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 Poincar´e 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 ThisdescriptionoftheSchubertcellsallowsonetocalculatethePoincar´epolynomialofSchubert 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≤p<q ≤n and w (1) q occurs in a box below p and in the same column or in any column strictly to the left of p in T, and (2) if the box directly to the right of p in T is filled by r , then q <r . p p ThedimensionformulafortheintersectionC ∩BX generalizestheformulafor(cid:96)(w)=dim(C ) w w astheinversionsofw. Toseethis, readthenumbersintheYoungdiagramofshapeλintheorder given by the base filling: the pairs (p,q) described by Condition (1) are precisely the inversions of w. These pairs (p,q) are used enough to warrant their own terminology. Definition 2.6. If (p,q) is a pair with 1 ≤ p < q ≤ n that satisfies Conditions (1) and (2) of Lemma 2.5 for a row-strict tableau T then we call (p,q) a Springer dimension pair of T. IfT isarow-stricttableauofshapeλ, letT[i]beobtainedfromT bydeletingtheboxeslabeled byi+1,...,n. SinceT isrow-stricttherearenogapsintherowsofT[i],meaningifaboxisdeleted then all boxes in the same row and to the right must also be deleted. Therefore the diagram of T[i] forms a composition of i. This gives another way to count Springer dimension pairs. Lemma 2.7. Let (cid:96) denote the number of Springer dimension pairs of the form (p,q) where q−1 2≤q ≤n. Then (cid:96) is the sum of q−1 • the number of rows in T[q] above the row containing q and of the same length, plus • the total number of rows in T[q] of strictly greater length than the row containing q. Proof. The tableau T[q] has no boxes filled with numbers greater than q so Condition (2) above is satisfied only when p fills a box at the end of a row in T[q]. The rest of the claim follows from imposing Condition (1). (cid:3) Remark 2.8. When T is a standard tableau, the formula above reduces even further. The entries in both rows and columns are increasing so there are no rows below the row containing q in T[q] of length greater than or equal to the row containing q. (In other words the diagram of T[q] is a partition.) Therefore (cid:96) simply counts the number of rows above the row containing q. q−1 3. Schubert Points and combinatorial results about Springer permutations We begin by describing a canonical factorization of permutations and some of its properties. Using this factorization, we define Schubert points, which are permutations corresponding to row- strictfillingsofYoungdiagramsinadifferentwaythanSpringerpermutations. Wethengivesome properties of Schubert points, including many that were observed by Garsia and Procesi and by Mbirika in their earlier studies of essentially the same objects [GP, M]. Each element of the symmetric group can be factored canonically into monotone-increasing strings of simple reflections, as detailed below [BB, Corollary 2.4.6]. Lemma 3.1. Each w ∈W can be written uniquely as w =w w ···w w where n−1 n−2 2 1 w =s s ···s s for each i=1,...,n−1 i ki ki+1 i−1 i and either w =e or k is a fixed integer with 1≤k ≤i. Moreover i i i • (cid:96)(w)=(cid:96)(w )+(cid:96)(w )+···+(cid:96)(w )+(cid:96)(w ) and n−1 n−2 2 1 • if w (cid:54)=e then (cid:96)(w )=i−k +1. i i i 6 MARTHAPRECUPANDJULIANNATYMOCZKO The monomial in Z[x ,x ,...,x ] associated to this factorization is x(cid:96)(wn−1)x(cid:96)(wn−2)···x(cid:96)(w1). 1 2 n n n−1 2 Wecallw theithstringofw. Forexample,thelongestwordinS canbewrittenass s s s s s . i 4 1 2 3 1 2 1 In this case the strings are: • w =s s s 3 1 2 3 • w =s s 2 1 2 • w =s 1 1 so k =1 for each i=1,2,3. i Given a row-strict tableau T we construct the associated Schubert point w by using Springer T dimension pairs to determine k for each i. This produces a permutation w whose length is the i T dimension of the affine cell associated to T in the Springer fiber. Definition 3.2 (Schubert points). Let wB ∈ BX and let T denote the corresponding row-strict tableau as in Lemma 2.5. For each 2≤q ≤n let (cid:96) be the number of Springer dimension pairs q−1 of the form (p,q) of T. Define a string w by q−1 (cid:26) s s ···s s if (cid:96) (cid:54)=0 w = q−(cid:96)q−1 q−(cid:96)q−1+1 q−2 q−1 q−1 q−1 e if (cid:96) =0 q−1 so w is a string of length (cid:96) by construction. Then q−1 q−1 w =w w ···w w T n−1 n−2 2 1 is the Schubert point associated to wB ∈ BX. We also refer to w as one of the Schubert points T associated to the partition λ. Our definition together with the properties of the canonical factorization and Lemma 2.5 imply that (cid:96)(w )=(cid:96) +(cid:96) +···+(cid:96) =dim(C ∩BX). T n−1 n−2 1 w Example2gaveonesetofSchubertpoints. ThenextexamplelistsSchubertpointscorresponding to row-strict fillings other than the standard tableaux. Example 3.3. As in Example 2, let λ = (2,2,1). Below are a few of the row-strict diagrams of this shape and the corresponding Schubert points. Note that each of the following examples is smaller in Bruhat order than one of the permutations in Example 2. 2 3 1 3 3 4 1 5 2 4 2 5 3 5 3 5 T row-strict 1 4 4 5 1 2 2 4 1 3 3 4 1 4 2 4 5 2 5 3 5 1 2 1 w ∈S s s s s s s s s s s s s s s s s e T 5 3 4 3 4 3 1 3 4 2 1 2 1 3 4 2 1 The association between row-strict tableaux and Schubert points is unique, as Mbirika proved [M, Section 2] using results of Garsia-Procesi [GP]. Lemma 3.4 (Mbirika). Given either wB ∈BX or a row-strict tableau T the corresponding Schu- bert point w is unique. T Proof. Foreach1<q ≤nlet(cid:96) bethenumberofdimensionpairs(p,q)ofT asinDefinition3.2. q−1 MbirikashowedthemapT (cid:55)→(cid:81)n x(cid:96)i−1 fromrow-stricttableauxtomonomialsisaninjectionthat i=2 i surjectsontoasetofmonomialsdefinedbyGarsiaandProcesi[M,Theorem2.2.9]. EachSchubert point w is uniquely determined by the numbers (cid:96) for 2≤q ≤n so the claim follows. (cid:3) T q−1 SPRINGER FIBERS AND SCHUBERT POINTS 7 The main theorem in Section 4 proves that the set of Schubert points for various λ is closed under the Bruhat order. We end this section with three results that prove special cases of this main theorem. The first of these results proves that Schubert points corresponding to standard tableaux are maximal with respect to Bruhat order in the set of all Schubert points for a partition λ. (This is independent of the partition λ.) Theorem 3.5. Let St(λ) denote the set of standard tableaux of shape λ. Then the Schubert points {w :T ∈St(λ)} are maximal with respect to Bruhat order in the set of all Schubert points for λ. T Proof. Let T be a row-strict tableau with corresponding Schubert point w . We will construct a T standard tableau T(cid:48) ∈St(λ) such that w ≤w . In fact let T(cid:48) be the tableau we obtain from T T T(cid:48) by reordering the entries in each column so that they increase from top to bottom. We first show that the tableau T(cid:48) is row-strict. Suppose r is the entry in row i and column i k >1ofT(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 <i−(cid:96) −1 (cid:96) p(cid:48) p i i−1 i−1 i i 2 p =i−(cid:96) −1 (cid:96) +(p −p(cid:48) +1) NA NA i i−1 i−1 i i 3 p(cid:48) ≤i−(cid:96) ≤p (cid:96) −1 p(cid:48) p −1 i i−1 i i−1 i i 4 i−(cid:96) <p(cid:48) (cid:96) p(cid:48) −1 p −1 i−1 i i−1 i i Proof. Ifp <i−(cid:96) −1theneachsimplereflectionins s ···s commuteswitheachsimple i i−1 p(cid:48)i p(cid:48)i+1 pi reflection in s s ···s which proves the first line of the table. If p = i−(cid:96) −1 i−(cid:96)i−1 i−(cid:96)i−1+1 i−1 i i−1 then the strings glue together to form s ···s proving the second line of the table. If s is a p(cid:48) i−1 j i simple reflection with i−(cid:96) <j ≤i−1 then i−1 (cid:0) (cid:1) (cid:0) (cid:1) s s s ···s = s s ···s s j i−(cid:96)i−1 i−(cid:96)i−1+1 i−1 i−(cid:96)i−1 i−(cid:96)i−1+1 i−1 j−1 using the braid relations. Repeating this proves the fourth line of the table. Combining this with the fact that (cid:0) (cid:1) (cid:0) (cid:1) s s s ···s = s ···s i−(cid:96)i−1 i−(cid:96)i−1 i−(cid:96)i−1+1 i−1 i−(cid:96)i−1+1 i−1 proves the third line of the table. (cid:3) We will prove the main theorem by deleting a simple reflection and then rewriting the resulting permutation in monotone-increasing form, one step at a time. Indeed suppose T is a row-strict SPRINGER FIBERS AND SCHUBERT POINTS 9 tableauofshapeλwithSchubertpointw =w w ···w . Whenwedeleteasimplereflection T n−1 n−2 1 s from the initial monotone-increasing string in w we obtain pn+1 T (4.2) s ···s sˆ s ···s w ···w w =w(cid:48) (cid:70) w ···w w n−(cid:96)n−1 pn pn+1 pn+2 n−1 n−2 2 1 n−1 n−1 n−2 2 1 wherew(cid:48) =s ···s s and(cid:70) =s s ···s . Ontheonehand,ifthere n−1 pn+2 n−2 n−1 n−1 n−(cid:96)n−1 n−(cid:96)n−1+1 pn is a row-strict tableau T(cid:48) of shape λ corresponding to this permutation, it must have n in the box at the end of row (cid:96)(w(cid:48) )+1=(cid:96)(s ···s s )+1=(n−1)−(p +2)+2=n−p −1. n−1 pn+2 n−2 n−1 n n Lemma 4.1 then allows us to write (cid:70) w = w(cid:48) (cid:70) for (possibly empty) monotone- n−1 n−2 n−2 n−2 increasingstringsw(cid:48) and(cid:70) . Thelengthofw(cid:48) determinestheboxinT(cid:48) wheren−1must n−2 n−2 n−2 go, if possible. Continuing this process, the ith step produces the permutation w(cid:48) w(cid:48) ···w(cid:48) (cid:70) w ···w w n−1 n−2 i i i−1 2 1 where (cid:70) =s s ···s s i p(cid:48)i p(cid:48)i+1 pi−1 pi forsomep ,p(cid:48) determinedbythisprocess. Intheproofsinthenexttwosectionsweshowthatthis i i process terminates and that it results in a row-strict tableau T(cid:48) of the same shape as T, namely that T(cid:48) is large enough to accommodate each i according to the specifications of w(cid:48) . i−1 We now prove our main result given the following lemma, which will be proven in the next two sections. Lemma 4.3. Fix a Schubert point w associated to a partition λ with at most three rows or two columns. Suppose that w(cid:48) is a permutation obtained from w by erasing one simple reflection s . j Then w(cid:48) is also a Schubert point associated to the partition λ. The main theorem shows that this lemma implies our claim. Theorem 4.4. Suppose that w is a Schubert point associated to a partition λ with at most three rows or two columns and v ≤w. Then v is also a Schubert point associated to the partition λ. Proof. Since v ≤w we can find a string of simple reflections s ,s ,...,s so that j1 j2 jk • foreach1≤i≤kthepermutationv isobtainedfromv byerasingonesimplereflection i i−1 s and ji • the initial and terminal permutations are v =w and v =v respectively. 0 k Lemma4.3saysthatifv isaSchubertpointassociatedtothepartitionλthensoisv . Inducting i−1 i on i we conclude that v =v is a Schubert point associated to λ as well. (cid:3) k Corollary 4.5. Let X ∈gl (C) be a nilpotent matrix whose Jordan type is given by the partition n λ with at most three rows or two columns. Then the Poincar´e polynomial of the Springer fiber BX equals the Poincar´e polynomial of the union of Schubert varieties for Schubert points corresponding to standard tableaux of shape λ: P(BX,t)=P(∪ C ,t)=P(∪ C ,t). wB∈BX wT T∈St(λ) wT Proof. Theorem4.4showsthatthesetofSchubertpointscorrespondingtoλisalowerorderideal with respect to Bruhat order, so the set of these permutations corresponds to a union of Schubert varieties in the flag variety G/B. The second equality follows from Theorem 3.5. (cid:3) 10 MARTHAPRECUPANDJULIANNATYMOCZKO Example 4.6. When λ=(2,2,1) we have P(BX,t)=5t4+11t3+9t2+4t+1 by Corollary 4.5 together with Example 2.1. The reader can independently verify this fact using the inductive formula for the Poincar´e polynomial given in [S] or [Fr]. The following example shows that Lemma 4.3 does not hold if λ is a partition containing the shape µ=(3,1,1,1) as a subdiagram. Example 4.7. Let T be following standard tableau of shape µ. 1 3 5 2 4 6 T has associated Schubert point w = s s s s s s . Let w(cid:48) = s sˆ s s s s = s s s s s so by T 3 4 5 2 3 1 3 4 5 2 3 1 5 2 3 2 1 construction w(cid:48) ≤ w . However there exists no row-strict filling of µ corresponding to w(cid:48)! While T Lemma 4.3 fails, it is still possible that the Springer fibers have the same Poincar´e polynomials as a union of other Schubert varieties. We have attempted computer calculations to confirm or refute this in the case of λ=(3,1,1,1) but so far have not found an algorithm that terminates in reasonable time. Finally the following example demonstrates that these results do not always hold in arbitrary Lie type, not even for partitions with at most two rows. Example 4.8. Let sp (C) denote the symplectic Lie algebra of Lie type C . The corresponding 6 3 root system has three simple roots so its Weyl group is generated by three simple reflections. This means there are precisely three Schubert cells of dimension 1 in B. However a well-known result states that if X ∈ g is a subregular nilpotent element, then BX is a Dynkin curve [H, Theorem 6.11]. ForLietypeC theDynkincurveconsistsof4projectivelines(becausetheassociatedDynkin 3 diagram is a path with four edges). In particular the Poincar´e polynomial of BX is 1+4t. There is no Schubert variety or union of Schubert varieties in this flag variety with the same Poincar´e polynomial. Therefore our results in this paper do not extend exactly as stated to all Springer fibers in arbitrary Lie type. 5. The three row case The following theorem proves Lemma 4.3 for the three row case. Recall that if T denotes a row-strict filling of λ then T[i] denotes the diagram obtained from T by deleting the boxes labeled by i+1,...,n. We let λ[i] denote the partition of i obtained from the composition corresponding to T[i] by reordering the rows in decreasing order. Theorem 5.1. Let λ be a partition of n with at most three rows and T be a row-strict tableau of shape λ. Suppose w(cid:48) is obtained from w by deleting a simple reflection. Then there exists a T row-strict tableau T(cid:48) of shape λ such that w(cid:48) =w . T(cid:48) Proof. Our proof is by induction on n. We start with the base cases n≤2. The cases when λ is a single row are trivial, since the Springer fibers in those cases consist of the single flag eB and the only Schubert point is e. The cases when λ is a single column are also trivial, since every filling of the diagram is row-strict and hence every permutation flag is in the Springer fiber. Indeed, the Springer fiber in that case is the full flag variety. Therefore the Schubert points are also the set of all permutations, namely {e,s } in the case n=2. 1