ∅ (( w ′ ) − ) is rationally smooth. Finally,since u is the maximal element of W J ′ , we have w ′ = u v is a left sided BP decomposition.Thus X ∅ (( w ′ ) − ) is rationally smooth if and only if X J ′ ( v − ) is rationally smooth. (cid:3) Proof of Proposition 5.4.
Clearly (ii) ⇒ (i) in the proposition. We will show (iii) ⇒ (ii)and (i) ⇒ (iii). We start with the proof of (iii) ⇒ (ii). Suppose v ∈ W J is an element listedin parts (a)-(c) of Theorem 5.1. If v is of type G , then it is easy to see that v is almost-maximal. For types B, C and F , it follows from Lemmas 5.6, 5.7, and 5.8 that all elementslisted in parts (a)-(c) are almost-maximal. As mentioned previously, the elements listed inparts (a)-(c) of Theorem 5.1 are precisely the elements listed in the table of Theorem 3.8for which s is a leaf of the Dynkin diagram. By the second parts of Lemmas 5.6, 5.7, and5.8, if v ′ is any another element listed in Theorem 3.8, then there is a leaf s of the Dynkindiagram, and a rationally smooth element v ∈ W S \{ s } listed in Theorem 5.1, such that thelongest element w ′ in vW S ( v ) ∩ J has left-sided parabolic decomposition w ′ = u v , where v ′ = v − . Lemma 5.9 now implies that all the elements listed in Theorem 3.8 are almost-maximal and the corresponding Schubert varieties are rationally smooth. This proves (iii) ⇒ (ii) of the proposition.Now we prove (i) ⇒ (iii). Suppose v ∈ W J is not maximal and let w ′ be the longestelement of vW J . Theorem 5.1 implies there exists a leaf s ′ ∈ S such that either w ′ or( w ′ ) − has a BP decomposition v ′ u ′ where v ′ appears on the list given in parts (a)-(c) ofTheorem 5.1. If w ′ = v ′ u ′ , then the fact that D R ( w ′ ) = J and w ′ is not maximal impliesthat s ′ = s and hence, u ′ = u and v ′ = v . Now suppose that ( w ′ ) − = v ′ u ′ and let J ′ = S \ { s ′ } . By Proposition 4.2, part (e) we have S ( v ′ ) ∩ J ′ = J ′ ⊆ D L ( u ′ ) . Since w ′ isnot maximal, we must have J ′ = D L ( u ′ ) and that u ′ is the longest element of W ′ J . Thus w ′ is almost-maximal. Lemma 5.9 together with Lemmas 5.6, 5.7, and 5.8 imply that v isan element listed in Theorem 3.8. (cid:3) The equivalence of Proposition 5.4 parts (ii) and (iii) gives the following rephrasingTheorem 5.1.
Corollary 5.10.
Let X ∅ ( w ) be rationally smooth, where | S ( w ) | ≥ . Then there is a leaf s ∈ S ( w ) of the Dynkin diagram of W S ( w ) such that either w or w − has a BP decomposition vu with respect to J = S \ { s } . Furthermore, s can be chosen so that v is either maximalor almost-maximal in W S ( v ) ∩ JS ( v ) . Corollary 5.10 plays an important role in the proof of Theorem 3.6 in the next section.We finish the proof of Theorem 3.8 by determining which Schubert varieties listed inTheorem 3.8, part (2) are smooth.
Lemma 5.11.
Let v ∈ W S ( v ) ∩ JS ( v ) and let w ′ be the longest element in vW S ( v ) ∩ J . Let w ′ = u v be the left sided parabolic decomposition of w ′ with respect to J ′ := D L ( w ′ ) .Then X J ( v ) is smooth if and only if X J ′ ( v − ) is smooth.Proof. Let u be maximal in W S ( v ) ∩ J , so that w ′ = vu , and let Z = [ x ≤ w ′ BxB be the inverse image of X J ( v ) in G . Then Z is a principal P J -bundle over X J ( w ), so Z issmooth if and only if X J ( w ) is smooth. But Z is isomorphic to the inverse image [ x ≤ ( w ′ ) − BxB of X J ′ ( v − ) in G , so the lemma follows. (cid:3) By Lemma 5.11 and Lemmas 5.6, 5.7, and 5.8, it suffices to determine when X J ( w )is smooth for J = S \ { s } , s a leaf. Note that Kumar has given a general criterion forsmoothness of Schubert varieties, and by this criterion the smoothness of Schubert varietiesis independent of characteristic [Kum96].If W is of type B n or C n , then the singular locus of X J ( w ) when J = S \ { s } , s a leaf, iswell-known (see pages 138-142 of [BL00]). In type G , the smooth Schubert varieties arealso well-known (see the exercise on page 464 of [Kum02]). Finally, for type F we use acomputer program to apply Kumar’s criterion to the Schubert varieties in question. ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 17
Remark 5.12.
Note that we only use prior results on smoothness for the rationally smoothalmost-maximal elements, which do not occur in the simply-laced case. Hence the proof ofPeterson’s theorem (Corollary 3.11) depends only on Theorem 5.1. Existence of Billey-Postnikov decompositions
In this section, we prove Theorem 3.6 and Corollary 3.7. The main theorem, statedbelow, is an extension of Theorem 5.1, wherein we show that if X ∅ ( w ) is rationally smooththen w has a right-sided Grassmannian BP decomposition. Theorem 6.1.
Let W be a finite Weyl group. Suppose X ∅ ( w ) is rationally smooth forsome w ∈ W . Then one of the following is true:(a) The element w is the maximal element of W S ( w ) .(b) There exists s ∈ S ( w ) \ D R ( w ) such that w has a right sided BP decomposition w = vu with respect to J = S \ { s } , where v is either the maximal or an almost-maximal element of W S ( v ) ∩ JS ( v ) . Note that if w is almost-maximal (relative to J = ∅ ) then w satisfies part (b) of Theorem6.1 by definition. The requirement that s not belong to D R ( w ) in part (2) of the theorem iscritical both for the inductive proof of the theorem, and for showing that BP decompositionsexist in the relative case.We introduce some terminology for subsets of the generating set S . Definition 6.2.
A subset T ⊆ S is connected if the Dynkin diagram of W T is connected.The connected components of a subset T ⊆ S are the maximal connected subsets of T . Definition 6.3.
We say that s and t in S are adjacent if s and t are adjacent in theDynkin diagram. We also say that s is adjacent to a subset T if s is adjacent to some t ∈ T , and that two subsets T , T are adjacent if there is some element of T adjacent to T . We use this terminology for the following lemma:
Lemma 6.4.
Let w = vu be a right sided parabolic decomposition with respect to some J ⊆ S . If s ∈ S \ S ( v ) is adjacent to S ( v ) , then s / ∈ D L ( w ) .Similarly, let w = uv be a left sided parabolic decomposition with respect to some J ⊆ S .If s ∈ S \ S ( v ) is adjacent to S ( v ) , then s / ∈ D R ( w ) .Proof. Clearly the second statement of the lemma follows from the first by considering w − . We proceed by induction on the length of v . If ℓ ( v ) = 1 then the proof is obvious.Otherwise take a reduced decomposition v = v tv , where s is adjacent to t , but notadjacent to S ( v ). (Note that v could be equal to the identity.) Then tv ∈ W J and since s / ∈ S ( v ), we have s ∈ D L ( w ) only if s ∈ D L ( u ). Assume that s ∈ D L ( u ) and write w = ( v tv )( u ) = ( v tsv )( su ) . Once again, we have s ∈ D L ( w ) only if s ∈ D L ( tsv ( su )). But since t and s are adjacent,we have s ∈ D L ( tsv ( su )) only if t ∈ D L ( v ( su )). If t / ∈ J , then t / ∈ S ( v ( su )), and we aredone. Otherwise, if t ∈ J , then t must be adjacent to S ( v ) since tv ∈ W J . But v ∈ W J ,so by induction, we get t / ∈ D L ( v ( su )). (cid:3) We now proceed to the proof of Theorem 6.1. The proof uses multiple reduced decom-positions for the same element, so we have included certain schematic diagrams to aid thereader in keeping track of the reduced decompositions under consideration. For example, if w = w w w is a reduced decomposition with v = w w and u = w w , then we diagramthis relation by wv w w uw w w Proof of Theorem 6.1.
We proceed by induction on | S ( w ) | . It is easy to see that thetheorem is true if | S ( w ) | = 1 ,
2. Hence we can assume that | S ( w ) | >
2. We can alsoassume without loss of generality that the Dynkin diagram of W S ( w ) is connected. Since w is rationally smooth, we can apply Corollary 5.10 to get that w has either a right sidedBP decomposition w = vu , or a left sided BP decomposition w = uv , with respect to J = S \ { s } where s ∈ S is a leaf of the Dynkin diagram of W S ( w ) . Note that in bothcases, X ∅ ( u ) is rationally smooth by Lemma 4.4. We now consider four cases, dependingon whether we get a right or left BP decomposition from Corollary 5.10, and dependingon whether or not u is maximal in W S ( u ) . Case 1 : w has a right sided BP decomposition w = vu as in Corollary 5.10, where u is maximal in W S ( u ) . If s ∈ D R ( w ) , then w is the maximal element of W . Otherwise, if s / ∈ D R ( w ) , then the decomposition w = vu satisfies condition (b) in Theorem 6.1, since v is maximal or almost-maximal in W S ( v ) ∩ JS ( v ) by Corollary 5.10. Case 2 : w has a left sided BP decomposition w = uv as in Corollary 5.10, where u is maximal in W S ( u ) , and v is either maximal or almost-maximal in S ( v ) ∩ J W S ( v ) . If S ( u ) = S ( v ) ∩ J , then w is either maximal or almost-maximal respectively, in which casewe are done.Since S ( w ) is connected, if S ( v ) ∩ J ( S ( u ) then we can choose s ′ ∈ S ( u ) \ S ( v ) adjacentto S ( v ). Since u is maximal in W S ( u ) , the parabolic decomposition u = v ′ u ′ with respect to J ′ := S ( u ) \{ s ′ } is a BP decomposition with v ′ is maximal in W J ′ S ( u ) . Moreover, s ′ / ∈ D R ( w )by Lemma 6.4. Hence the decomposition w = v ′ ( u ′ v ) satisfies condition (b) in Theorem6.1. Case 3 : w has a left sided BP decomposition w = uv as in Corollary 5.10, where u isnot maximal in W S ( u ) . Then by induction, we have a right BP decomposition u = v ′ u ′ with respect to J ′ := S ( u ) \ { s ′ } for some s ′ ∈ S ( u ), satisfying the conditions of Theorem6.1. Since s ′ / ∈ D R ( u ), we must have s ′ / ∈ S ( v ) \ { s } ⊆ D R ( u ), so w = v ′ ( u ′ v ) is a parabolicdecomposition, and s ′ D R ( w ). But S ( v ′ ) ∩ J ′ ⊆ D L ( u ′ ) ⊆ D L ( u ′ v ) , and thus w = v ′ ( u ′ v ) is a BP decomposition with respect to S ( w ) \ { s ′ } , by Proposition4.2 (d). Since v ′ is either maximal or almost-maximal by the inductive hypothesis, thedecomposition w = v ′ ( u ′ v ) satisfies condition (b) of Theorem 6.1. wu vv ′ u ′ v Figure 1: w = uv in Cases 2 and 3. ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 19
Case 4 : w has a right BP decomposition w = vu as in Corollary 5.10, where u is notmaximal in W S ( u ) . Note that if s / ∈ D R ( w ) then condition (b) of Theorem 6.1 is satisfiedimmediately, and we would be done. We consider several subcases where we either prove s / ∈ D R ( w ) or we find s ′ / ∈ D R ( w ) that satisfies condition (b) of Theorem 6.1.First, suppose that v is almost maximal in W S ( v ) ∩ JS ( v ) . Take a reduced decomposition u = u u , where u is the maximal element in W S ( v ) ∩ J . Then S ( v ) ∩ J ⊆ D R ( vu ) ( S ( v ) , implying that s / ∈ D R ( vu ). It follows that s D R ( w ). wv uv u u Figure 2: w = vu with u not maximal and v almost maximal.Now assume v is maximal in W S ( v ) ∩ JS ( v ) , and apply induction to get a BP decomposition u = v ′ u ′ with respect to S ( u ) \{ s ′ } , satisfying condition (b) of Theorem 6.1. By Proposition4.2, S ( v ) \ { s } ⊆ D L ( u ), and hence if t ∈ S ( v ) \ { s } is adjacent to S ( v ′ ), then t ∈ S ( v ′ )by Lemma 6.4. Now since v is Grassmannian, the set S ( v ) is connected, and since s isa leaf, the set S ( v ) \ { s } is also connected. We conclude that if S ( v ) \ { s } is adjacent to S ( v ′ ), then S ( v ) \ { s } ⊆ S ( v ′ ). Furthermore s is adjacent to S ( v ) \ { s } , and hence s isadjacent to S ( v ′ ). Conversely, if S ( v ) \ { s } is non-empty and s is adjacent to some elementof S ( v ′ ), then this element must be contained in S ( v ) \ { s } , since S ( v ) is connected and s is adjacent to a unique element of S ( w ). We conclude that either S ( v ) is not adjacent to S ( v ′ ), or S ( v ) \ { s } ⊆ S ( v ′ ) with s adjacent to S ( v ′ ).In the former case, when S ( v ) is not adjacent to S ( v ′ ), the elements of S ( v ) pairwisecommute with the elements of S ( v ′ ). Hence the decomposition w = v ′ ( vu ′ ) is a BP decom-position with respect to S ( w ) \ { s ′ } . The element v ′ is either maximal or almost-maximalin W S ( v ′ ) \{ s ′ } S ( v ′ ) by induction. Since s ′ / ∈ D R ( u ) and s ′ ∈ J , we conclude that s ′ / ∈ D R ( w ). wv uv v ′ u ′ v ′ v u ′ Figure 3: w = vu with u not maximal, v maximal, and S ( v ) , S ( v ′ ) not adjacent.This leaves the case that s is adjacent to S ( v ′ ) and S ( v ) \ { s } ⊆ S ( v ′ ). Take a reduceddecomposition u ′ = u ′ u ′ , where u ′ is maximal in W S ( v ′ ) \{ s ′ } . Suppose first that v ′ ismaximal, so that v ′ u ′ is maximal W S ( v ′ ) . If S ( v ) \ { s } = S ( v ′ ), then vv ′ u ′ is the maximalelement of W S ( v ) , and hence there is a BP decomposition vv ′ u ′ = x x , where x isthe maximal element of W S ( v ) \{ s ′ } S ( v ) and x is the maximal element of W S ( v ) \{ s ′ } . Since s ′ / ∈ D R ( u ) and s ′ ∈ J , we have s ′ / ∈ D R ( w ), and thus w = x ( x u ′ ) is a BP decompositionwith respect to S ( w ) \ { s ′ } satisfying the condition of Theorem 6.1. If S ( v ) \ { s } is astrict subset of S ( v ′ ), then find t ∈ S ( v ′ ) such that t / ∈ S ( v ) \ { s } . Consider the parabolic decomposition v ′ u = y y , where y ∈ W S ( v ′ ) \{ t } and y ∈ S ( v ′ ) \{ t } W S ( v ′ ) . Since t / ∈ S ( v )and s is adjacent to S ( v ′ ) = S ( y ), we conclude from Lemma 6.4 that s / ∈ D R ( vv ′ u ′ ). Since s / ∈ S ( u ′ ), we get that s / ∈ D R ( w ). wv uv v ′ u ′ v v ′ u ′ u ′ v y y u ′ x x u ′ Figure 4: w = vu with u not maximal, v maximal, and S ( v ) , S ( v ′ ) adjacent.We use a similar argument for the case that v ′ is almost-maximal in W S ( v ′ ) \{ s } S ( v ′ ) . Bydefinition v ′ u ′ is almost maximal in W S ( v ′ ) , meaning that D L ( v ′ u ′ ) = S ( v ′ ) \ { t } for some t ∈ S ( v ′ ). Since s ′ / ∈ S ( u ) and u ′ is maximal in W S ( v ′ ) \{ s ′ } , the element u ′ belongsto S ( v ′ ) W S ( u ) . Thus t does not belong to D L ( u ), since otherwise t ∈ D L ( v ′ u ′ ), and weconclude that t / ∈ S ( v ). Take the unique left sided BP decomposition v ′ u ′ = y y , where y is the maximal element in W D L ( v ′ ) , and y ∈ S ( v ′ ) \{ t } W S ( v ′ ) . Then S ( y ) = S ( v ′ ) by thedefinition of almost-maximal, and since s is adjacent to S ( y ), we conclude from Lemma6.4 that s / ∈ D R ( vv ′ u ′ ). Consequently s / ∈ D R ( w ). (cid:3) Using Theorem 6.1, we can prove Theorem 3.6 for relative Schubert varieties.
Proof of Theorem 3.6.
Suppose X J ( w ) is rationally smooth and that | S ( w ) \ J | ≥
2. Let w be maximal in W J , so w ′ = ww is the longest element of wW J . Then w ′ is rationallysmooth by Lemma 4.4, so we can apply Theorem 6.1. If w ′ is maximal, then w is the max-imal element in W J ∩ S ( w ) S ( w ) , and hence by choosing any s ∈ S ( w ) \ J we get a GrassmannianBP decomposition of w with respect to K = S \ { s } as required.If w ′ is not maximal, then there exists s ∈ S ( w ) \ D R ( w ) such that w ′ has a BPdecomposition w ′ = vu with respect to K = S \ { s } . Since s / ∈ D R ( w ), s must notbe in J , so u has parabolic decomposition u = u w with respect to W J , and w hasparabolic decomposition w = vu with respect to K . This latter decomposition is a BPdecomposition by Proposition 4.2. (cid:3) The Ryan-Wolper theorem.
In this section we use our results to prove Corollary3.7 on iterated fibre bundle structures of smooth and rationally smooth Schubert varieties.We assume that X J ( w ) is a rationally smooth Schubert variety of finite type. Proof of Corollary 3.7.
Suppose that W is a finite Weyl group, and that X J ( w ) is ratio-nally smooth. By repeatedly applying Theorems 3.3 and 3.6, we can write w = v m · · · v , where v i ∈ W J i − J i for J i := J ∪ S j ≤ i S ( v j ), and v i ( v i − · · · v ) is a Grassmannian BPdecomposition. Let w i := v m · · · v i +1 ∈ W J i , so w = w and w m is the identity . By Lemma4.3, w i = w i +1 v i +1 is a BP decomposition with respect to ( J i , J i +1 ), so the morphism(4) X J i ( w i ) → X J i +1 ( w i +1 ) ILLEY-POSTNIKOV DECOMPOSITIONS AND FIBRE BUNDLES 21 is a locally-trivial fibre bundle with fibre X J i ( v i +1 ). Hence the sequence X J ( w ) = X J ( w ) → X J ( w ) → · · · → X J m − ( w m − ) = X J m − ( v m ) → Spec k meets the required conditions in Corollary 3.7. If X J ( w ) is smooth, then by Theorem 3.3all the fibres X J i ( v i ) are smooth, and hence so are the morphisms.Conversely, given a locally-trivial morphism X → Y with fibre F such that the coho-mology H ∗ ( Y ) of the base and the cohomology H ∗ ( F ) of the fibre are concentrated ineven degrees, we can argue as in the proof of Corollary 3.5 that H ∗ ( X ) is isomorphic to H ∗ ( Y ) ⊗ H ∗ ( F ). Thus H ∗ ( X ) will be concentrated in even degrees, and if H ∗ ( Y ) and H ∗ ( F ) satisfy Poincar´e duality, then so does H ∗ ( X ). If there is a sequence X J ( w ) = X → X → · · · → X m = Spec k in which all the morphisms are locally trivial, and all the fibres are rationally smoothSchubert varieties, then H ∗ ( X i ) satisfies Poincar´e duality for all i = 0 , . . . , m . In particular, X J ( w ) will be rationally smooth by the Carrell-Peterson theorem [Car94]. (cid:3) Only limited results are known about BP decompositions outside of finite type. Billeyand Crites have shown that if X ( w ) is a rationally smooth Schubert variety of affine type˜ A , then (a la Theorem 5.1) either w or w − has a BP decomposition [BC12]. Using thisresult, the proof of Theorem 6.1 extends to the affine setting with minor modifications,and thus the Ryan-Wolper theorem also holds in affine type ˜ A . The framework of BPdecompositions also allows us to prove [BC12, Conjecture 1], that X ( w ) is smooth in affinetype ˜ A if and only if w avoids (as an affine permutation) 3412 and 4231.In [RS14], the authors show that right-sided BP decompositions exist for rationallysmooth Schubert varieties in the full flag varieties of a large class of non-finite Weyl groups.Hence the Ryan-Wolper theorem also holds for Schubert varieties X ∅ ( w ) in this class viathe application of Theorem 3.3. However, with the exception of ˜ A , all of the Coxetergroups in this class are of indefinite type. It is an open problem to prove the existence ofBP decompositions for this class when J is non-empty.Based on this evidence, the following conjecture seems plausible: Conjecture 6.5. If W is any Coxeter group, and w belongs to W J with P Jw ( t ) palindromic,then w has a Grassmannian BP decomposition. As a result, the Ryan-Wolper theorem holdsin any Kac-Moody flag variety. References [BC12] Sara Billey and Andrew Crites,
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