Smooth Schubert varieties in the affine flag variety of type A ~
aa r X i v : . [ m a t h . C O ] F e b SMOOTH SCHUBERT VARIETIES IN THE AFFINE FLAGVARIETY OF TYPE ˜ A EDWARD RICHMOND AND WILLIAM SLOFSTRA
Abstract.
We show that every smooth Schubert variety of affine type ˜ A is aniterated fibre bundle of Grassmannians, extending an analogous result by Ryanand Wolper for Schubert varieties of finite type A . As a consequence, we finish aconjecture of Billey-Crites that a Schubert variety in affine type ˜ A is smooth if andonly if the corresponding affine permutation avoids the patterns 4231 and 3412.Using this iterated fibre bundle structure, we compute the generating function forthe number of smooth Schubert varieties of affine type ˜ A . Introduction
Let X be a Kac-Moody flag variety, and let W be the associated Weyl group.Although X can be infinite-dimensional, it is stratified by finite-dimensional Schu-bert varieties X ( w ), where w ∈ W . It is natural to ask when X ( w ) is smoothor rationally smooth, and this question is well-studied [BL00]. For the finite-typeflag variety of type A n , the Weyl group is the permutation group S n , and theLakshmibai-Sandhya theorem states that X ( w ) is smooth if and only if w avoids thepermutation patterns 3412 and 4231 [LS90]. From another angle, the Ryan-Wolpertheorem states that X ( w ) is smooth if and only if X ( w ) is an iterated fibre bundleof Grassmannians of type A [Rya87, Wol89]. Haiman used the Ryan-Wolper theo-rem to enumerate smooth Schubert varieties [Hai, B´on98]. The Lakshmibai-Sandhyatheorem, the Ryan-Wolper theorem, and the enumeration of smooth and rationallysmooth Schubert varieties has been extended to all finite types (see [Bil98, BP05],[RS16], and [RS15] respectively). The latter enumeration uses a data structure called staircase diagrams , which keeps track of iterated fibre bundle structures.There are also characterizations of smoothness and rational smoothness that applyto all Kac-Moody types [Car94, Kum96]. For instance, a theorem of Carrell andPeterson states that X ( w ) is rationally smooth if and only if the Poincare polynomial P w ( q ) of X ( w ) is palindromic, meaning that the coefficients read the same from top-degree to bottom-degree and vice-versa [Car94]. However, much less is known aboutthe structure of (rationally) smooth Schubert varieties in general Kac-Moody type.The one exception is affine type ˜ A , where Billey and Crites have characterized theelements w for which X ( w ) is rationally smooth [BC12]. In this case, the Weyl group W is the affine permutation group ˜ S n . As part of their characterization, they provethat if X ( w ) is smooth, then w must avoid the affine permutation patterns Note that we use S to refer to groups, and S to refer to sets of simple reflections. and 4231. They conjecture the converse, that X ( w ) is smooth if w avoids these twopatterns.The purpose of this paper is to extend what we know about finite-type Schubertvarieties to affine type ˜ A . For smooth Schubert varieties, we show: Theorem 1.1.
Let X ( w ) be a Schubert variety in the full flag variety of type ˜ A .Then the following are equivalent:(a) X ( w ) is smooth.(b) w avoids the affine permutation patterns and .(c) X ( w ) is an iterated fibre bundle of Grassmannians of finite type A . In particular, this finishes the proof of Billey and Crites’ conjecture. We notethat the proof relies heavily on ideas from both [BC12] and [RS16]. One corollary(explained in Section 3) is that there is a bijection between smooth Schubert varietiesin the full flag variety of type ˜ A n , and spherical staircase diagrams over the Dynkindiagram of type ˜ A n . This allows us to enumerate smooth Schubert varieties in affinetype ˜ A n : Theorem 1.2.
Let A ( t ) = P a n t n , where a n is the number of smooth Schubertvarieties in the full flag variety of type ˜ A n . Then A ( t ) = P ( t ) − Q ( t ) √ − t (1 − t )(1 − t ) (1 − t + 8 t − t ) where P ( t ) = (1 − t ) (cid:0) − t + 18 t − t + 10 t − t (cid:1) and Q ( t ) = (1 − t )(2 − t ) (cid:0) − t + 6 t (cid:1) . In Table 1, we list the number of smooth Schubert varieties of type ˜ A n , or equiv-alently, the number of affine permutations in ˜ S n which avoid 3412 and 4231 for n ≤ n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 95 31 173 891 4373 20833 97333 448663 Table 1.
Number of smooth Schubert varieties of type ˜ A n .Using the generating series, we can also determine the asymptotics of a n . Let α := 16 (cid:18) − q
17 + 3 √
33 + q −
17 + 3 √ (cid:19) ≈ . − t +8 t − t from the denominatorof the generating function A ( t ). Corollary 1.3.
Asymptotically, we have a n ∼ α − n . MOOTH SCHUBERT VARIETIES IN THE AFFINE FLAG VARIETY OF TYPE ˜ A Proof.
The singularity of A ( t ) with smallest modulus is the root α of the polynomial1 − t t − t . Since this occurs with multiplicity one, [FS09, Theorem IV.7]states that a n ∼ C/α n +1 , where C := lim t → α A ( t )( α − t ) . In this case, C = α (at themoment we do not have an explanation for this interesting coincidence). (cid:3) In finite type A , every rationally smooth Schubert variety is smooth. In affine type˜ A , this is not true [BM10, BC12]. For the full flag variety, Billey and Crites showthat there is just one infinite family of rationally smooth Schubert varieties whichare not smooth. Theorem 1.4 (Theorem 1.1, Corollary 1.2 and Remark 2.16 of [BC12]) . A Schubertvariety X ( w ) in the full flag variety of type ˜ A n is rationally smooth if and only ifeither(a) w avoids the affine permutation patterns and , or(b) w is a twisted spiral permutation (in which case, X ( w ) is not smooth). The twisted spiral permutations are defined in the next section. As we will explain,it is easy to see that if w is a twisted spiral permutation, then X ( w ) is a fibre bundleover a rationally smooth Grassmannian Schubert variety, with fibre equal to the fullflag variety of type A n − . The base of this fibre bundle is a spiral Schubert variety ,a family of Schubert varieties in the affine Grassmannian introduced by Mitchell[Mit86]. Thus it follows from Theorems 1.1 and 1.4 that, just as for finite-typeSchubert varieties, a Schubert variety in the full flag variety of type ˜ A n is rationallysmooth if and only if it is an iterated fibre bundle of rationally smooth GrassmannianSchubert varieties. As we explain in the next section, this also holds for Schubertvarieties in the partial flag varieties of affine type ˜ A n . Finally, we note that Theorem1.1 was first proved in a preprint version of [RS16], but was removed during thepublication process. Here we give a variant of the original proof, along with anadditional proof using staircase diagrams.The rest of the paper is organized as follows. In the next section, we give thefirst proof of Theorem 1.1, along with the related results for partial flag varieties. InSection 3, we review the notion of a staircase diagram, and give the second proof ofTheorem 1.1. Finally, in Section 4 we prove Theorem 1.2.1.1. Acknowledgements.
We thank Erik Slivken for suggesting the bijection be-tween increasing staircase diagrams and Dyck paths given in the proof of Proposition4.1. We thank Sara Billey for helpful discussions.2.
BP decompositions in affine type ˜ A As in the introduction, let W denote the Weyl group of a Kac-Moody group G . Let S be the set of simple reflections of W , and let ℓ : W → Z ≥ be the length function.A parabolic subgroup of W is a subgroup W J generated by a subset J ⊆ S . Every(left) W J -coset has a unique minimal-length element, and the set of minimal-lengthcoset representatives is denoted by W J (similarly, the set of minimal length rightcoset representatives is denoted by J W . The partial flag variety X J is stratified EDWARD RICHMOND AND WILLIAM SLOFSTRA by Schubert varieties X J ( w ) for w ∈ W J , each of complex dimension ℓ ( w ). ThePoincar´e polynomial of w ∈ W J is P Jw ( q ) = X x ≤ w and x ∈ W J q ℓ ( x ) , where ≤ is Bruhat order. As mentioned in the introduction, a theorem of Carrell andPeterson states that X J ( w ) is rationally smooth if and only if P Jw ( q ) is palindromic,meaning that q ℓ ( w ) P Jw ( q − ) = P Jw ( q ) [Car94]. In the case that J = ∅ , let X ( w ) := X ∅ ( w ) and P w ( q ) := P ∅ w ( q ).Given J ⊆ K ⊆ S , every element w ∈ W J can be written uniquely as w = vu where v ∈ W K and u ∈ W JK := W K ∩ W J . This is called the parabolic decom-position of w with respect to K . A parabolic decomposition w = vu is a Billey-Postnikov (BP) decomposition (relative to J ) if P Jw ( q ) = P Kv ( q ) · P Ju ( q ). There areother equivalent combinatorial characterizations which are computationally easy tocheck. In particular, if J = ∅ , then w = vu is a BP decomposition if and only if S ( v ) ∩ K ⊆ D L ( u ), where S ( w ) := { s ∈ S : s ≤ w } is the support set of w ∈ W , and D L ( w ) := { s ∈ S : sw ≤ w } is the left descent set of w (see [RS16, Proposition 4.2]for more details).The following result from [RS16] gives a geometric interpretation for BP decom-positions. Theorem 2.1. ([RS16, Theorem 3.3])
Given J ⊆ K ⊂ S , let w = vu , where w ∈ W J , v ∈ W K , and u ∈ W JK . Then the following are equivalent:(a) The decomposition w = vu is a BP decomposition.(b) The natural 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 X J ( w ) is (rationally) smooth ifand only if X J ( u ) and X K ( v ) are (rationally) smooth. A Grassmannian BP decomposition is a BP decomposition w = vu with respectto a set K with | S ( w ) \ K | = 1. The main technical result of [BC12] is: Proposition 2.2 ([BC12]) . Let W be the Weyl group of type ˜ A n . If w ∈ W , as anaffine permutation avoids and then either w or w − has a GrassmannianBP decomposition vu , where both v and u belong to proper parabolic subgroups of ˜ A n . Proposition 2.2 is proved implicitly in [BC12]; in particular, see the proof of Theo-rem 3.1, and the discussion before Corollary 7.1 in [BC12]. The following extension ofProposition 2.2 shows that we don’t need to look at w − to find a BP decomposition: Proposition 2.3.
Let W be a Weyl group of type ˜ A n . If w ∈ W avoids both and , then w has a Grassmannian BP decomposition w = vu where both v and u belong to proper parabolic subgroups of ˜ A n .Furthermore, one of the following is true:(a) w is the maximal element of a parabolic proper subgroup of ˜ A n or,(b) w = vu is a BP decomposition with respect to S ( w ) \{ s } , for some s D R ( w ) . MOOTH SCHUBERT VARIETIES IN THE AFFINE FLAG VARIETY OF TYPE ˜ A The proof is similar to that of [RS16, Theorem 6.1]; for the convenience of thereader, we give a complete proof for the ˜ A n case. Proof. If S ( w ) is a proper subset of S , then W S ( w ) is finite of type A , and theproposition is exactly Theorem 6.1 of [RS16]. Hence, we assume that S ( w ) = S throughout. By Proposition 2.2, it suffices to prove that if w − has a GrassmannianBP decomposition with both factors belonging to proper parabolic subgroups, then w has a Grassmannian BP decomposition with respect to some K = S \ { s } where s / ∈ D R ( w ). Assume that w − has a Grassmannian BP decomposition with bothfactors belonging to proper parabolic subgroups. Hence there is a subset K = S \ { s } such that w = uv with u ∈ W K , v ∈ K W , and S ( v ) ∩ K ⊆ D R ( u ), the right descentset of u . Furthermore, S ( v ) is a proper subset of S = S ( w ); consequently, S ( u ) = K and K \ S ( v ) is non-empty.Next, we claim that u has a Grassmannian BP decomposition u = v ′ u ′ with respectto some K ′ = S \{ s ′ } , where s ′ S ( v ). Indeed, u is rationally smooth of type A . If u is the maximal element of W K then we can take s ′ to be any element of K \ S ( v ). If u is not maximal, then u has a Grassmannian BP decomposition u = v ′ u ′ with respectto K ′ = K \ { s ′ } , where s ′ D R ( u ) [RS16, Theorem 6.1], and hence s ′ S ( v ).Since s ′ S ( v ), we have w = v ′ ( u ′ v ) is the parabolic decomposition of w withrespect to K ′ . Since v ′ u ′ is a BP decomposition, and u ′ v is reduced, we have S ( v ′ ) ∩ K ′ ⊆ D L ( u ′ ) ⊆ D L ( u ′ v ) , and thus w = v ′ ( u ′ v ) is a BP decomposition. Finally S ( v ′ ) ⊆ K ( S , completingthe proof of the first part of the proposition.We now show we can choose s ′ / ∈ D R ( w ). First, if u ′ is not maximal, then choosea Grassmannian BP decomposition with respect to s ′ D R ( u ). Since s ′ S ( v ),we get that s ′ D R ( w ) as well (this follows, for instance, from the fact that if a = b ∈ D R ( w ), then b ∈ D R ( wa )). If u ′ is instead the maximal element, then wecan take s ′ ∈ K \ S ( v ) such that s ′ is adjacent (in the Dynkin diagram) to someelement of S ( v ). It follows from [RS16, Lemma 6.4] that s ′ D R ( w ). (cid:3) Corollary 2.4.
Suppose w ∈ W J , where W is the Weyl group of type ˜ A n . If X J ( w ) is a smooth Schubert variety and S ( w ) \ J = ∅ , then w has a Grassmannian BPdecomposition with respect to some J ⊆ K ( S ( w ) , in which each factor belongs toa proper parabolic subgroup of W .Proof. Let u be the maximal element of W J ∩ S ( w ) . Then w ′ := wu is a BP decom-position for w ′ with respect to J . By Theorem 2.1, X ( w ′ ) is a fibre bundle over X J ( w ) with fibre X ( u ). Since X ( u ) is a full flag variety of type A (and henceis smooth), it follows that X ( w ′ ) is smooth. By Theorem 1.4, w ′ must avoid 3412and 4231. We claim that w ′ has a Grassmannian BP decomposition w ′ = vu ′ withrespect to some K = S ( w ) \ { s } such that s J . Indeed, if w ′ is the maximalelement of W S ( w ′ ) , then w ′ has a Grassmannian BP decomposition with respect toany s ∈ S ( w ′ ), including any element of S ( w ) \ J . If w ′ is not maximal, then byProposition 2.3, w ′ has a Grassmannian BP decomposition where s D R ( w ′ ), andsince D R ( w ′ ) contains S ( u ) = J ∩ S ( w ), we conclude that s J . EDWARD RICHMOND AND WILLIAM SLOFSTRA
Since s J , we conclude that K = S \ { s } contains J , and thus that u ′ = uu forsome u ∈ W JK . It is easy to see that w = vu is a BP decomposition of w with respectto K , as desired. (cid:3) When combined with Theorem 2.1, Corollary 2.4 implies that every smooth ele-ment in type ˜ A has a complete BP decomposition in the following sense: Definition 2.5 ([RS15]) . Let W be a Coxeter group. A complete BP decompositionof w ∈ W J (with respect to J ) is a factorization w = v · · · v m , where m = | S ( w ) \ J | , and, if we let u i := v i · · · v m ∈ W J , then u i = v i u i +1 is a Grassmannian BPdecomposition with respect to K i := S ( u i +1 ) ∪ J for all ≤ i < m .A complete maximal BP decomposition w = v · · · v m is a complete BP decomposi-tion w = v · · · v m as above such that v i is maximal in W S ( v i ) K i − ∩ S ( v i ) for all ≤ i ≤ m . Hence (and similarly to [RS16, Corollary 3.7]), Theorem 2.1 and Corollary 2.4imply that a Schubert variety X J ( w ) of type ˜ A n is smooth if and only if it is aniterated fibre bundle of Grassmannians. Corollary 2.6.
A Schubert variety X J ( w ) in a partial flag variety of type ˜ A n issmooth if and only if there is a sequence X J ( w ) = X m → X m − → · · · → X → X = pt , where each map X i → X i − is a Zariski-locally trivial fibre bundle whose fibre is aGrassmannian variety of type A .Proof. A Zariski-locally trivial fibre bundle with a smooth base and fibre is itselfsmooth, so it follows that if X J ( w ) is an iterated fibre bundle in the above sense,then X J ( w ) is smooth.For the converse, let W be the Weyl group of type ˜ A n , and suppose that X J ( w )is smooth, where w ∈ W J . If S ( w ) ⊆ J , then X J ( w ) is a point, and the theorem istrivial. If | S ( w ) \ J | = m ≥
1, then by Corollary 2.4, w has a complete BP decompo-sition w = v · · · v m in which S ( v i ) is a strict subset of S . Let u i := v i · · · v m ∈ W J ,and let w i := v · · · v i , so that u = w m = w and u m +1 = w = e . In addition, set K i := S ( u i +1 ) ∪ J , so that • J = K m ( K m − ( · · · ( K ( K := S ( w ), • | K i − \ K i | = 1, • v i ∈ W K i K i − , • u i = v i u i +1 is a BP decomposition with respect to K i (and relative to J ).By Theorem 2.1, X K i ( v i ) is smooth for all 1 ≤ i ≤ m . Since S ( v i ) is a strict subset of S , it follows that X K i ( v i ) is a smooth Grassmannian Schubert variety of finite type A . But it is well-known (see for instance [BL00]) that the only smooth GrassmannianSchubert varieties of finite type A are themselves Grassmannians (specifically, v i mustbe maximal in W K i ∩ S ( v i ) S ( v i ) , so w = v · · · v m is a complete maximal BP decomposition).By [RS16, Lemma 4.3], w i = w i − v i is a BP decomposition with respect to K i − (and relative to K i ), so if we set X i = X K i ( w i ), then by Theorem 2.1 the standardprojection X i → X i − is a Zariski-locally trivial fibre bundle with fibre X K i ( v i ). (cid:3) MOOTH SCHUBERT VARIETIES IN THE AFFINE FLAG VARIETY OF TYPE ˜ A s s s s n − s n − Figure 1.
The Dynkin diagram of type ˜ A n Proof of Theorem 1.1. If X ( w ) is smooth, then w avoids 3412 and 4231 by Theorem1.4, so part (a) implies part (b).Suppose w avoids 3412 and 4231. If w = vu is a parabolic decomposition, then u also avoids 3412 and 4231 [BC12, Lemma 3.10]. Thus Proposition 2.3 impliesthat w has a complete BP decomposition, in which every factor belongs to a properparabolic subgroup of W . Every rationally smooth Grassmannian Schubert varietyin finite type A is smooth by the Carrell-Peterson theorem [CK03], and hence aGrassmannian. Thus the proof of Corollary 2.6 implies that X ( w ) is an iteratedfibre bundle of Grassmannians. Hence part (b) implies part (c).Similarly, if X ( w ) is an iterated fibre bundle of Grassmannians, then X ( w ) issmooth as in the proof of Corollary 2.6, so part (c) implies part (a). (cid:3) We finish the section by looking at rationally smooth Schubert varieties, start-ing with the twisted spiral permutations. Suppose W has type ˜ A n , and let S = { s , . . . , s n − } , where s i is the simple reflection corresponding to node i in the Dynkindiagram of ˜ A n as shown in Figure 1. Given 0 ≤ i < n and k ≥
1, define x ( i, k ) = s i + k − s i + k − · · · s i and y ( i, k ) = s i − k +1 s i − k +2 · · · s i , where the indices of the s j ’s are interpreted modulo n . Both x ( i, k ) and y ( i, k ) belongto W S \{ s i } . A spiral permutation is an element of W of the form x ( i, k ( n − y ( i, k ( n − k ≥ The spiral permutations were first studied by Mitchell[Mit86], who showed that the corresponding Grassmannian Schubert varieties, called spiral Schubert varieties , are rationally smooth.A twisted spiral permutation is an element of W of the form w = vu , where v is aspiral permutation x ( i, k ( n − y ( i, k ( n − u is the maximal element of W S \{ s i } . Note that w = vu is a BP decomposition with respect to J = S \ { s i } . Inaddition, D R ( w ) contains D R ( u ) = S \{ s i } , and since ˜ A n is infinite, D R ( w ) cannot beequal to S , so D R ( w ) must be equal to S \ { s i } . Thus we get a version of Proposition2.3 for all rationally smooth elements of W : if X ( w ) is rationally smooth, then either • w is the maximal element of a proper parabolic subgroup of W , or • w has a Grassmannian BP decomposition with respect to K = S ( w ) \ { s } ,where s D R ( w ). Note that the length of these elements is a multiple of n −
1, even though the rank of W is n .In particular, if k = 1 then these elements belong to a parabolic subgroup of finite type A , so weexclude this case. EDWARD RICHMOND AND WILLIAM SLOFSTRA
This means that we can repeat the proofs of Corollaries 2.4 and 2.6 to get:
Corollary 2.7.
A Schubert variety X J ( w ) in a partial flag variety of type ˜ A n isrationally smooth if and only if there is a sequence X J ( w ) = X m → X m − → · · · → X → X = point , where each map X i → X i − is a Zariski-locally trivial fibre bundle whose fibre is arationally smooth Grassmannian Schubert variety of type A or ˜ A . It is implicit in the proof of Corollary 2.7 that if X J ( w ) is rationally smooth, thenwe can construct such a sequence where all the fibres are either Grassmannians offinite type A , or spiral Schubert varieties. In fact, these are the only rationally smoothGrassmannian Schubert varieties, by a theorem of Billey and Mitchell [BM10].In the finite-type analogue of Corollary 2.7, every rationally smooth GrassmannianSchubert variety is almost-maximal [RS16]. Although we don’t need that fact here,it is interesting to note that the spiral permutations are also almost-maximal.3. Staircase diagrams for affine type ˜ A The main fact we had to establish in the previous section was that every 3412-and 4231-avoiding element of type ˜ A has a complete maximal BP decomposition.To prove this fact, we showed that the existence of BP decompositions for w or w − implies the existence of BP decompositions for w . The same proof strategy was usedin [RS16]. Both results are instances of a more general result, which we formulateas follows: Proposition 3.1.
Let F be a family of Coxeter groups which is closed under parabolicsubgroups (i.e. if W ∈ F , then W J ∈ F for every J ⊆ S ). Let C be a class of elementsof the groups of F such that if w ∈ C , then(1) w − ∈ C , and(2) u ∈ C for all parabolic decompositions w = vu .(3) either w or w − has a maximal Grassmannian BP decomposition, i.e. aGrassmannian BP decomposition vu where v is maximal in W S ( u ) ∩ S ( v ) S ( v ) .Then every w ∈ C has a complete maximal BP decomposition. The point of this section is to show that Proposition 3.1 has a short proof usingstaircase diagrams. Taking F to be the Weyl groups of finite type A and affine type˜ A , and C to be the class of permutations avoiding the pattern 3412 and 4231, weget a proof of Theorem 1.1 by a somewhat different route. Our proof of Proposition3.1 will still hold if we replace “maximal” by “maximal or almost-maximal” BPdecompositions, and hence Proposition 3.1 also gives an alternate path to the resultson existence of BP decompositions in [RS16]. For simplicity, we restrict ourselves tomaximal BP decompositions.As we will also use staircase diagrams in the next section, we briefly review thedefinition from [RS15]. Let G be a graph with vertex set S , and recall that a subset MOOTH SCHUBERT VARIETIES IN THE AFFINE FLAG VARIETY OF TYPE ˜ A B ⊆ S is connected if the subgraph of G induced by B is connected. If D is acollection of subsets of S and s ∈ S , we set D s := { B ∈ D | s ∈ B } . Staircase diagrams are then defined as follows:
Definition 3.2 ([RS15]) . Let Γ be a graph with vertex set S . Let D = ( D , (cid:22) ) bea partially ordered subset of S not containing the empty set. We say that D is a staircase diagram if the following are true:(1) Every B ∈ D is connected, and if B covers B ′ then B ∪ B ′ is connected.(2) The subset D s is a chain for every s ∈ S .(3) If s adj t , then D s ∪ D t is a chain, and D s and D t are saturated subchains of D s ∪ D t .(4) If B ∈ D , then there is some s ∈ S (resp. s ′ ∈ S ) such that B is the minimumelement of D s (resp. maximum element of D s ′ ). The definition is symmetric with respect to the partial order, so if D = ( D , (cid:22) ) isa staircase diagram, and (cid:22) ′ is the reverse order to (cid:22) , then ( D , (cid:22) ′ ) is also a staircasediagram, called flip( D ). Finally, if D is a staircase diagram over the Coxeter graphof a Coxeter group W , we say D is spherical if W B is a finite group for all B ∈ D .The main result about staircase diagrams is: Theorem 3.3 ([RS15], Theorem 5.1, Theorem 3.7, and Corollary 6.4) . Let Γ be theCoxeter graph of a Weyl group W . Then there is a bijection between spherical stair-case diagrams over Γ , and elements of w with a complete maximal BP decomposition(relative to ∅ ).Furthermore, if a staircase diagram D corresponds to w ∈ W , then flip( D ) corre-sponds to w − .Second proof of Proposition 2.3. The proof is by induction on | S ( w ) | . Suppose that w has a maximal Grassmannian BP decomposition w = vu . Since | S ( u ) | < | S ( v ) | ,we can conclude by induction that u has a complete maximal BP decomposition.But these means that w has a complete maximal BP decomposition, and we aredone. Similarly, if w − has a maximal Grassmannian BP decomposition, then weconclude that w − has a complete maximal BP decomposition. But this means that w − comes from a staircase diagram D , so w corresponds to flip( D ). In particular, w must also have a complete maximal BP decomposition. (cid:3) Enumeration of smooth Schubert varieties
Let ˜Γ n be the Coxeter graph of type ˜ A n with vertices ˜ S n = { s , s , . . . , s n − } as inFigure 1. In this case, proper, connected subsets of ˜ S n are simply intervals on thecycle graph ˜Γ n . Define the interval −−−→ [ s i , s j ] := ( { s i , . . . , s j } if i ≤ j { s i , . . . , s n − , s , . . . , s j } if i > j We represent a staircase diagram D pictorially with a collection of “blocks” whereif B lies above B and B ∪ B is connected, then B ≺ B in D . For example, thestaircase diagram D = {−−−−→ [ s , s ] ≺ −−−−→ [ s , s ] ≺ −−−−→ [ s , s ] ≺ −−−−→ [ s , s ] } over ˜Γ could be represented in Figure 2. Figure 2.
Staircase diagram on ˜Γ However, for the sake of convenience we will represent the cycle graph ˜Γ n as a linegraph with vertex s each end point, as follows. s s s s n − s n − s We can then draw the staircase diagram in two-dimensions; for instance, the stair-case diagram in Figure 2 is represented as:
Note that if s ∈ B ∈ D , then B appears as a “disconnected” block in the pictorialrepresentation of D .By Corollary 2.6 and Theorem 3.3, to enumerate smooth Schubert varieties of type˜ A n , it suffices to enumerate spherical staircase diagrams over the graph ˜Γ n . Sinceevery proper subgraph of ˜Γ n is a Dynkin diagram of finite type, the only non-sphericalstaircase diagram is D = { ˜ S n } .We first consider staircase diagrams over the Dynkin graph of finite type A n . LetΓ n denote the line graph with vertex set S n = { s , . . . , s n } . In this case, we denoteinterval blocks on line graph Γ n as simply [ s i , s j ] for i < j . s s s n − s n MOOTH SCHUBERT VARIETIES IN THE AFFINE FLAG VARIETY OF TYPE ˜ A A staircase diagram D is said to have full support if every vertex of the underlyinggraph appears in some B ∈ D . We say a staircase diagram D over Γ n of full supportis increasing if we can write D = { B ≺ B ≺ · · · ≺ B m } with s ∈ B . Similarly, a staircase diagram is decreasing if if we can write D = { B ≻ B ≻ · · · ≻ B m } with s ∈ B . Let M ± ( n ) denote the set of fully supportedincreasing/decreasing staircase diagrams over Γ n . Define the generating series(1) A M ( t ) := ∞ X n =1 m n t n where m n := | M + ( n ) | = | M − ( n ) | . Proposition 4.1.
The generating function A M ( t ) = 1 − t − √ − t t .Proof. The proposition can be proved by modifying the proof of [RS15, Proposition8.3]. In this paper we present an alternate proof by giving a bijection between fullysupported increasing staircase diagrams and Dyck paths. Indeed, let D = { B ≺ B ≺ · · · ≺ B m } be a fully supported increasing staircase diagram on Γ n . For each B i ∈ D , define the numbers r ( B i ) := { s ∈ B i \ B i − } and u ( B i ) := { s ∈ B i \ B i +1 } where we set B = B m +1 = ∅ . Let P ( D ) denote the lattice path in Z from (0 ,
0) to( n, n ) which takes r ( B ) steps to the right, then u ( B ) steps going up, followed by r ( B ) steps to the right, then u ( B ) steps going up and so forth (See Example 4.2).Since D is fully supported, we have that m X i =1 r ( B i ) = m X i =1 u ( B i ) = n and hence P ( D ) terminates at ( n, n ). Definition 3.2 implies r ( B i ) , u ( B i ) > i X k =1 r ( B k ) ≥ i X k =1 u ( B k )for all i ≤ m. Thus P ( D ) is a Dyck path. Conversely, any Dyck path is given bya sequence of positive pairs ( r i , u i ) giving steps to the right followed up steps goingup. Set u := 0 and define¯ B i := ( s j ∈ S | i − X k =1 u k < j ≤ i X k =1 r k ) and ¯ D := { ¯ B ≺ ¯ B ≺ · · · ≺ ¯ B m } . It is easy to see that ¯ D is a fully supportedstaircase diagram and that this construction is simply the inverse of the map P . Theproposition now follows from the generating function for Dyck paths which is givenby Catalan numbers. (cid:3) Example 4.2.
Consider the staircase diagram D = ( s ≺ [ s , s ] ≺ [ s , s ]) on Γ .The sequence of pairs ( r i , u i ) is ((1 , , (4 , , (1 , and corresponding Dyck path P ( D ) is given below. The idea behind enumerating staircase diagrams over ˜Γ n is to partition a staircasediagram into a disjoint union of increasing and decreasing staircase diagrams of finitetype A . To do this precisely, we introduce the notion of a broken staircase diagram.We say that a partially ordered collection of subsets ( B , ≺ ) of vertices of the graphΓ n is a broken staircase diagram if B = { B ∩ S n | B ∈ D} for some D ∈ M + ( n + 1) ∪ M − ( n + 1) where the partial order on B is induced from D .Note that broken staircase diagrams are allowed to violate part (4) of Definition 3.2,and must be either increasing or decreasing. In particular, if B = { B ≺ · · · ≺ B m } is broken, then it may be possible for B m ⊂ B m − . Define the generating series A B ( t ) = ∞ X n =1 b n t n where b n denotes the number of increasing (or equivalently, decreasing) broken stair-case diagrams on Γ n . Proposition 4.3.
The generating function A B ( t ) = (1 − t ) A M ( t ) t − . Proof.
Clearly b = 0 and b = 1, so we assume that n ≥
2. Let D = { B ≺ · · · ≺ B m } ∈ M + ( n +1) and let B ( D ) = { B ∩ S n | B ∈ D} denote the corresponding brokenstaircase diagram. If block index k ≤ m −
2, then B k ⊆ S n . Hence B ( D ) determines D up to the last two blocks B m − , B m . If B m ⊂ B m − , then B ( D ) uniquely determines D as shown in Figure 3. If B m B m − , then there are two possibilities for D given Figure 3.
The broken staircase diagram B ( D ) determining D . B ( D ). Either D s n = { B m } and thus s n , s n +1 ∈ B m , or D s n = { B m − } which implies B m = { s n +1 } (see Figure 4). In the latter case, removing the last block from D gives MOOTH SCHUBERT VARIETIES IN THE AFFINE FLAG VARIETY OF TYPE ˜ A or Figure 4.
Two possibilities for D given B ( D ).a unique staircase diagram in M + ( n ). Hence b n = m n +1 − m n and t + tA B ( t ) = A M ( t ) − tA M ( t ) . This proves the proposition. (cid:3)
We can now state the main bijection:
Proposition 4.4.
There is a bijection between fully-supported spherical staircasediagrams on ˜Γ n , and pairs [( B , . . . , B k ) , v ] , where • B i is a broken staircase on Γ n i , n i ≥ , • P ki =1 n i = n , • for all ≤ i ≤ k − , if B i is increasing (resp. decreasing) then B i +1 isdecreasing (resp. increasing), and • v is a distinguished vertex in B k .Proof. For the purpose of this proof, we let s j + nk = s j for any k and 0 ≤ j < n .Suppose D is a fully-supported staircase diagram on ˜Γ n , and B is any block of D , say B = −−−→ [ s i , s j ]. By Definition 3.2, part (3), all the blocks of D s j +1 are comparable with B . If B has an upper cover B ′ (cid:23) B in D s j ∪D s j +1 , then there are no elements of D s j +1 below B , since then D s j +1 would not be saturated in D s j ∪ D s j +1 . And vice-versa,if B has a lower cover in D s j ∪ D s j +1 then there are no elements of D s j +1 above B .Consequently we can say that B has a unique cover B ′ containing s j +1 . We call B ′ the right cover of B .Choose some block B , and let B , . . . , B m be a sequence where B i +1 is the rightcover of B i for 1 ≤ i < m , and B is the right cover of B m . Then every block of D must appear in this sequence. Indeed, every vertex of ˜Γ n appears in some block inthis sequence, so every block of D is comparable to some element of the sequence.It follows that if there is a block of D not in the sequence, then there is a block B not in the sequence which has an upper or lower cover B ′ in the sequence. Theneither B will be the right cover of B ′ , or B ′ will be the right cover of B . But thesame argument as above shows that B ′ has a unique left cover, and this is the onlyelement with B ′ as a right cover. So in both cases, B must also be in the sequence,a contradiction.Let B i , . . . , B i m denote the subsequence of extremal blocks, i.e. blocks whichare maximal or minimal. Note that if B i j is maximal then B i j +1 must be minimal,and vice-versa. Since ˜Γ n is a cycle, the same must apply to B i m and B i , andin particular m must be even. By Definition 3.2, part (4), every extremal block contains a vertex which does not belong to any block. Let 1 ≤ c j ≤ n be the indexof the leftmost such vertex in B i j . By cyclically shifting the indices, we can assumethat 1 ≤ c < c < . . . < c m < n . Finally, set J j = −−−−−−−−→ [ s c j , s c j +1 − ] 1 ≤ j < m −−−−−−−→ [ s c m , s c − ] j = m , so that J , . . . , J m partitions ˜ S n , and let B j = { B ∩ J j : B ∈ D and B ∩ J j = ∅} with the induced partial order. Since B i j is the only block containing s c j , and noblock of D contains any other [RS15, Lemma 2.6(b)], the block B i j can meet at mosttwo of the intervals J k . Hence B j is either an increasing or decreasing chain. Itfollows that B , . . . , B m is a sequence of broken staircases as required. We set v tobe the vertex s n − , which is always in B m by construction.This construction gives a map from staircase diagrams to sequences of brokenstaircases with a marked vertex. To show that this map has an inverse, supposethat B = { B ≺ · · · ≺ B m } is an increasing broken staircase. If B m ⊂ B m − thenwe can think of B as the staircase diagram { B ≺ · · · ≺ B m − } with an additionalbroken block B m on top of B m − . If B m B m − , so B is a staircase diagram in itsown right, then we think of B as a staircase diagram with an empty broken blockon top of the block B m , starting and ending after the rightmost vertex of B m . If B ′ is then a decreasing staircase diagram, we can glue B and B ′ together by attachingthe broken block of B to the first block of B ′ . We can similarly glue a decreasingbroken staircase to an increasing broken staircase. Given a sequence B , . . . , B k ofalternately increasing and decreasing broken staircases, we can glue them togetherin order, and then glue B k to B to get a staircase diagram on a cycle. Labellingthe vertices of the cycle with s , . . . , s n − starting to the right of the marked vertex v , we get a staircase diagram on ˜Γ n , and this process inverts the above map. (cid:3) Example 4.5.
The staircase diagram D = {−−−−→ [ s , s ] ≺ −−−−→ [ s , s ] ≺ −−−−→ [ s , s ] ≻ −−−−→ [ s , s ] ≻ −−−−→ [ s , s ] ≺ −−−−→ [ s , s ] } has four extremal blocks and partitions into an alternating sequence of increasing anddecreasing broken staircase diagrams as follows: Define the generating series ¯ A ( t ) = ∞ X n =1 ¯ a n t n where ¯ a n denotes the number of fully supported spherical staircase diagrams on ˜Γ n . MOOTH SCHUBERT VARIETIES IN THE AFFINE FLAG VARIETY OF TYPE ˜ A Corollary 4.6.
The generating function ¯ A ( t ) = 2 A B ( t ) · t ddt A B ( t )1 − A B ( t ) . Proof.
Follows immediately from Proposition 4.4, and the fact that t ddt A B ( t ) is thegenerating series for broken staircases with a marked vertex. Note that we get afactor of two because the first broken staircase can be increasing for decreasing. (cid:3) If a staircase diagram on ˜Γ n is not fully supported, then it is a disjoint union offully supported staircase diagrams over a collection of subpaths of the cycle. Let f n denote the number of fully supported staircase diagrams on the path Γ n and definethe generating series A F ( t ) := ∞ X n =0 f n t n . The following proposition is proved in [RS15, Proposition 8.3]. We give an alternateproof using broken staircase diagrams.
Proposition 4.7. ([RS15, Proposition 8.3])
The generating function A F ( t ) = A M ( t )1 − A B ( t ) .Proof. We emulate the proof of Proposition 4.4 as follows: Given a staircase diagram D on Γ n , let B , . . . , B k be the maximal and minimal blocks in order from left toright. Let 1 ≤ m ≤ k be the largest index such that B m is minimal in D (so actually, m ∈ { k − , k } ). For every 1 ≤ j ≤ m , let a j be the index of the leftmost element of B j which is not contained in any other block, and let b j = ( a j +1 − j < mn j = m . Let B i = [ s a i , s b i ]. Then B i is a broken staircase for 1 ≤ i ≤ m −
1, while B m is anincreasing staircase. This also implies that B m − is decreasing, B m − is increasing,and so on. It is not hard to see that this map is a bijection, and hence everystaircase diagram over Γ n decomposes into a sequence of broken staircases, followedby an increasing staircase. (cid:3) One subtlety of the above bijection is that it seems to miss the case when D is de-creasing, or more generally, ends with a decreasing staircase. However, a decreasingstaircase decomposes into a decreasing broken staircase followed by a single block.Since a single block is an increasing staircase, the bijection will in fact count decreas-ing staircases correctly. Since single blocks are both increasing and decreasing, theseemingly more straightforward approach of allowing B m to be increasing or decreas-ing will lead to overcounts. The reason this problem doesn’t arise in Proposition 4.4is that in that bijection every B i is a broken staircase. We can always tell whether abroken staircase is increasing or decreasing based on where it is glued to the adjacentbroken staircase. Finally, let a n denote the number of spherical staircase diagrams over the graph˜Γ n and define A ( t ) = ∞ X n =1 a n t n . Proposition 4.8.
The generating series A ( t ) = ¯ A ( t ) + t ddt ( A ∗ ( t ))1 − A ∗ ( t ) + t − t where A ∗ ( t ) = tA F ( t )1 − t . Proof.
First note that the generating function fully supported staircase diagramsover ˜Γ n is ¯ A ( t ). If a staircase diagram is not fully supported and nonempty, thenit partitions into a sequence ( D , E , . . . , D r , E r ) where each D k is a nonempty, fullysupported staircase diagram over a path and E k is an empty staircase diagram overa path of at least length one. Moreover, we can choose such a partition such that s is in the support of D or E . Thus we get a bijection between non-empty non-fully-supported staircase diagrams on ˜Γ n and sequences ( D , E , . . . , D r , E r ) where( D , E ) has a marked vertex corresponding to s . The generating series for staircasediagrams corresponding to pairs ( D k , E k ) over Γ n is A ∗ ( t ) := A F ( t ) · t − t . Thus the generating function for the number non-fully support staircase diagramsover ˜Γ n is t ddt ( A ∗ ( t ))1 − A ∗ ( t ) + t − t where the second summand corresponds to the generating function of empty staircasediagrams. This completes the proof. (cid:3) Proof of Theorem 1.2.
Combine Propositions 4.1, 4.3, 4.7, and 4.8, along with Corol-lary 4.6. (cid:3)
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