SPRINGER FIBERS AND SCHUBERT POINTS MARTHAPRECUPANDJULIANNATYMOCZKO 7 1 0 2 Abstract. Springerfibersaresubvarietiesoftheflagvarietyparametrizedbypartitionsandare centralobjectsofstudyingeometricrepresentationtheory. Schubertvarietiesaresubvarietiesof n a theflagvarietythatinduceawell-knownbasisforthecohomologyoftheflagvariety. Thispaper J relates these two varieties combinatorially. We prove that the Betti numbers of the Springer 2 fiber associated to a partition with at most three rows or two columns are equal to the Betti 1 numbersofaspecificunionofSchubertvarieties. O] 1. Introduction C This paper proves an explicit combinatorial topological relationship between two families of h. varieties: certain Springer fibers and certain Schubert varieties. Both are subvarieties of the flag t variety B, whose elements in Lie type A can be described as the collection of nested subspaces a m V = ({0} ⊆ V ⊆ ··· ⊆ V = V) where each V is an i-dimensional subspace of a fixed complex • 1 n i n-dimensionalvectorspaceV. TheflagvarietycanalsobewrittenasthequotientB =GL (C)/B [ n where B is the subgroup of upper-triangular matrices. 1 Springer fibers are the fibers of a particular desingularization of the nilpotent cone inside the v 2 space of n×n matrices. Explicitly, if X is a nilpotent n×n matrix then the flag V is in the • 0 Springer fiber BX if and only if X(V ) ⊆ V for all i = 1,...,n. In other words BX consists of all i i 5 flagsthatarestableundertheoperatorX. ThecohomologyofeachSpringerfibercarriesanatural 3 0 action of the symmetric group that is one of the seminal constructions of geometric representation 1. theory. Let λ be the partition of n determined by the Jordan blocks of X. Springer showed that 0 the top-dimensional cohomology of BX is the irreducible representation of S corresponding to λ n 7 [Sp, Sp2] and in fact every irreducible representation of S can be obtained this way. 1 n Schubert varieties are subvarieties of the flag variety parametrized by permutations that induce : v animportantbasisforthecohomologyoftheflagvariety. Theirgeometryisintrinsicallyconnected i X to the combinatorics of the symmetric group [F, BL]. To start, permutations index the double- r coset decomposition B = (cid:70)BwB/B. Each double coset is an open affine cell C = BwB/B in a w the flag variety that contains the permutation flag wB and is called a Schubert cell. The closure relationsbetweenSchubertcellsaredeterminedbytheBruhatorderonS andthedimensionofthe n Schubert variety C is given by the number of inversions of w. Moreover the cohomology classes w ofSchubertvarietiesaregivenbytheSchubertpolynomials[BGG],whichareimportantobjectsin algebraiccombinatoricsandrepresentationtheory. ThestudyofSchubertvarietiesandpolynomials fundamentally relates results in geometry, combinatorics, Lie theory, and representation theory; see [BP, CK, K] for just a few examples. In this paper we show that if the partition λ has at most three rows or two columns then the Betti numbers of BX coincide with the Betti numbers of a particular union of Schubert varieties. We make this correspondence explicit using row-strict tableaux. The second author showed that the Betti numbers of Springer fibers are enumerated by row-strict Young tableaux of shape λ and gave a combinatorial rule to compute the Betti number corresponding to a given tableau T in [T]. ThenumbercomputedfromT bythiscombinatorialruleiscalledthedimensionofT. Usingwork 1 2 MARTHAPRECUPANDJULIANNATYMOCZKO of Garsia-Procesi and Mbirika [GP, M], we identify a unique permutation w ∈ S associated to T n each row-strict tableau T with the property that the length (cid:96)(w ) is the dimension of T. These T permutations are called Schubert points. Our main result is that the Betti numbers of the Springer fibers are the same as the Betti numbers of the union of Schubert varieties corresponding to Schubert points. Theorem 1. Let X ∈gl (C) be a nilpotent matrix with Jordan form corresponding to a partition n λ with at most three rows or two columns. There is an equality of Poincaré polynomials P(BX,t)=P (cid:0)∪ C ,t(cid:1) wB∈BX wT where T denotes the row-strict tableau associated to wB ∈ BX and w ∈ S is the corresponding T n Schubert point. The union simplifies as (cid:91) (cid:91) C = C wT wT wB∈BX T∈St(λ) where St(λ) denotes the set of standard tableaux of shape λ. ThisprovesaconjecturearisingoutofworkbyHaradaandthesecondauthor,whichsuggesteda similarresultforalargerfamilyofvarietiescallednilpotentHessenbergvarieties[HT]. Earlierwork of Mbirika proves the conjecture for regular nilpotent Hessenberg varieties, whose Betti numbers equal those of a particular Schubert variety [M]. The second part of Theorem 1 recovers one of the key conclusions of Springer theory: that the top-dimensional components of BX are indexed by standard tableaux, implied both by Springer’s original work and geometrically by later work of Spaltenstein [S] and others. In fact our main result is stronger: it is a bijection on all Betti numbers, not just the top- dimensional ones. While this paper does not stress the geometric context, this is a dimension- (cid:83) preserving bijection between the Schubert cells in the union C and a set of affine cells T∈St(λ) wT C ∩BX that partition the Springer variety. w Thisbijectionisparticularlysimpleforthestandardtableaux,ortop-dimensionalcase. Toeach standard tableau T of shape λ we associate the following Schubert point w . If i occurs in the kth T row of T set (cid:26) s ···s s if k >1 w := i−k+1 i−2 i−1 i−1 e if k =1 where s denotes the simple transposition (i,i+1). Then w =w w ···w w is the product i T n−1 n−2 2 1 of these strings. (Definition 3.2 describes Schubert points for general row-strict tableaux.) Example 2. Consider the partition λ = (2,2,1) of 5. Below we list all standard tableaux of this shape and the corresponding Schubert points. 1 2 1 2 1 3 1 3 1 4 T ∈St(2,2,1) 3 4 3 5 2 4 2 5 2 5 5 4 5 4 3 w ∈S s s s s s s s s s s s s s s s s s s s s T 5 3 4 3 2 4 2 3 2 3 4 3 1 4 2 3 1 4 1 2 1 ByTheorem1,ifX hasJordanblocksofdimensions(2,2,1)thentheBettinumbersoftheSpringer fiber BX are equal to those of the union X ∪X ∪X ∪X ∪X . s3s4s3s2 s4s2s3s2 s3s4s3s1 s4s2s3s1 s4s1s2s1 Our results also strengthen related work of Garsia-Procesi [GP]. Garsia and Procesi defined a monomial basis for the cohomology ring of each Springer fiber BX that Mbirika later bijectively associatedtotherow-stricttableauxofshapeλ. InMbirika’sbijection,thedegreeofeachmonomial corresponds to the dimension of the corresponding tableau [M]. (Mbirika’s work also provides a SPRINGER FIBERS AND SCHUBERT POINTS 3 readable summary of Garsia-Procesi’s algorithm that uses the notation of this paper.) Schubert points are related to Garsia-Procesi’s monomials by the rule n (cid:89)x(cid:96)i−1 ←→w :(cid:96)(w )=(cid:96) . i T i−1 i−1 i=2 For instance, the monomials in the previous example are x2x x ,x x2x ,x2x x ,x x2x , and 5 4 3 5 4 3 5 4 2 5 4 2 x x2x . Garsia-Procesiprovedthatthesemonomialsareclosedunderthepartialorderofdivision: 5 3 2 if xβ is a monomial corresponding to a row-strict tableau and xα|xβ then xα corresponds to a row-strict tableau, too. Similarly, we prove that if w is the Schubert point corresponding to a T row-strict tableau and w(cid:48) ≤w in Bruhat order then w(cid:48) is also the Schubert point corresponding T to a row-strict tableau. What makes our result more powerful is that Bruhat order is stronger than the monomial order. For instance x (cid:54) |x x2x but s < s s s s in Bruhat order. Indeed, 3 5 4 2 2 4 2 3 1 the monomial order is equivalent to erasing some initial simple transpositions from the string w i while Bruhat order permits erasing simple transpositions in arbitrary locations. The methods and results in this paper are combinatorial. In a second paper we extend these results to parabolic nilpotent Hessenberg varieties [PT]. Despite results in these cases and in the regular nilpotent Hessenberg case, we see no straightforward way of extending these methods to general nilpotent Hessenberg varieties or other Springer fibers. The fact that results of this nature hold in so many cases may indicate some deeper geometric phenomenon, such as the degeneration given by Knutson and Miller in [KM] from a Schubert variety to a collection of line bundles. However, the geometry of Springer fibers is much less well understood than that of the Schubert varieties, so new methods will be necessary to find such a degeneration. The next section covers background information on the geometry of Springer fibers and the dimensions of row-strict tableaux. Section 3 describes Schubert points and preliminary properties relating them to the permutation flags in Springer fibers. We prove the main result, Theorem 4.4, in Section 4 using a lemma that is proven for three-row tableaux in Section 5 and for two-column tableaux in Section 6. Acknowledgements. The first author was partially supported by an AWM-NSF mentoring grant. The second author was partially supported by National Science Foundation grants DMS- 1248171 and DMS-1362855. 2. Geometric background on Schubert varieties and Springer fibers This section establishes notation and key definitions about Springer fibers. LetB betheBorelsubgroupofGL (C)consistingofupper-triangularmatrices. Theprojective n variety B = GL (C)/B is the flag variety. As noted in the introduction, the flag variety can be n identified with the set of full flags V ⊆ V ⊆ ··· ⊆ V ⊆ V in a complex n-dimensional vector 1 2 n−1 spaceV. TheWeylgroupW isthesubgroupofpermutationmatricesinGL (C). Wecanidentify n W with the symmetric group on n letters S via the action on column vectors. The Weyl group n is generated by the simple transpositions s =(i,i+1). The Bruhat order on W is defined by the i rulethatv ≤w ifv canbewrittenasasubwordofw wheneachisexpressedintermsofthesimple transpositions. If w factors minimally into simple transpositions as w = s s ···s then (cid:96)(w) i1 i2 i(cid:96)(w) is the length of w. The length of w is also equal to the number of inversions of w. (cid:70) The Bruhat decomposition partitions the flag variety B = C into a union of Schubert w∈Sn w cells, each of which is induced by a double coset. The Schubert cell indexed by w ∈ S is the n collection of flags C = BwB/B. This is in fact a CW-decomposition and it can be proven that w C = (cid:70) C where ≤ denotes the Bruhat order and C ∼= C(cid:96)(v) for all v ∈ S . (See [BL] for a w v≤w v v n more thorough introduction.) 4 MARTHAPRECUPANDJULIANNATYMOCZKO ThisdescriptionoftheSchubertcellsallowsonetocalculatethePoincarépolynomialofSchubert varieties using the combinatorics of permutations, as shown in the following example. Example 2.1. Let G = GL (C) and consider the union of Schubert varieties from Example 2, 5 X ∪X ∪X ∪X ∪X . The set of all permutations less than or s3s4s3s2 s4s2s3s2 s3s4s3s1 s4s2s3s1 s4s1s2s1 equal to each of s s s s , s s s s , s s s s , s s s s , and s s s s respectively in Bruhat order 3 4 3 2 4 2 3 2 3 4 3 1 4 2 3 1 4 1 2 1 is • s s s s ,s s s ,s s s ,s s s ,s s ,s s ,s s ,s s ,s ,s ,s ,e, 3 4 3 2 3 4 3 3 4 2 4 3 2 3 4 4 3 4 2 3 2 4 3 2 • s s s s ,s s s ,s s s ,s s s ,s s ,s s ,s s ,s s ,s ,s ,s ,e, 4 2 3 2 4 2 3 2 3 2 4 3 2 4 2 4 3 3 2 2 3 4 3 2 • s s s s ,s s s ,s s s ,s s s ,s s ,s s ,s s ,s s ,s ,s ,s ,e, 3 4 3 1 3 4 3 3 4 1 4 3 1 3 4 4 3 3 1 4 1 4 3 1 • 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 ,s s ,s ,s ,s ,s ,e, and 4 2 3 1 4 2 3 4 2 1 4 3 1 2 3 1 4 2 4 3 2 3 4 1 3 1 2 1 4 3 2 1 • s s s s ,s s s ,s s s ,s s s ,s s ,s s ,s s ,s s ,s ,s ,s ,e. 4 1 2 1 4 1 2 4 2 1 1 2 1 4 2 4 1 2 1 1 2 4 2 1 Therefore P(X ∪X ∪X ∪X ∪X ,t)=5t4+11t3+9t2+4t+1. s3s4s3s2 s4s2s3s2 s3s4s3s1 s4s2s3s1 s4s1s2s1 We now define the subvariety of B that is the main focus of this manuscript. Definition 2.2 (Springer fiber). Let X be an n×n nilpotent matrix. The Springer fiber BX consists of all flags gB ∈ B such that g−1Xg is upper-triangular, or equivalently the flags V ∈ B • with XV ⊆V for all i∈{1,2,...,n}. i i Instead of a CW-decomposition, Springer fibers have a partition called an affine paving. The closure conditions are weaker in an affine paving than a CW-decomposition but the cells and their dimensions still compute Betti numbers. (Surveys like Fulton’s text have more details [F2].) If X ischosenappropriatelyinitsconjugacyclass, anaffinepavingoftheSpringerfiberBX isobtained byintersectingwiththeSchubertcells. IfwB isapermutationflaginBX thenwecallwaSpringer permutation. The Springer fibers corresponding to X and to any conjugate of X are homeomorphic [T, Proposition 2.7] so the Betti numbers of BX are an invariant of the conjugacy class of X. When X is nilpotent its conjugacy class is given by the sizes of its Jordan blocks, which we encode as a partition λ of n. For this reason we refer to the Betti numbers of Bλ in much of this paper. We now give a combinatorial description of the Springer permutations and the Betti numbers of Bλ. We start with some basic definitions. Definition 2.3 (Partitions and base fillings). Let λ = (λ ,λ ,...,λ ) be a partition of n drawn 1 2 k as a Young diagram, namely with k rows of boxes so that the ith row from the top has λ boxes. i The base filling of λ is obtained as follows. Fill the boxes of λ with integers 1 to n starting at the bottom of the leftmost column and moving up the column by increments of one. Then move to the lowest box of the next column and so on. Example 2.4. Let n=5 and λ=(3,2). The base filling of λ is: 2 4 5 1 3 In fact the row-strict tableaux of shape λ parameterize Springer permutations, and a quantity like the inversions of a permutation describe the dimensions of the corresponding affine cell [T, Theorem 7.1]. Lemma 2.5 (Tymoczko). Fix a partition λ of n and consider its base filling. Suppose that X is the matrix such that X =1 if j fills a box directly to the right of k and X =0 otherwise. The kj kj SPRINGER FIBERS AND SCHUBERT POINTS 5 Springer fiber BX is paved by affines C ∩BX. The intersection C ∩BX is nonempty if and only w w if wB ∈ BX or equivalently if and only if the filling of λ given by labeling the ith box in the base filling of λ by w−1(i) is row-strict. If T denotes that row-strict tableau of shape λ, the dimension of C ∩BX is equal to the number of pairs (p,q) such that 1≤p1ofT(cid:48). Thenr isgreaterthani−1otherentriesofthekth columninT. SinceT isrow-strict i r is greater than at least i distinct entries in the k−1st column of T. Thus r is greater than the i i box to its immediate left in T(cid:48). So T(cid:48) is row-strict and by construction also standard. We claim that w ≤ w . Consider T[q] and T(cid:48)[q] for 2 ≤ q ≤ n. The number of rows of T T(cid:48) each length is the same in T[q] as in T(cid:48)[q] because we obtained T(cid:48) from T by reordering entries withincolumns. InparticulartherowsinT[q]andT(cid:48)[q]containingq haveequallength. Thusboth T[q] and T(cid:48)[q] have the same number of rows of strictly greater length than the row containing q. Additionally any row in T[q] above the row containing q and of equal length will end in a box in the same column of T as q and be labeled by a value p < q. Since T(cid:48) reorders the entries of each column of T to increase from top to bottom, this row will also occur above the row containing q in T(cid:48)[q]— and there may be more rows of this type in T(cid:48)[q]. Lemma 2.7 implies that the number of Springer dimension pairs (p,q) in T is at most the number of Springer dimension pairs (p,q) in T(cid:48). Inotherwords(cid:96)(w )≤(cid:96)(w(cid:48) )forall2≤q ≤n. Byconstructionw ≤w asdesired. (cid:3) q−1 q−1 T T(cid:48) The second claim is a special case of our main theorem, and a slight modification of results of Garsia-Procesi and Mbirika [GP, M]. Lemma 3.6. Let T be a row-strict tableau of shape λ and denote the corresponding Schubert point byw =w w ···w w . Supposethatw(cid:48) isapermutationoftheformw(cid:48) =w(cid:48) w(cid:48) ···w(cid:48)w(cid:48) T n−1 n−2 2 1 n−1 n−2 2 1 where w(cid:48) ≤w in Bruhat order for all i=1,..,n−1. Then w(cid:48) is a Schubert point associated to λ. i i Proof. We have only to show that there exists a row-strict diagram T(cid:48) of shape λ such that w =w(cid:48). The monomials associated to w(cid:48) and w according to Lemma 3.1 are T(cid:48) T x(cid:96)n(wn(cid:48)−1)xn(cid:96)(−w1n(cid:48)−2)···x3(cid:96)(w2(cid:48))x(cid:96)2(w1(cid:48)) and xn(cid:96)(wn−1)xn(cid:96)(−w1n−2)···x3(cid:96)(w2)x2(cid:96)(w1) The assumption that w(cid:48) ≤ w implies (cid:96)(w(cid:48)) ≤ (cid:96)(w ) for all i = 1,..,n−1 so the first monomial i i i i divides the second. Garsia-Procesi proved that it follows that x(cid:96)n(wn(cid:48)−1)xn(cid:96)(−w1n(cid:48)−2)···x3(cid:96)(w2(cid:48))x2(cid:96)(w1(cid:48)) is an element in their monomial basis for the cohomology of BX where X is a nilpotent matrix with Jordan blocks of size λ [GP, Proposition 4.2]. Let T(cid:48) denote the row-strict tableau of shape λ associated to this monomial by Mbirika [M, Theorem 2.2.9]. Lemma 3.4 thus gives w =w(cid:48). (cid:3) T(cid:48) The final result of this section uses dominance order on partitions, which we define below. Definition 3.7 (Dominance order). Suppose that λ and µ are two partitions of n. We say λ≥µ if for each row i we have λ +λ +···+λ ≥µ +µ +···+µ 1 2 i 1 2 i 8 MARTHAPRECUPANDJULIANNATYMOCZKO Garsia and Procesi showed that divisibility of their monomials respects the dominance order [GP, Proposition 4.1], which we restate in our notation below. Lemma 3.8. Suppose λ,µ are partitions of n with λ≥µ. If T is a row-strict tableau of shape λ associated to Schubert point w then there exists a row-strict tableau T(cid:48) of shape µ whose Schubert T point satisfies w =w . T(cid:48) T Proof. Let xα denote the monomial corresponding to w . Garsia-Procesi proved that if λ ≥ µ T then xα is also a monomial in the basis for the cohomology of BX where X is a nilpotent matrix with Jordan blocks of size µ [GP, Proposition 4.1]. Mbirika showed how to construct a row-strict tableau T(cid:48) of shape µ with monomial xα [M, Proof of Theorem 2.2.9]. Let w be the unique T(cid:48) Schubert point associated to the tableau T(cid:48) of shape µ by Lemma 3.4. Then w =w since both T(cid:48) T have the same monotone-increasing factorization. (cid:3) 4. Outlining the main theorem In this section we outline and prove the essential lemmas of the main theorem. The key step in the proof of the main theorem is to carefully follow what happens after one simple reflection is erased from the monotone-increasing factorization of a Schubert point. In general erasing one simple reflection produces an extra monotone-increasing string and a factorization that no longer has the form w w ···w as in Lemma 3.1. This is the basic situation that the following n−1 n−2 1 lemma addresses. Lemma 4.1. Suppose p ≤i−1. Then the product i (cid:0) (cid:1)(cid:0) (cid:1) (cid:16) (cid:17)(cid:16) (cid:17) s s ···s s s ···s = s s ···s s s ···s p(cid:48)i p(cid:48)i+1 pi i−(cid:96)i−1 i−(cid:96)i−1+1 i−1 i−(cid:96)(cid:48)i−1 i−(cid:96)(cid:48)i−1+1 i−1 p(cid:48)i−1 p(cid:48)i−1+1 pi−1 where (cid:96)(cid:48) ,p(cid:48) ,p are given by the following table: i−1 i−1 i−1 Case Condition (cid:96)(cid:48) = p(cid:48) = p = i−1 i−1 i−1 1 p

Springer fibers and Schubert points

Similar Springer fibers and Schubert points