aa r X i v : . [ m a t h . AG ] J un THE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS
JULIANNA TYMOCZKO
Abstract.
This survey paper describes Springer fibers, which are used in one of the earliestexamples of a geometric representation. We will compare and contrast them with Schubertvarieties, another family of subvarieties of the flag variety that play an important role inrepresentation theory and combinatorics, but whose geometry is in many respects simpler.The end of the paper describes a way that Springer fibers and Schubert varieties are related,as well as open questions. Introduction
The flag variety
G/B is a critical object at the heart of geometry, combinatorics, algebra.The interplay of these fields makes all of them richer. In this survey article, we describetwo families of subvarieties of the flag variety in some depth: Springer fibers and Schubertvarieties. We will compare the two families as well as point towards how they are related ona structural level.Schubert varieties are the more foundational objects. The double cosets
BwB partition G into a finite number of subsets that the elements of the Weyl group W naturally index.These cosets induce a paving BwB/B of the flag variety into Schubert cells, and the clo-sure of
BwB/B is the Schubert variety corresponding to the Weyl group element w . Thecombinatorics of w is intimately related to the geometry of the associated Schubert vari-ety. Moreover the Schubert varieties induce a cohomology basis in H ∗ ( G/B ) that plays animportant role in representation theory.Springer first defined the varieties now known by his name, as the fixed flags under theinfinitesimal action of a nilpotent element X [42]. More precisely, the Springer fiber S X associated to X is the collection of flags S X = { gB ∈ G/B : g − Xg ∈ b } Springer used them to construct an action of the Weyl group on the cohomology H ∗ ( S X ) thathas many remarkable properties, described further in Section 4.2. For this reason, and tounderstand more deeply the geometry underlying the construction, many have constructedSpringer’s representation (or its dual) [22, 4, 20, 28, 15]. These constructions often reliedon geometric properties of Springer fibers like purity [39, 40], closed formulas for dimensions[38, 10], or identification of the components [39]. The geometry is interesting in its own right,as well, and is connected to interesting combinatorial and representation-theoretic propertiesof W .In what follows, we will describe these properties and connections in more detail, culmi-nating in Section 4.2 with a result that the Betti numbers of the Springer variety S X are the same as a particular union of Schubert varieties determined by X . We end with openquestions.Throughout we include many examples, focusing on examples in the type A case becauseit is the most important geometric case.The author was partially supported by NSF grant DMS-1362855 and gratefully acknowl-edges the organizers of the conference at Orsay for the opportunity to discuss this work.2. Definitions and basic examples of Schubert varieties and Springer fibers
Let G be any complex semisimple connected linear algebraic group and let B be a Borelsubgroup of G . The flag variety is the quotient G/B . It is the ambient variety for everythingthat we consider in this paper.The most important geometric example is the type A case, when G = GL n ( C ) and B isthe subgroup of upper-triangular matrices. In this case, each flag gB can be thought of asa nested collection of subspaces gB ←→ V ⊆ V ⊆ V ⊆ · · · ⊆ V n − ⊆ V n = C n where V i is an i -dimensional complex subspace for each i ∈ { , , . . . , n } . This correspondenceis given by the rule that the first i columns of g span V i for each i ∈ { , , . . . , n } .Now let X be an element in the Lie algebra g of G . In type A we can take X to be alinear operator X : C n → C n . We write E ij for the basis element in gl n that has one in the( i, j ) position and zero elsewhere.The Springer(-Grothendieck) fiber over X is defined to be the collection of flags S X = { gB : g − Xg ∈ b } Geometrically in type A this can be rephrased as the collection of flags fixed by X in thefollowing sense: S X = { V • : XV i ⊆ V i for all i } The geometry of the flag variety is concrete and relatively well-adapted to direct compu-tations. For instance one classical way to analyze the flag gB is via traditional “Gaussianelimination”, or column reduction, from undergraduate linear algebra. Using column reduc-tion, we can confirm the following. Proposition 2.1.
Each coset gB has a unique representative of the following form: • a permutation matrix • with zeros below or to the right of its ones • and nonzero entries only above and to the left of its ones. HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 3
For instance one set of these representatives follows: ∗ ∗ ∗ The entries marked ∗ can be nonzero, and in fact can be any arbitrary element of C . Thisprocess hints at why the set of representatives that share a fixed permutation w form a cellisomorphic to C d w for some integer d w . The cells { C w } are called Schubert cells. In generaltypes, this decomposition is precisely the one given by the double cosets: C w = BwB/B
Springer fibers are more complicated but similar arguments tell us quite a bit about theirgeometry. If we choose a basis of C n well with respect to X then column reduction givesfree entries in some of the free positions of C w . Example 2.2.
Let X = E + E and consider the set of flags in the Schubert cell ∗ ∗ ∗ We can compute which of these flags are in S X by calculating a b c = c and then determining the conditions that guarantee that the i th column in the matrix on theright is in the span of the first i columns of the matrix on the left. In this case, the conditionis that a = c . (In general the conditions on the entries can be more complicated.) This basic approach allows us to determine entire Springer fibers, as in the followingexamples.
Example 2.3.
The first three examples are more general; the last is very specific. • Suppose X = 0 . The condition that g − Xg ∈ b is always satisfied so the Springer fiber S = G/B . JULIANNA TYMOCZKO • Suppose that X is regular semisimple. (In type A this means that X has n distinct eigenvalues.)The condition that g − Xg ∈ b is satisfied if and only if gB is a Weyl flag. (In type A we note that the first column of g must be an eigenvector of X and similarly the i th column of g must be an eigenvector of X for each i . Putting this together impliesthat gB is a permutation flag.) This means S X = { wB : w ∈ W } . • Suppose that X regular nilpotent. (In type A this means that X has a singleJordan block.)The condition that g − Xg ∈ b is now satisfied if and only if gB is the identity flag eB . (In type A the first column of g must be an eigenvector of X and similarly the i th column of g must be an eigenvector of X i for each i . Since X has a unique eigenvectorand more generally X i has i eigenvectors there is a unique flag gB satisfying theconstraints.) This means that S X = { eB } . • Suppose that X = E in gl . Concrete arguments like the previous tell us that the flag g − Xg ∈ b if and only if g is in one of the cosets , a , b
10 1 0 where a, b ∈ C . Inspecting what happens to the flags as each of a, b gets large, we cansee the closure of each one-(complex-)dimensional cell is the single point eB . Thismeans the Springer fiber is S X = (cid:8) two spheres joined at a single common point (cid:9) Unless otherwise indicated, we assume henceforth that X is nilpotent. Comparing the geometry of Schubert varieties and Springer fibers
In the previous section we used linear algebraic techniques to identify the flags in specificSchubert varieties and Springer fibers. In each case we found that the variety was parti-tioned into cells that were indexed by permutation flags. In fact the combinatorics of thepermutation at the heart of each cell determines quite a bit of the geometry of the celland its closure. The key difference between Schubert varieties and Springer fibers is thatall entries in the cells of Schubert varieties are free while entries in cells of Springer fibershave conditions on them. In this section we describe the ramifications of this key difference,from its impact on the dimension of the cells and how they intersect, to the kinds of openquestions we have about each. We start by sketching the main properties of Schubert cellsand Schubert varieties, and then proceed to describe the cells in Springer fibers. The readerinterested in learning more about the geometry of Schubert varieties is referred to texts likethose of Brion [5], Fulton [13], or Billey and Lakshmibai [2].3.1.
Geometry of Schubert cells in the flag variety.
Decomposing the flag variety intoSchubert cells C w = BwB/B is very natural from an algebraic perspective. Amazingly, this
HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 5 decomposition is more than just a partition of the flag variety: the cells actually form aCW-decomposition of the flag variety. In other words • each Schubert cell C w is isomorphic to C ℓ ( w ) for some integer ℓ ( w ) (as evident fromthe construction above) and • the closure C w of each Schubert cell is a union of Schubert cells of smaller dimension(less evident, but since each C w is a B -orbit, its closure is a union of B -orbits, namelyother Schubert cells C v ).More amazingly yet, the statistics that guarantee the CW-decomposition are completelynatural combinatorial statistics. The dimension ℓ ( w ) is the length of the permutation w ,namely the smallest number of simple reflections s i = ( i, i + 1) into which w can be factored.(Length is defined similarly for the other Weyl groups.) The closure relation is given by C w = [ v ≤ w C v where the partial order v ≤ w is the Bruhat order . The combinatorial definition of v ≤ w inBruhat order is that v is a subword of w when w is written in terms of the simple reflections s i = ( i, i + 1). For instance if we consider the permutation w = 3142 in one-line notation(meaning that w ( i ) is in the i th position) then we can factor w = s s s . The Weyl groupelement s s s s has length 4 and is greater than the following set of Weyl group elements: { e, s , s , s , s s , s s , s s , s s , s s s , s s s , s s s } From a topological perspective, the most useful fact about CW-decompositions is thatthey induce cohomology bases. In particular the closures { C w : w ∈ W } induce a modulebasis for the cohomology H ∗ ( G/B, Q ). The closure C w of the Schubert cell C w is calleda Schubert variety and the classes they induce in H ∗ ( G/B, Q ) are called Schubert classes.They are one of the most important bases of the cohomology of the flag variety (and moregenerally partial flag varieties G/P when P is a parabolic subgroup) for reasons we willdescribe in Section 4.1.Moreover the geometry of Schubert varieties can be used to determine the cohomology ringstructure directly. From a geometric perspective, the key point of intersection theory is thatwhen varieties lie in appropriate relative positions, their intersection induces the productof the corresponding cohomology classes. This basic principle is true but more complicatedwhen the varieties have singularities, so understanding singularities is an important part ofgeometric calculations in cohomology.For this reason many researchers have studied the singularities of Schubert varieties. Whatthey discovered was that these singularities are also deeply entwined with the combinatoricsof permutations, specifically patterns inside permutations. Lakshmibai and Sandhya provedthat the variety C w is smooth if and only if the permutation w avoids the patterns 3412 and4231 [27]. The permutation w avoids a pattern if, when w is written in one-line notation, JULIANNA TYMOCZKO no four numbers have the same relative positions as the pattern. (Both Ryan [36] andWolper [45] characterized the singular Schubert varieties before Lakshmibai and Sandhya,and without this particular combinatorial formulation; all three results are independent.)
Example 3.1.
The permutations w = 624351 and v = 324651 both avoid the pattern because only , , or can be the fourth-largest number in asubset of { , , , , , } and none of them can have three smaller numbers in the appropriaterelative positions. The permutation v also avoids the pattern and so C v is smooth. Thepermutation w = does not, so C w is singular. Pattern avoidance is an important condition in computer science and enumerative combi-natorics. It also turns out to be related to many other properties of Schubert varieties, forinstance the components of the singular locus of a Schubert variety [3, 9, 21, 30], or whetherthe Schubert variety C w is Gorenstein (a geometric criterion that is not as strong as beingsmooth) [46]. (See Woo and Yong’s work for a unified presentation and extension of theseresults [47].)3.2. Geometry of cells in Springer fibers.
Recall from Example 2.2 that column reduc-tion of flags in Springer fibers produced cells that satisfied the same conditions for Schubertcells described in Proposition 2.1 except that not all entries needed to be free . This is be-cause each of these cells in the Springer fiber is an intersection with a specific Schubert cell.Of course, intersections can be fantastically complicated depending on the conditions onthe non-free entries. In this section, we describe how this affects what is known about thegeometry of the cells in Springer fibers.The first complication is that the intersections { C w ∩S X } do not form a CW-decompositionof the Springer fibers, unlike the Schubert cells for the flag variety. Instead these intersections { C w ∩S X } form a topological decomposition called a paving by affines . To be paved by affines means having a partition into cells that are each affine, namely isomorphic to C d w for someinteger d w . (In other applications, it can be useful to study pavings by other geometricspaces.) A paving restricts the closure conditions on each cell more loosely than in a CW-decomposition. Cells form a paving if they can be ordered by an index set I so that theclosure C i ⊆ S j ≤ i C j for each i in the index set. That containment—which would be anequality if the cells formed a CW-decomposition—changes the intuition around pavings byaffines. The classic example of a paving by affines that is not a CW-decomposition is a“string of pearls”, namely a collection of copies of P with the north pole of one glued to thesouth pole of the next. The homologically natural way to partition this space is by pullingoff the leftmost north pole, then the copy of C left in the leftmost P , then the copy of C left in the next P , and so on, as shown in Figure 1. This paving has one affine cell for each(co)homology basis class, with dimensions of the cells matching the degrees of the classes.However it is not a CW-decomposition since the closure of most of the C cells contains asingle point in the middle of another C cell. Historically Spaltenstein first described the HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 7 irreducible components of each Springer fiber [39]; Shimomura refined his analysis to paveSpringer fibers with affines [38]. In both works, the dimension of the cells was naturallyassociated to certain combinatorial objects that we describe next. ✫✪✬✩✫✪✬✩✫✪✬✩✫✪✬✩✫✪✬✩ . .................. ................. ................ ................ ................ ................ ................. .................. . .................. ................. ................ ................ ................ ................ ................. .................. . .................. ................. ................ ................ ................ ................ ................. .................. . .................. ................. ................ ................ ................ ................ ................. .................. . .................. ................. ................ ................ ................ ................ ................. .................. ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣✉ s s s s
Figure 1.
Paving a “string of pearls” by affinesThe dimension d w of each cell can be determined combinatorially for Springer fibers butwe only count part of the dimension ℓ ( w ) of the corresponding Schubert cell. Another way ofdefining the length of a permutation w is as the number of inversions of w . If the permutationis written in one-line notation, an inversion is a pair of numbers with the bigger number tothe left of the smaller number. For instance the permutation w = 513264has 6 inversions, one from the pair 3 >
2, four from the numbers less than 5, and one fromthe pair 6 >
4. (The definition of inversions can be extended to arbitrary Lie type using thecombinatorics of roots.)To count the dimension of the cells in Springer fibers, we need to incorporate both in-versions and the structure of the matrix X into the combinatorics. To do this, we use aYoung diagram, which is a top-aligned and left-aligned collection of n boxes. (This is knownas English notation rather than French notation, which traditionally uses bottom-alignedand left-aligned boxes. To borrow Macdonald’s quip [29, page 2], the reader accustomedto French notation should read this paper upside down in a mirror.) In our case we take λ ( X ) to be the Young diagram with the same number of rows as Jordan blocks in X andthe same number of boxes in each row as the dimension of the corresponding Jordan block.“Top-aligned” means that we arrange the diagram so that rows decrease in size. Figure 2shows examples of Young diagrams, in this case filled with numbers. We fill our Young dia-grams bijectively with the integers { , , . . . , n } , meaning that each integer appears exactlyonce. A row-strict filling is one in which the integers increase from left-to-right in each row,without any conditions on columns.1 42 53 1 23 4 1 32 4 2 31 41 42 3 3 41 2 2 41 3(a) Young diagram example (b) All row-strict Young tableauxof shape (2 , Figure 2.
Young tableaux
JULIANNA TYMOCZKO
Theorem 3.2. If X is chosen carefully within its conjugacy class, the nonempty intersections C w ∩ S X are bijective with the set of row-strict fillings of λ ( X ) . For certain X the cells C w ∩ S X are in fact described geometrically by the fillings. Looselyspeaking, reading the filling in a specific order gives a permutation, which tells in what orderto add basis vectors to the flag (equivalently which permutation matrix appears in the cosetrepresentative). A more precise argument can be found in [38] and [43] or [10]. (Little isknown about the intersections C w ∩S X for other X except that they can be very complicated.For instance in the analogous case of Peterson varieties, if X is a particular lower-triangularmatrix then the coordinate ring of the largest intersection is the quantum cohomology of theflag variety [26, 35], while if X is the opposite upper-triangular matrix then the intersectionsare affine cells [43].)Moreover the row-strict Young diagrams can be used to compute the dimensions of thecorresponding intersection C w ∩ S X . For each integer i in the filling, count the integers j that satisfy these conditions: • i > j • j is in any of the columns to the right of i or is in the same column as i but above i • either j is the rightmost box in a row or the box to the right of j is filled with anumber k that satisfies i < k The first two conditions together are the conditions for i and j to form an inversion, if youread the entries in the Young diagram starting at the bottom of the left column, proceedingup each column, and then moving to the bottom of the next-right column. The last conditiongives a way to eliminate certain inversions—exactly ones that correspond to the dependententries in the cell [43, 10]. For example, the leftmost tableau in Figure 2 corresponds to a cellof dimension 4. The three tableaux on the top row of the right side of Figure 2 correspondto cells of dimension 2 , , , , C w ∩ S X . Even relatively simple questions about the closureconditions are mysterious: for instance, we do not know which permutation flags lie in theboundary of the cell C w ∩ S X . Most of what is known about smoothness conditions consistsof criteria determining when all components are smooth, which is of course a high standardto meet. Indeed all components are smooth very rarely, including when λ ( X ) has a hookshape [44], when X has exactly two Jordan blocks [14], and a small list of other cases [11].The case when X has exactly two Jordan blocks was particularly interesting to Khovanovbecause of connections to categorification in knot theory and quantum representations (seeSection 4.3 and [23]). Of course, if that little is known about which components are singular,it is unsurprising to learn that nothing is known about more refined conditions like beingGorenstein. The exception is when λ ( X ) has two columns, in which case the componentsare normal, Cohen-Macaulay, and rationally smooth [32] and smooth precisely when theirPoincar´e polynomials are symmetric [12].Figure 3 summarizes the discussion in this section. HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 9
Flag variety
G/B
Springer fiber S X Partitioned into cells Partitioned into affine pieces C w = BwB/B
CW-decomposition into C w Paved by affines C w ∩ S X { C w } induce basis of H ∗ ( G/B ) { C w ∩ S X } induce basis of H ∗ ( S X ) C w = S v ≤ w C w ?smooth iff w avoids 1324 and 2143 all components smooth: • if λ ( X ) has a hook shape • if X has two Jordan blocks • for small list of shapes of X Gorenstein iff w avoids31524 and 24153 (+ conditions) ?
Figure 3.
Comparing Schubert varieties and Springer fibers4.
Schubert varieties and Springer fibers in representation theory
Combinatorics, geometry, and representation theory collide in the representations associ-ated to Schubert varieties and Springer fibers. Despite the similarities, the representationsinvolve very different constructions—and groups!—using very different tools and approaches.This section introduces three representations: the representation of GL n arising from theSchubert basis in the cohomology of the Grassmannian, the representation of S n on thecohomology of Springer fibers (or in general Lie type the representation of the Weyl group W ), and more recent research into a quantum representation encoded by the components ofcertain Springer fibers. Fulton’s text is an excellent introduction to Schubert calculus [13]while Chriss and Ginzburg’s is an expansive introduction to Springer theory [8].4.1. Classical representation theory of Schubert varieties.
The most well-known andwell-understood geometric representation involving Schubert varieties does not consider theSchubert classes within the cohomology of the flag variety. Rather, it uses the image of theSchubert classes under the projection
G/B →→ G/P where P is a maximal parabolic of type A . In that case G/P is a Grassmannian G ( k, n ), namely the collection of k -dimensionalsubspaces of the vector space C n . The surjection G/B →→ G/P collapses many of theSchubert varieties in the full flag variety, in the sense that their images are not distinct.Instead of being indexed by the set of all permutations, the Schubert cells { C λ } in theGrassmannian are indexed by the Young diagrams λ with at most k columns and at most n − k rows. Figure 4 shows the Young diagrams that index the Schubert cells for G (2 , GL ( C ) /B has 24 = 4! Schubert classes instead of just 6. ∅ , , , , , Figure 4.
The Young diagrams indexing Schubert classes in H ∗ ( G (2 , σ λ represents the cohomology class of C λ in G ( k, n ). We can write theproduct σ λ · σ µ in the cohomology ring H ∗ ( G ( k, n ) , C ) in terms of its basis of Schubertclasses as so: σ λ · σ µ = X ν c νλ,µ σ ν It turns out that the coefficients c νλ,µ are precisely the same coefficients obtained in the tensorproduct decomposition V λ · V µ = X ν c νλ,µ V ν where V λ are the irreducible representations of GL n ( C ).Moreover the combinatorics of Young diagrams determines many properties of these coef-ficients. For instance if µ is not contained in ν then the coefficient c νλ,µ must be zero; similarlycertain fillings of the Young diagrams λ and µ determine the coefficients c νλ,µ . (The fillings inSchubert calculus are not the same fillings that we used for Springer fibers.) Some of theseproperties can be proven with very elegant geometry, while for others the only known proofsrely on technical results from the theory of symmetric functions. Indeed the classical proofthat the structure constants of H ∗ ( G ( k, n )) equal the tensor product multiplicities simplyobserves that both objects are determined by a recurrence relation counted by certain com-binatorial objects, and then shows that the recurrence and the initial cases are the same inboth contexts. In other words, the proof is not terribly illuminating either from a geometricor from a combinatorial perspective.Modern Schubert calculus seeks to construct the coefficients c νλ,µ explicitly for variousdifferent groups G , parabolic subgroups P , or cohomology theories. One of the most vexingopen questions in the field is the one closest to our starting point: to find an explicit, positiveconstruction of the coefficients in the cohomology ring H ∗ ( GL n ( C /B )) of the flag variety oftype A .4.2. Classical representation theory of Springer fibers.
The combinatorial parametriza-tions of the cells in Springer varieties hint at the representations that arise in the Springercontext. We open this section with a variation of Theorem 3.2 that makes this hint more
HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 11 explicit. Recall that a standard Young tableau is one that is filled so that rows increaseleft-to-right and columns increase top-to-bottom.
Theorem 4.1 (Spaltenstein 1976) . The top-dimensional cells C w ∩ S X are bijective with theset of standard fillings of λ ( X ) . Standard Young tableaux of shape λ also count a classical representation-theoretic object:the dimension of the irreducible representation of S n associated the partition λ of n . This isnot coincidence. Theorem 4.2.
Fix a nilpotent matrix X : C n → C n . The cohomology H ∗ ( S X ) of theSpringer fiber carries an S n -action. The top-dimensional cohomology is irreducible and infact is the irreducible representation associated to λ ( X ) . Each irreducible representation of S n can be obtained uniquely in this way by varying over the conjugacy classes of nilpotentmatrices. Originally proven by Springer [41], the theorem has many different proofs using differentapproaches and perspectives, some recovering only parts of the theorem as we have statedit and others much stronger. For instance there are proofs due to Kazhdan and Lusztig[22], Borho and MacPherson [4], Lusztig [28], Garsia and Procesi [15], and many others.Of course there is a second irreducible S n -representation of the same dimension as the irre-ducible representation associated to λ , namely its dual (obtained by tensoring with the signrepresentation). Interestingly the literature on Springer’s representation is ambiguous onthis point: different constructions of “the” Springer representation use either λ or its dual.Hotta appears to be the first to recognize this subtlety and classify different constructionsup to that moment [20]. The representation described in this theorem generalizes to all Lietypes, though Lusztig showed that outside of type A the top-dimensional cohomology neednot be bijective with irreducible representations; he defined cuspidal representations to bethose that do not appear.A proof of Theorem 4.2 is outside the scope of this survey but we include a sketch ofGrothendieck’s approach (see Grinberg’s exposition for more [16]). We consider two sub-spaces of the Lie algebra g : • the nilpotent subalgebra N consisting of all nilpotent elements of g and • the subalgebra g rs consisting of all regular semisimple elements of g .Now define the subspace e g of the product space g × G/B by e g = { ( X, gB ) : g − Xg ∈ b } and define subspaces e N and f g rs analogously. By projecting to the first factor we get thefollowing commutative diagram. e N ֒ → e g ← ֓ f g rs ↓ ↓ ↓N ֒ → g ← ֓ g rs This diagram has many useful and surprising features. First the projection e N → N is aresolution of singularities, which we can see by projecting e N to the second factor and notingthat the fiber over each flag gB is isomorphic to b . Moreover the fiber of the map e N → N over the element X is the Springer fiber S X . (This explains the name Springer fiber .) Onthe right-hand side, the projection f g rs → g rs is an n !-sheeted cover on which the group S n acts as deck transformations. (The regular semisimple case in Example 2.3 gives a smallexample of this cover.)More subtlely, the map e N → N has a geometric property called semismall , which is acondition that constrains the size of the fibers over the singular part of N . It allows usto use the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber [1], which in aninformal sense breaks the total cohomology into pieces incorporating geometric subspacesfrom g paired with S n -actions on other pieces of g . From a dimension count, the only piecesthat survive are the geometric subspaces corresponding to conjugacy classes in N togetherwith S n -representations from the generic part of the Lie algebra. The generic element of g is regular semisimple, so the representations come from g rs which we observed carries theregular representation and thus decomposes as desired. We emphasize that this proof relieson being able to calculate dimensions and other geometric properties of Springer fibers inorder to prove hypotheses of the theorem, including that the map is semismall.We end this section with a remark about one way that this representation generalizes. Hes-senberg varieties are a larger family of varieties than Springer fibers, in which b is replacedby different subspaces H of the Lie algebra (more general even than parabolic subalgebras).The Weyl group acts on the cohomology of nilpotent Hessenberg varieties by the monodromy representation, though the structure of the monodromy representation is more mysterious.Shareshian-Wachs conjectured that the representation arises in the combinatorics of cer-tain quasisymmetric functions that they studied for independent reasons [37]; recently bothBrosnan and Chow [6] and Guay-Paquet [17] proved this conjecture independently and withdifferent methods. Another open question asks for explicit combinatorial descriptions of therepresentations that arise for various X and H .4.3. Non-classical representation theory and components of Springer fibers.
Theprevious two sections described geometric representations arising from the total space of avariety, namely in H ∗ ( G ( k, n ) , C ) in the Schubert case and on H ∗ ( S X ) in the Springer case.In this section we describe interesting new work relating certain quantum representations tothe components of S X in the type A case. This section is less detailed than the previous two,partly because this representation was more recently discovered and our understanding of itcontinues to evolve.Spiders are diagrammatic categories encoding representations; the spider for A n encodesthe representations of U q ( sl n ). The objects in this category represent tensors of represen-tations. The morphisms are planar graphs with boundary called webs , and the spider isequipped with skein-theoretic braiding morphisms that allow us to interpret tangles as webs. HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 13
In some descriptions the diagrams are drawn like tangles: the top and bottom representthe weight of the source and target representation and the twists in the strands indicate anaction on the corresponding tensor products.For instance in the A case the webs are just Temperley-Lieb diagrams or crossinglessmatchings. When the partition λ ( X ) has two rows, namely X has two Jordan blocks, thecomponents of the Springer fiber S X naturally index a basis for the webs of A spiders.“Naturally” here means that the combinatorics and the geometry of the components reflectinformation encoded in the diagram of the corresponding morphism. More precisely Kho-vanov proved that each component of the Springer fiber S X is homeomorphic to an iteratedtower of P -bundles, where the structure of the tower corresponds to the nesting of arcs inthe web corresponding to the component [23]. Figure 5 gives an example of web for A andits corresponding tableau; the corresponding Springer component is the product of P witha P -bundle over P × P . Khovanov discovered this connection between Springer fibers andwebs when he was analyzing a ring that arose in his construction of Khovanov homology fortangles; he happened to observe that the center of his ring was isomorphic to the cohomology H ∗ ( S X ) using work of Fung [14] and Garsia-Procesi [15]. ................. ............ .......... ........ ........ .......... ............ ................ ................. ............ .......... ........ ........ .......... ............ ................. ........................ ....................... ..................... .................... ................... .................. .................. ................... .................... ..................... ....................... ........................ ................. ............ .......... ........ ........ .......... ............ ................ Figure 5.
A web when λ ( X ) has two rows and the corresponding Young tableauWhen λ ( X ) has three rows Khovanov and Kuperberg showed that the standard Youngtableaux combinatorially index the morphisms for A spiders [24]. It seems that there is ageometric relationship between the combinatorics of the diagram and the components of thecorresponding Springer fiber: for instance, a diagram without any crossings corresponds toa smooth component (and in fact an iterated fiber product of copies of GL ( C ) /B )). Theexample on the left in Figure 6 shows a component that is homeomorphic to GL ( C ) /B × GL ( C ) /B while the example on the right is a more complicated component. One openquestion asks for a deeper analysis of how combinatorial features in the webs (crossings,nesting, etc.) correspond to geometric properties of the components (singularities, nestedproduct structures, etc.). .................... ................ ............ ........... .......... .......... ........... ............ ................ ................... .................... ................ ............ ........... .......... .......... ........... ............ ................ ................... ................. ............ .......... ........ ........ .......... ............ ................ ................. ............ .......... ........ ........ .......... ............ ................ ................. ............ .......... ........ ........ .......... ............ ................. ........................... .......................... ........................ ....................... ....................... ........................ .......................... ........................... Figure 6.
Webs when λ ( X ) has three rows and their corresponding Young tableauxWe close this section by sketching a categorical description of the connection betweenSpringer fibers and A n spiders, one which extends to a larger theory connecting similar vari-eties to other representation-theoretic constructions. In the case of spiders, we want a mapbetween ( n, m )-tangles and isomorphism classes of certain exact functors D n → D m that pre-serves knot-theoretic and algebraic structures of the tangles. Cautis and Kamnitzer provedthat if D n is the derived category of certain equivariant coherent sheaves on Springer fibersof shape ( n, n ) then braid moves and other tangle operations correspond to Fourier-Mukai transforms [7]. Part of their insight is that Springer fibers naturally arise in the geometricLanglands program, and in fact are associated via the geometric Satake correspondence tothe same sl n representations that arise from the knot-theoretic perspective; this is the coreidea that they generalize to other varieties.5. Connecting Springer fibers with Schubert varieties
The previous sections have used the analogy between Springer fibers and Schubert varietiesto describe their geometry and representation theory in more depth. In this section, wedescribe a web of new results and conjectures that connects the two kinds of varieties moredeeply and directly.To begin we define permutations w T associated to standard Young tableaux T . Definition 5.1.
Fix a standard Young tableau T with n boxes. For each i with ≤ i ≤ n • let d i denote the number of rows strictly above i in T and • let w i denote the increasing product of simple transpositions s i − d i s i − d i +1 · · · s i − s i − where each s i = ( i, i +1) . (Our convention is that if i = 0 then w i = e is the identity.)Then the Schubert point associated to T is the permutation w T = w n w n − w n − · · · w Example 5.2.
For instance we have the following: ←→ d = 1 d = 0 d = 2 d = 1 d = 0 ←→ ( s ) ( s s )( s )It turns out that these permutations w T index a set of Schubert varieties whose union hasthe same Betti numbers as a Springer fiber. More precisely we have the following [34]. Theorem 5.3 (Tymoczko-Precup) . The Betti numbers of the Springer variety associated to X agree with the Betti numbers of the union of Schubert varieties H ∗ ( S X ) ∼ = H ∗ (cid:16)[ C w T (cid:17) where the union is taken over all standard tableaux T of shape λ ( X ) . Example 5.4.
To show the subtlety of this result, we will calculate the Euler characteristicof each side of this equation in the case when λ = (2 , , . In other words we will count the HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 15 number of nonempty cells C w ∩ S X and the number of nonempty cells in the union S C w T not worrying about degrees except when it is unavoidable.To find the Euler characteristic of the Springer fiber, we need to count the number ofrow-strict tableaux of shape (2 , , by Theorem 3.2. The bottom box can be filled with anynumber, so there are five choices for that entry. The four numbers left must then fill theshape (2 , so that each row increases. In other words we just partition the four numbersinto pairs and order each pair in the unique increasing way. There are (cid:0) (cid:1) = 6 ways to dividefour distinct numbers into pairs and so · row-strict tableaux overall.To get the Euler characteristic of the union of Schubert varieties, we need to computeall standard tableaux of shape (2 , , . There are five in this case, which we found usingbrute force and the fact that must go in the top-left box while must go in one of the twobottom-right corners. We created w T for the first of these tableaux in Example 5.2. All of the Schubert points w T are found in the same way. We list them below in the same order as their correspondingtableaux: s s s s , s s s s , s s s s , s s s s , s s s s To enumerate the cells in a particular Schubert variety, we just need to determine all thesubwords of each permutation. The challenge of determining the cells in a union of Schubertvarieties is that many of these subwords can coincide. For instance the Schubert varietycorresponding to the permutation s s s s is the union of the Schubert cells corresponding to e, s , s , s , s s , s s , s s , s s , s s s , s s s , s s s , s s s s (omitting duplicates and using relations in the permutation group to simplify where possible).But all of those except s s s s and s s s index a Schubert cell in one of the Schubertvarieties earlier in our list. In other words, the Schubert variety for s s s s only increasesthe Euler characteristic of the union by two. (In this calculation, the dimension of each cellcomes “for free” since it’s simply the number of simple reflections in the word.) Performingthis calculation for the whole union, we get the following Betti numbers: , , , , Their sum is 30, consistent with the theorem.
This result is one piece of a more general collection of conjectures and results involvingHessenberg varieties, the same generalization of Springer fibers mentioned in Section 4.2.Indeed it appears that the parameters of a nilpotent X and a Hessenberg space H determinea union of Schubert varieties whose homology is the same as the corresponding Hessenbergvariety. Harada and the author proved this conjecture for Peterson varieties , when X isregular nilpotent (meaning has a single Jordan block) and H consists of the subspace of matrices that are zero below the subdiagonal [19]. Mbirika proved the conjecture when X isregular nilpotent and H is arbitrary [31]. The case of the Springer fiber is more complicatedand is also the a key tool in a more recent result by the author and Precup, which extendsit to the case when H is parabolic (the so-called parabolic Hessenberg varieties) [33]. Someof these results also extend to general Lie type [19].We want to stress that this result only describes an enumerative and combinatorial prop-erty of Springer fibers. It says nothing about, for instance, the cohomology class inducedby S X in H ∗ ( G/B ) or the multiplicative structure of the ring H ∗ ( S X ), though those areboth interesting questions. Indeed this conjecture came about in part because of work todetermine the equivariant cohomology ring of Hessenberg varieties. In particular, it appearsthat the equivariant cohomology of the Hessenberg variety corresponding to X and H canbe determined by restricting a set of equivariant Schubert classes Y to certain fixed pointsin the Hessenberg variety. The equivariant cohomology of the specific union of Schubertvarieties identified by these conjectures appears to be computed by restricting the same setof equivariant Schubert classes Y instead to fixed points in the Schubert varieties. That isan exciting conjecture though attempts to prove it have so far been limited to special caseslike that of Peterson varieties [19] by algebraic and combinatorial challenges.For these reasons we conjecture that an underlying geometric principle determines theseresults, even though all known proofs are combinatorial. Indeed we conjecture that thereis a degeneration of nilpotent Hessenberg varieties to unions of Schubert varieties, perhapssimilar to Knutson-Miller’s degeneration of Schubert varieties to unions of line bundles [25].6. Open questions
For convenience we list here the open questions throughout the whole paper. • Characterize the intersections C w ∩ S X for arbitrary nilpotent X . For which X in afixed conjugacy class is C w ∩ S X affine? For which X are the nonempty intersections C w ∩ S X enumerated by row-strict tableaux of shape λ ( X )? (See Section 3.2.) • Give a complete list of the singular cells of Springer fibers, namely for each nilpotent X and permutation flag wB ∈ S X , give a closed condition to determine if the closure C w ∩ S X is singular or not. (See Section 3.2.) • Determine which fixed points are in the closure of each Springer cell, namely for eachnilpotent X and pair of permutation flags vB, wB ∈ S X give a closed condition todetermine if vB is in the closure of the cell C w ∩ S X . (See Section 3.2.) • Determine which Springer Schubert cell closures C w ∩ S X are Cohen-Macaulay (Goren-stein, etc.). (See Section 3.2.) • Find an explicit, positive construction of the coefficients in the cohomology ring H ∗ ( GL n ( C /B ) of the flag variety of type A . (See Section 4.1.) • Find an explicit combinatorial description of the representation of S n on the coho-mology of Hessenberg varieties for different X and H . (See Section 4.2.) • Identify the combinatorial features in web diagrams (crossings, nesting, etc.) thatcorrespond to geometric properties in the closures of the corresponding componentsof the Springer fiber (singularities, nested product structures, etc.). (See Section 4.3.)
HE GEOMETRY AND COMBINATORICS OF SPRINGER FIBERS 17 • Extend Theorem 5.3 to all nilpotent Hessenberg varieties. (See Section 5.) • Identify closed combinatorial formulas for the cohomology class induced by the Springerfiber S X in H ∗ ( G/B ). (See Section 5.) • Identify the structure constants of the cohomology H ∗ ( S X ) of the Springer fiber interms of the basis of Springer Schubert classes. (See Garsia-Procesi’s work on H ∗ ( S X )[15] as well as Section 5.) • Identify the equivariant cohomology H ∗ T ( S X ) of the Springer fiber according to theoutline in [18], or any other way. (See Section 5.) • Give a geometric explanation for Theorem 5.3. (See Section 5.)
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