An equivariant basis for the cohomology of Springer fibers

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for $GL_n(\mathbb{C})$ using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.


introduction
This paper analyzes the equivariant cohomology of Springer fibers in Lie type A. Springer fibers are fibers of a desingularization of the nilpotent cone in gl n (C). Springer showed that the symmetric group acts on the cohomology of each Springer fiber, the topdimensional cohomology is an irreducible representation, and each irreducible symmetric group representation can be obtained in this way [32,33]. As a consequence, Springer fibers frequently arise in geometric representation theory and algebraic combinatorics; see [31,12,13,16,18,29] for just a few examples.
There is also an algebraic approach to the Springer representation for GL n (C), as we now explain. Motivated by a conjecture of Kraft [24], De Concini and Procesi [7] gave a presentation for the cohomology of a type A Springer fiber as the quotient of a polynomial ring. Furthermore, this identification is S n -equivariant so Springer's representation can also be constructed as the symmetric group action on the quotient of a polynomial ring. These results were generalized to the setting of other algebraic groups by Carrell in [5].
The generators of the ideal defining the presentation of the cohomology of a type A Springer fiber were further simplified by Tanisaki [34]. Finally, Garsia and Procesi used the aforementioned results to study the graded character of the Springer representation in [17]. Their work gives a linear algebraic proof that this character is closely connected to the so-called q-Kostka polynomials. As part of their analysis, Garsia and Procesi study a monomial basis for the cohomology ring, originally defined by De Concini and Procesi in [7], with many amenable combinatorial and inductive properties. We refer to the collection of these monomials as the Springer monomial basis.
Let GL n (C) denote the algebraic group of n × n invertible matrices with Lie algebra gl n (C) of n × n matrices. Denote by B the Borel subgroup of upper triangular matrices, and by b its Lie algebra. Given a nilpotent matrix X ∈ gl n (C), let λ be the partition of n determined by the sizes of the Jordan blocks of X. The flag variety of GL n (C) is the quotient B := GL n (C)/B and the Springer fiber corresponding to λ is defined as the subvariety Let T denote the maximal torus of diagonal matrices in GL n (C) and L be the Levi subgroup of block diagonal matrices with block sizes determined by the partition λ. We may assume without loss of generality that X is in Jordan canonical form, and hence X is regular in the Lie algebra of L. Moreover, the subtorus S := Z G (L) 0 ⊆ T acts on the Springer fiber B λ . We consider the equivariant cohomology H * S (B λ ). The goal of this manuscript is provide a combinatorial framework to study this equivariant cohomology.
There is a known presentation for H * S (B λ ) given by Kumar and Procesi [25], and the equivariant Tanisaki idea has been determined by Abe and Horiguchi [1]. Our work below initiates a study of H * S (B λ ) which parallels the analysis of the ordinary cohomology by Garsia and Procesi in [17]. We define a collection of polynomials in H * S (B λ ) using the combinatorics of row-strict tableaux. Since these polynomials map onto the Springer monomial basis under the natural projection map H * S (B λ ) → H * (B λ ), we call them equivariant Springer monomials. We prove that a basis of equivariant Springer monomials exists for any Springer fiber, and provide a determinant formula (see Theorem 4.5 below) for the structure constants of any element of H * S (B λ ) with respect to this basis. As an application, we use the algebraic and combinatorial framework developed in this manuscript to study the images of Schubert classes in H * (B λ ). Let φ : B λ → B denote the inclusion of varieties, and φ * 0 : H * (B) → H * (B λ ) the induced map on ordinary cohomology. We prove that for every partition λ, there is a natural collection of Schubert classes whose images under φ * 0 form an additive basis of H * (B λ ). This result appears as Theorem 5.9 in Section 5 below and Corollary 5.14 contains the equivariant version of the statement. Phrased in terms of the work of Harada-Tymoczko in [21], the equivariant version of Theorem 5.9 says that there exists a successful game of Betti poset pinball for each type A Springer fiber. As a result, we can do computations in the (equivariant) cohomology ring more easily, as combinatorial properties of (double) Schubert polynomials are well-studied (c.f., for example, [26]). Bases of this kind have been used to do Schubert calculus style computations in the equivariant cohomology rings of other subvarieties of the flag variety [19,9]; the authors will explore analogous computations for Springer fibers in future work.
Our Theorem 5.9 generalizes results of Harada-Tymoczko [21] and Dewitt-Harada [8] which address the case of λ = (n − 1, 1) and λ = (n − 2, 2), respectively. The main difficulty in generalizing the methods used in those papers is that the equivariant cohomology classes in H * S (B λ ) constructed via poset pinball may not satisfy upper triangular vanishing conditions (with respect to some partial ordering on the set of S-fixed points of B λ ). The methods used to prove Theorem 5.9 side-step this difficulty by making use of the equivariant Springer monomials. Combining our determinantal formula for the structure coefficients of this basis with known combinatorial properties of the Schubert polynomials yields the desired result.
Recall that the Schubert polynomial S w (x) for w ∈ S n represents the fundamental cohomology class of the Schubert variety B w := BwB/B. That is, S w (x) is a polynomial representative for the cohomology class σ w ∈ H * (B λ ) defined uniquely by the property that In this paper, we study the polynomials φ * 0 (S w (x)) in H * (B λ ) from a combinatorial perspective. On the other hand, each is a polynomial representative for the cohomology class φ * 0 (σ w ) and one could ask if these classes have geometric meaning. In the last section, we show that the classes φ * 0 (σ w ) play an analogous role with respect to the homology of B λ as that played by the Schubert classes with respect to the homology of B. More precisely, we prove in Proposition 6.1 that Since B λ is typically not smooth, this map is not an isomorphism of groups. The remainder of the paper is structured as follows. The next section covers the necessary background information and notation needed in later sections, including a presentation of the equivariant cohomology of the Springer fiber due to Kumar and Procesi. The third and fourth sections of this paper establish the combinatorial groundwork for our study of H * S (B λ ). We use row-strict composition tableaux to define an equivariant generalization of the Springer monomial basis in Section 3, called the equivariant Springer monomials, and develop the structural properties of these polynomials further in Section 4. In particular, we give a determinant formula for the structure coefficients of H * S (B λ ) with respect to the basis of equivariant Springer monomials in Theorem 4.5 of Section 4. Finally, Section 5 uses the equivariant Springer monomials to study the images of monomials and Schubert polynomials in the cohomology of Springer fibers. Our main result in Section 5 is Theorem 5.9, which was discussed above. We conclude with an analysis of the geometric meaning of the classes φ * 0 (σ w ) in Section 6. Acknowledgements. The authors are grateful to Alex Woo, Jim Carrell, Dave Anderson, Anand Patel, Prakash Belkale and Jeff Mermin for helpful conversations and feedback. The first author was supported by a Oklahoma State University CAS summer research grant. The second author was supported by an AWM-NSF travel grant during the course of this research.

Background
As in the introduction, let G = GL n (C) and g = gl n (C) denote its Lie algebra. Denote by T the maximal torus of diagonal matrices in G and by B the Borel subgroup of upper triangular matrices. Let b denote the Lie algebra of B. The Weyl group of G is W S n . We let s i denote the simple transposition exchanging i and i + 1. Throughout this manuscript, α = (α 1 , . . . α k ) denotes a (strong) composition of n.
The composition α uniquely determines a standard Levi subgroup L in G, namely the subgroup of block diagonal matrices such that the i-th diagonal block has dimension α i ×α i . We denote the Weyl group for L by W L . Let X α : C n → C n be a principal nilpotent element of l, the Lie algebra of L. Note that by construction, X α ∈ g is a nilpotent matrix of Jordan type λ, where λ is the partition of n obtained from α by reordering its parts in weakly decreasing order.
Let B := G/B denote the flag variety. The Springer fiber of X α is defined to be If two compositions have the same underlying partition shape, then the corresponding Springer fibers are isomorphic. However, taking different compositions corresponding to the same partition shape yields actions of different sub-tori of T on the corresponding Springer fibers. This ultimately leads to the construction of different bases for the equivariant cohomology ring of B α . Let S denote the connected component of the centralizer of L in G, so S ⊆ T . Since X α ∈ l, we get that S centralizes X α and therefore S acts on B α by left multiplication. As indicated above, the purpose of this manuscript is to study the equivariant cohomology ring H * S (B α ). We begin by reviewing a presentation for H * S (B α ) due to Kumar and Procesi [25].
Recall from the introduction that φ : B α → B denotes the inclusion map of B α into the flag variety and consider the induced map on equivariant cohomology, φ * : H * T (B) → H * S (B α ). In this paper, we work with singular and equivariant cohomology with coefficients in C. Note that φ * naturally factors through H * S (B), . Let t denote the Lie algebra of T . The coordinate ring of t × t is the polynomial ring where J is the ideal J = i y i , i x i . It is well known that the T -action on B by left multiplication is equivariantly formal, implying Let s ⊆ t denote the Lie algebra of S and Z α be the reduced closed subvariety of t × t defined by Note that we may also view Z α as a subvariety of s × t ⊆ t, and we use this perspective in our computations below. The coordinate ring is naturally an S(s * )-algebra via the projection Z α → s onto the first factor. Moreover, the ring A inherits a non-negatively graded structure from C[t × t]. We also define the graded C-algebra where C is considered an S(s * )-module under evaluation at 0. Note that if α = (1, 1, . . . , 1), then B α = B and S = T . In this special case, we denote the corresponding coordinate ring by A := C[Z (1,1,..., 1) ]. The next theorem from [25] gives a presentation of H * S (B α ).
, making the following diagram commute.
The graded S(s * )-algebra isomorphism of Theorem 2.1 implies A is a free S(s * )-module with rank equal to the number of S-fixed points of B α , namely |W/W L | (c.f. [25, Lemma 2.1]).

2.2.
Maps of polynomial rings. Since L is a standard Levi subgroup and S = Z G (L) 0 , if t = diag(t 1 , . . . , t n ) ⊆ g and s has coordinates (z 1 , . . . , z k ), then the embedding of the subalgebra s into t is given by This embedding induces a map i * : , then F and i * (F ) have the same values on Z α . This implies where I s (Z α ) = i * (I(Z α )) denotes the ideal of Z α as a subvariety of s × t. We let π : C[s × t] → A denote the canonical projection map. By a slight abuse of notation, we will also denote the quotient C[t × t] → A by π; the isomorphism of (2.5) tells us that we may do so without loss of generality.
As in (2.1), there are isomorphisms and we make this identification below whenever it is convenient (and similarly for C[t × t]). The ring A inherits the graded structure of C[s × t]. In particular, the degree k component of C[s × t] is i+j=k S i (s * ) ⊗ S j (t * ) and we denote its image under the canonical projection map π : denote the positive degree and degree zero components of A k with respect to the grading of S(s * ). It is easy to see that There is a surjective map ev : S(s * ) ⊗ S(t * ) → S(t * ) given by evaluation at 0. More explicitly, if F = p ⊗ q with p ∈ S(s * ) and q ∈ S(t * ), then ev(F ) := p(0) ⊗ q (then extend linearly to all of S(s * ) ⊗ S(t * )). This map induces an evaluation map ev : A → A 0 giving the commutative diagram: where, by Theorem 2.1, A H * S (B α ) and A 0 H * (B α ). Note that, under these identifications, the evaluation map is simply the usual restriction map from equivariant to ordinary cohomology. The following lemma relates bases of the ordinary and equivariant cohomology rings of B α . Proof. To begin, we prove that the S(s * )-span of It suffices to assume that F ∈ A k for some k. We proceed by induction on k.
Observe that the second term of the above sum belongs to A k + and is therefore of the form p ⊗ q where each q ∈ S j (t * ) from some j < k. By induction, each q is a S(s * )-linear combination of B 1 , . . . , B m and hence so is F . We next prove that {B 1 , . . . , B m } is S(s * )-linearly independent. As noted in Remark 2.2, A is a free S(s * )-module of rank m = dim H * (B α ). Let Q(s * ) denote the field of fractions of S(s * ) C[z 1 , . . . , z k−1 ]. Since A is a free module, the extension of scalars Q(s * ) ⊗ S(s * ) A is a free module of the same rank [10, §10.4, Cor. 18]. Furthermore, the polynomials B 1 , . . . , B m must also span Q(s * ) ⊗ S(s * ) A. Since the extension of scalars is an mdimensional vector space, {B 1 , . . . , B m } are Q(s * )-linearly independent. Any non-trivial linear relation among B 1 , . . . , B m with S(s * )-coefficients would also be a non-trivial linear relation over Q(s * ), contradicting the previous sentence. We conclude {B 1 , . . . , B m } is S(s * )-linearly independent, as desired.
2.3. The Springer monomial basis. We now recall the monomial basis of A 0 H * (B α ) defined by De Concini and Procesi in [7] and further analyzed by Garsia and Procesi in [17].
Let λ be a partition of n with k parts and λ[i] be the underlying partition shape of the composition of [n − 1] obtained from λ by decreasing the i-th part by 1. Define Sp λ ⊂ C[t] to be the collection of monomials constructed recursively as in [17, §1] by denotes the set of monomials obtained by multiplying each monomial in Sp λ[i] by x i−1 n . Observe that as defined in [17], the monomials in Sp λ are in the variables x 2 , . . . , x n . We define where the action of the longest permutation w 0 ∈ W on variables is given by w 0 · x i := x n−i+1 . Hence the monomials in Sp λ are in the variables x 1 , . . . , x n−1 . Since the ideal I(Z α ) is invariant under the action of W , it follows by results of De Concini and Procesi that, as graded vector spaces, A 0 Here we use standard monomial notation . We refer to the basis Sp λ of H * (B α ) as the Springer monomial basis, and to its elements as Springer monomials. We adopt the convention throughout this manuscript that if x δ ∈ Sp λ , then we denote both x δ ∈ C[t] and its image under the canonical projection π 0 : C[t] → A 0 by the same symbol.
Example 2.4. Let n = 4 and λ = (2, 2), then Sp (2,2) Remark 2.5. The Springer monomials have been generalized to study the cohomology rings of other subvarieties of the flag variety. In particular, Mbirika in [27] constructs an analogous set of monomials for nilpotent Hessenberg varieties (which include Springer fibers). In a later paper, Mbirika and Tymoczko give an analogue of the Tanisaki ideal in the Hessenberg setting [28].

Row-strict Tableaux
In this section we develop a combinatorial framework to study the ring A defined in (2.3) using row-strict composition tableaux.
A shifted row-strict composition tableau of shape β is a labeling Υ of the composition diagram with the m integers [n − m + 1, n] such that the values decrease from left to right in each row. For simplicity of notation, letm := n − m + 1. Let RSCT n (β) denote the collection of all shifted-standard row-strict tableaux of composition shape β with content [m, n]. Observe that ifm = 1 (i.e. β is a composition of n), then the content of β is the full standard content [n]; in this case, we say that β is a row-strict composition tableau.
Example 3.1. Consider the composition β = (1, 2, 0, 1) with n = 5. In this case m = 4 andm = 2. There are 12 row-strict composition tableaux in RSCT 4 (β). Indeed, note that there are 24 = 4! possible fillings of β using the content [2,5]. Furthermore, if we define two fillings to be equivalent up to the entries in each row, e.g. Given a composition β, let α = (α 1 , . . . , α k ) be the strong composition obtained from β by deleting any part equal to zero. By similar reasoning as in the example above, we have that |RSCT n (β)| = n! α 1 ! · · · α k ! which is precisely the number of S-fixed points in the Springer fiber B α . Notice that if m > 1, then each shifted row-strict composition tableau can be associated to a unique row-strict tableau in RSCT m (β) by the relabeling map i → i −m + 1. We use the "shifted" terminology since it simplifies the arguments below. Similarly, although we typically begin with a strong composition of n, our inductive procedures require the generality of weak compositions.
We now define a map where the union on the RHS is taken over all compositions β obtained from β by deleting one box from any nonzero row. Let η(Υ) be the composition tableau obtained by removing the box from Υ which contains its smallest entry, namelym. For example: In this case, the disjoint union in (3.1) is taken over β ∈ {(1, 2, 3), (2, 1, 3), (2, 2, 2)}. The map η plays an important role in the inductive arguments below; note that η is in fact a bijection.
Definition 3.2. Let Υ ∈ RSCT n (β). We say that (i, j) is an Springer inversion of Υ if there exists j in row j such that i < j and either: (1) j appears above i and in the same column, or (2) j appears in a column strictly to the right of the column containing i.

Denote the set of Springer inversions of Υ by Inv(Υ).
Example 3.3. Let n = 9 and β = (2, 0, 3, 2, 1). Consider Υ ∈ RSCT 9 (β) with content [2,9] Remark 3.4. Note that the definition above is closely related to the notion of a Springer dimension pair considered by the first author and Tymoczko in [29]. In that paper, the convention is that the row-strict tableaux have increasing entries (from left to right), while our convention is that the entries are decreasing (from left to right). This change in conventions is routine; to convert from one to the other, apply the permutation w 0 such that w 0 (i) = n − i + 1 for all i. A Springer inversion from this paper corresponds to a unique Springer dimension pair as defined in [29] (up to transformation under w 0 ). If (i, j) is a Springer inversion then (n − i + 1, n − j + 1) is a Springer dimension pair, where j denotes the smallest element in row j such that i < j .
The following lemma is a simple, but important fact about inversions.
Lemma 3.5. Let Υ, Ω ∈ RSCT n (β). Let j Υ , j Ω denote the indices of the rows containinḡ m in Υ and Ω, respectively. Then exactly one of the following is true: Proof. Without loss of generality, suppose that j Υ < j Ω and hencem is contained in different rows of the tableaux Υ and Ω. Sincem is the smallest number in the content, it must lie at the end its respective row of Υ and Ω. Moreover, the content of the row indexed by j Ω in Υ is strictly larger thanm and vice versa. If the size of row j Ω is at least the size of row j Υ , then (m, j Ω ) ∈ Inv(Υ). Otherwise, (m, j Υ ) ∈ Inv(Ω). Lemma 3.5 induces a total ordering on the set RSCT n (β) as follows.
In the next section we will associate a unique monomial to each element of RSCT n (α). We will see that the total ordering on the shifted row-strict composition tableaux defined above corresponds to the lex ordering on these monomials.

Equivariant Springer monomials.
In this section we define a collection of polynomials indexed by row-strict composition tableaux. The main purpose of defining these polynomials is to provide a combinatorial framework to study the cohomology ring H * S (B α ) A in the following sections. Indeed, the polynomials defined below will serve as an equivariant generalization of the Springer monomial basis.
If the inversion set of Υ is empty, then define P Υ = 1. We call the collection of polynomials obtained in this way equivariant Springer monomials.
While P Υ are not monomials in the traditional sense, we use the term "monomial" since P Υ is a product of equivariant factors (x i − z j ), a common generalization of monomials in ordinary cohomology. We adopt the convention throughout this manuscript that each equivariant Springer monomial P Υ ∈ C[s × t] and its image under the canonical projection map π : C[s × t] → A are denoted by the same symbol. This greatly simplifies the notation below.
Example 3.9. Let n = 9 and β = (2, 0, 3, 2, 1) as in Example 3.3. Then There is a simple inductive description of the equivariant Springer monomials, as explained in the next two paragraphs. Supposem labels a box in row j Υ of Υ and recall thatm must label the last box in row j Υ . Sincem is the smallest label that appears, we have Denote this set by Invm(Υ). Note in particular that | Invm(Υ)| is uniquely determined by the value of j Υ .
We obtain a corresponding decomposition formula for the polynomial P Υ given by if Invm(Υ) = ∅ and Q Υ = 1 otherwise. The next lemma show that the decomposition formula for the polynomials P Υ from (3.3) is compatible the recursive formula defining the Springer monomials given in equation (2.7).
Lemma 3.10. Let β be a composition of m ≤ n and λ denote its underlying partition shape. Then, In particular, the set {ev(P Υ ) | Υ ∈ RSCT n (β)} only depends on λ, the underlying partition shape of β.
Proof. First observe that if β = (β 1 , . . . , β k ) is a weak composition of m ≤ n, then β determines a unique strong compositionβ obtained by deleting the parts of β equal to 0. If Υ ∈ RSCT n (β), then there is also a corresponding Υ ∈ RSCT n (β ) obtained by upward justifying all rows. It is easy to see from the definitions that β and β have the same number of Springer inversions and that ev(P Υ ) = ev(P Υ ). Hence we may assume without loss of generality that β is a strong composition of m. We now proceed by (reverse) induction onm, the smallest value appearing in any Υ ∈ RSCT n (β). Ifm = n then m = 1 and λ = (1). In this case, RSCT n (β) contains a single element, namely the row-strict composition tableau consisting of a single box labeled by 1. Therefore Inv(Υ) = ∅ and Sp λ = {1} = {ev(1)}, as desired. Now supposem < n and β = (β 1 , . . . , β k ) has k non-zero parts. Let σ −1 denote the unique minimal length permutation of k such that λ = (β σ −1 (1) , . . . , β σ −1 (k) ). In other words, σ(i) − 1 is equal to the number of j ∈ [m] such that β j > β i plus the number of j ∈ [m] such that β j = β i and j < i.  Here we have While these polynomials are different, they correspond to the same Garsia-Procesi monomial, ev(Υ) = ev(Ω) = x 2 1 x 4 ∈ Sp (3,2,1) .
The next theorem tells us that the collection of equivariant Springer monomials is an S(s * )-module basis for the equivariant cohomology ring A H * S (B α ). We study the structure coefficients of A with respect to this basis in the next section.
Proof. The polynomials P Υ (z, x) are homogeneous elements of A. Lemmas 2.3 and 3.10 now imply the desired result.

Localization and Determinant Formulas
In this section, we explore algebraic properties of the equivariant Springer monomials. The results of this section establish methods for computing the expansion of any F ∈ A as an S(s * )-linear combination of the P Υ , Υ ∈ RSCT n (α). We begin by showing that the Springer polynomials satisfy upper triangular vanishing relations with respect to the total ordering defined on row-strict composition tableaux defined in the previous section. We then use these vanishing properties to give a determinant formula for the structure coefficients in Theorem 4.5 below.
4.1. Localization formulas. Suppose α = (α 1 , . . . , α k ) is a strong composition of n. Let h = (h 1 , . . . , h k ) be a regular element of s, which we identify as a point in t, by The condition that h be a regular element means that each of the h i are distinct. For every w ∈ W , there is a natural localization map, given by φ w (F (z, x)) = F (z, w · z). In other words, φ w (F )(h) := F (h, w · h) for any h ∈ s.
Here W acts on s * (and the coordinates of z) by permuting the entries; for example, if w = [2, 4, 1, 3] = s 1 s 2 and h = (h 1 , h 1 , h 2 , h 2 ) then w · h = s 1 s 2 · h = (h 2 , h 1 , h 1 , h 2 ). It is easy to see that F ∈ I(Z α ) if and only if φ w (F ) ≡ 0 for all w ∈ W . Hence any F ∈ A is uniquely determined by the collection of values {φ w (F ) | w ∈ W }. Recall that L is the Levi subgroup of GL n (C) determined by the composition α and W L denotes the Weyl group of L. Since L is standard, it is generated by a subset of simple reflections. Also, since W L acts trivially on s (because S = Z G (L) 0 ), it suffices to consider the maps φ w where w ∈ W L . Here W L denotes the set of minimal length coset representatives of W/W L . Recall that each permutation w ∈ W can be written uniquely as w = vy for v ∈ W L and y ∈ W L .
We now associate a coset representative w Υ ∈ W L to each Υ ∈ RSCT n (α) by constructing a vector h Υ ∈ t which is a particular permutation of the coordinates of h. Specifically, if i lies in the j-th row of Υ, then we require the i-th coordinate of h Υ equal to h j . Let w Υ to be the unique permutation in W L such that h Υ = w Υ h. Observe that the map from RSCT n (α) to W L given by Υ → w Υ is a bijection. Then h Υ = (h 1 , h 2 , h 3 , h 3 , h 1 ) with w Υ = [1, 5, 2, 3, 4] (in one-line notation). Note that in this case, W L = s 1 , s 4 and it easy to check that w Υ ∈ W L ; we have only to observe that w Υ (1) < w Υ (2) and w Υ (4) < w Υ (5). Also, in this example we have P Υ = (x 2 −z 1 )(x 4 −z 1 ) since Inv(Υ) = {(2, 1), (4, 1)}.
Our next proposition says that the equivariant Springer monomials satisfy upper triangular vanishing conditions with respect to the total order on row-strict composition tableaux defined in the previous section. Let Ω < Υ ∈ RSCT n (α). Then the following are true: Proof. Fix a regular element h ∈ s as in (4.1). We first prove part (1) of the proposition. By definition, if h j is the i-th coordinate of h Υ , then i is contained in the j-th row of Υ. We have Note that if (i, j) ∈ Inv(Υ), then i cannot be contained in the j-th row of Υ. Hence (h Υ ) i = h j for all (i, j) ∈ Inv(Υ) and φ w Υ (P Υ )(h) = 0 as claimed. We now prove part (2). Indeed, we have Since Ω < Υ, there exists (i, j) ∈ Inv(Υ) such that the content of j-th row of Ω contains i. This implies that (h Ω ) i = h j and hence φ w Ω (P Υ )(h) = 0. Since h ∈ s is an arbitrary regular element, we have φ w Ω (P Υ ) = 0 in S * (s). We conclude with a detailed example.
Example 4.4. Let n = 4 and α = (2, 2). A table of P Υ , w Υ and h Υ for all elements Υ ∈ RSCT 4 (α) is displayed in Figure 1 below. The matrix [φ w Ω (P Υ )] (Υ,Ω)∈RSCT 4 (α) 2 written with respect to the total ordering on RSCT 4 (α) given in Example 3.7 is: Proposition 4.2 implies this matrix is always upper triangular with respect to the total ordering in Definition 3.6 with non-vanishing polynomials in S(s * ) on the diagonal. where the Q(s * ) denotes the field of fractions of S(s * ). We index the set RSCT n (α) = {Υ 1 < · · · < Υ N } by the total ordering given in Definition 3.6 where N = |RSCT n (α)| = |W L |. For notational and computational simplicity, let P i := P Υ i and w i := w Υ i . Given any F ∈ A, we write implies c · P = v. Proposition 4.2 tells us that P is an upper triangular matrix with nonzero diagonal entries, and is therefore invertible as a matrix with entries in Q(s * ). Hence Our next theorem uses this equation to prove that each coefficient C k is the determinant of some matrix with entries determined by v and φ i (P j ). Normalize the polynomials P i by defining Q i := 1 φw i (P i ) · P i . Note that this definition makes sense, since φ w i (P i ) = 0 for all i by Proposition 4.2.  We now apply the localization map φ w k to both sides of Equation (4.5). Solving for D k and applying Equations (4.6) and (4.7) yields proving the theorem.
This theorem provides us with the computational tools to expand any polynomial of A in the basis of equivariant Springer monomials. It follows immediately that we can compute the expansion of any polynomial in A 0 H * (B α ) in the Springer monomial basis by simply applying the evaluation map ev : A → A 0 . We use these results in the next section to study the images of monomials and Schubert polynomials in H * (B α ).
Example 4.6. Let n = 4 and α = (2, 2). The polynomials P Υ for Υ ∈ RSCT 4 (α) are computed in Example 4.4 (see Figure 1 also). In this case N = 6 and the total order on RSCT 4 (α) is as in Example 3.7, so the rows of the table in Figure 1 list the polynomials in order: P 1 , · · · , P 6 , from top to bottom. We compute the expansion of using the determinant formula of Theorem 4.5. The reader may note that F is the image of the double Schubert polynomial S s 3 (y, The matrix [a(i, j + 1)] 5 0 from Theorem 4.5 is given by: Where the first row is the vector v = [φ w 1 (F ), . . . , φ w 6 (F )] with the rest of the matrix coming from first five rows of the matrix in Example 4.4 (normalized to φ w j (Q i )). If and hence F = P 2 + P 3 + P 4 . Note that we can also compute c by using the equation the inverse of the matrix from Example 4.4.

Monomials and Schubert polynomials
In this section, we study the images of the Schubert polynomials S w (x) for w ∈ W (α) under the map π 0 : C[t] → A 0 . We use Theorem 4.5 to identify a collection of permutations . This result is stated in Theorem 5.9. We obtain an analogous statement for equivariant cohomology in Corollary 5.14. Our analysis generalizes work of Harada-Tymoczko [21] and Harada-Dewitt [8] in the sense that Corollary 5.14 implies the existence of an explicit module basis for H * S (B α ) constructed by playing poset pinball. We prove Theorem 5.9 in two steps. First, we use the expansion formula of Theorem 4.5 to prove that the Springer monomial basis Sp λ of H * (B α ) defined in (2.7) above is uppertriangular in an appropriate sense. In particular, we study the expansion of any monomial in A 0 with respect to the Springer monomial basis. Since each Schubert polynomial is a sum of monomials, we are then able to leverage our results for monomials to prove the desired result for Schubert polynomials. More specifically, we prove that the transition matrix from {π 0 (S w ) | w ∈ W (α)} to Sp λ is invertible.
To begin, recall the commutative diagram from (2.6). In particular, recall that A C[t × t]/I(Z α ) and A 0 C[t]/ ev(I(Z α )) and the maps π and π 0 denote the canonical projection maps. δ → x δ := x δ 1 1 · · · x δ n−1 n−1 . We impose the lexicographical total ordering on monomials. In other words, x γ < x δ if and only if γ k < δ k where k denotes the smallest index where the entries of the compositions γ and δ differ. Note that lex order does not respect the degree of a monomial. For example x 2 1 x 4 2 < x 3 1 x 2 . If Υ ∈ RSCT n (α), then ev(P Υ ) is a monomial in C[t]. Hence we define the notation x Υ := ev(P Υ ).
By Lemma 3.10 the set of all monomials obtained in this way is precisely the set of Springer monomials Sp λ where λ is the underlying partition shape of α. Recall that, by convention, since x Υ ∈ Sp λ we also write x Υ to denote the image of the monomial x Υ in A 0 under π 0 . Observe that if γ is the associated exponent composition of x Υ , then γ i is simply the number of inversions in Inv(Υ) whose first factor is i. Hence we will call the composition γ the inversion vector of Υ. For Υ as in Example 3.3, the inversion vector is γ = (0, 2, 0, 0, 2, 1, 0, 0) and x Υ = x 2 2 x 2 5 x 6 . The next lemma follows immediately from the definition of the total order on RSCT n (α). Lemma 5.1 implies that the vanishing property given in Proposition 4.2 is, in some way, compatible with the total ordering on all monomials. To make this compatibility precise, for each x δ ∈ C[t] we construct a polynomial P δ (z, x) ∈ C[s × t] such that ev(P δ ) = x δ . This polynomial serves as an analogue of P Υ (z, x) for x Υ when Υ ∈ RSCT n (α).
Let δ = (δ 1 , . . . , δ n−1 ) be a composition of n. If δ is the inversion vector for some Υ ∈ RSCT n (α), then set P δ = P Υ . Otherwise, by Lemma 5.1 there is a unique maximal Υ ∈ RSCT n (α) such that x Υ < x δ . Let γ = (γ 1 , . . . , γ n−1 ) denote the inversion vector of Υ. By definition of the total ordering on monomials, there exists an index k such that where j denotes the index of the row containing k in Υ and the composition δ is defined by The following example illustrates the construction.
Proof. It easy to see by construction that ev(P δ ) = x δ which proves (1). If δ is the inversion vector for some Υ ∈ RSCT n (α), then (2) is an immediate consequence of Proposition 4.2.
Thus we have only to prove (2) in the case that δ is not the inversion vector for some row strict composition tableau. Let Υ be the maximal element of RSCT n (α) such that x Υ < x δ . First observe that if Ω = Υ, then φ w Ω (P δ ) = 0 since the factor (x k − z j ) in P δ evaluates to zero on any (h, w Υ · h) with h ∈ s. Now suppose Ω < Υ. By definition of the total order on RSCT n (α), there exists (i, j) ∈ Inv(Υ) such that the content of the j-th row of Ω contains i. Furthermore, we have that the numbers i + 1, . . . , n appear in the same rows (and the same exact position) of Υ and Ω. If (i, j) ∈ Inv ≤k (Υ), i.e. if i ≤ k, then φ w Ω (P δ ) = 0. Otherwise, if i > k then the tableaux Ω and Υ must contain k in the same row. This implies φ w Ω (P δ ) = 0 due to the factor (x k − z j ) again evaluating to zero.
The following proposition tells us that the expansion of π 0 (x δ ) in the Springer monomial basis contains only monomials x Υ for Υ ∈ RSCT n (α) such that x δ ≤ x Υ . This is what we mean when we say that the Springer monomial basis is compatible with the total ordering on all monomials. Then and note that if x δ = x Υ for some Υ ∈ RSCT n (α), then the proposition is trivial. We therefore assume that x δ = x Υ for any Υ ∈ RSCT n (α), i.e., that δ is not the inversion vector for any row strict composition tableaux of shape α. Consider the polynomial P δ ∈ C[s × t] as defined in equation (5.1) and write Let Υ ∈ RSCT n (α) be the unique maximal tableau for which x Υ < x δ . Theorem 4.5 and Lemma 5.3 together imply C Ω = 0 for all Ω ≤ Υ. (Note that this fact also follows from equation (4.4)). Again by Lemma 5.3, we have ev(P δ ) = x δ and hence c Ω = ev(C Ω ) = 0 for all Ω ≤ Υ.
We demonstrate Proposition 5.4 with an example.
Let S(s * ) >0 denote the subalgebra of polynomials with positive degree in S(s * ). One immediate consequence of Proposition 5.4 is the following.

Schubert polynomials. The set of Schubert polynomials
is an important collection of polynomials. Note that the map π 0 : C[t] → A 0 factors through φ * 0 : A 0 → A 0 where A 0 H * (B) (see Theorem 2.1) and hence S w (x) may be viewed as a polynomial in A 0 . It is well-known that Schubert polynomials are representatives for the Schubert classes in H * (B) and form a basis of the cohomology ring. The main result of this section is Theorem 5.9 which states there is a natural subset W (α) ⊆ W such that the set of images {π 0 (S w ) | w ∈ W (α)} form a basis for the cohomology of the Springer fiber A 0 H * (B α ). Corollary 5.14 in this section proves an equivariant version of this statement and generalizes results of Harada-Tymoczko [21] and Harada-Dewitt [8] which address the cases of α = (n − 1, 1) and α = (n − 2, 2), respectively.
We now describe the set W (α) ⊆ W . This subset is analogous to the set of Schubert points defined by the first author and Tymoczko in [29], although our conventions differ, as discussed in Remark 3.4 above. To any Υ ∈ RSCT n (α) we define u Υ to be the unique permutation (as defined in Lemma 5.7) such that the inversion vector of Υ equals the Lehmer code of u Υ . Define This collection of permutations has the property that the number of w ∈ W (α) with Bruhat length k is precisely dim (H 2k (B α )). Thus the set W (α) is the output of a successful game of Betti pinball in the sense of [21].
Before we prove the theorem, we review a relevant construction of the Schubert polynomials. For any w ∈ W , let R(w) denote the set of reduced words of w. In other words, R(w) is the collection of sequences ρ = (ρ 1 , . . . , ρ ) for which w = s ρ 1 · · · s ρ and (w) = , where (w) is the Bruhat length of w. A positive, weakly increasing sequence of integers β = (β 1 ≤ · · · ≤ β ) is said to be ρ-compatible if β j ≤ ρ j for all j ≤ and, whenever ρ j < ρ j+1 , we have β j < β j+1 . Let C(ρ) denote the set of ρ-compatible sequences. Note that C(ρ) can be empty.
The following theorem is originally due to Billey, Jockusch and Stanley [3]. It computes each Schubert polynomial in terms of reduced works and compatible sequences. For the purposes of this paper, this construction serves as the definition of a Schubert polynomial.
Let W a denote the subgroup of W generated by the simple reflections s a+1 , . . . , s n−1 and W a denote the set of minimal length coset representatives of W/W a . The next lemma is a key technical result needed to prove the main theorem, it shows that compatible sequences are well-behaved with respect to the coset decomposition of W a−1 corresponding to the quotient W a−1 /(W a−1 ∩ W a ), which has shortest coset representatives W a−1 ∩ W a . Lemma 5.12. Let w ∈ W a−1 and let w = vu be the unique parabolic decomposition of w, with v ∈ W a and u ∈ W a . Suppose β ∈ C(ρ) for some ρ ∈ R(w) such that β 1 ≥ a. Then β i = a for all 1 ≤ i ≤ (v).
Proof. Since w ∈ W a−1 , the smallest positive integer appearing in any reduced word for w is a. Note that if (v) = 0, that is, if s a does not appear in a reduced word for w, then the lemma is trivial. Thus we assume that (v) ≥ 1, i.e. v = e and a appears at least once in any reduced word for w.
If a appears more than once in ρ, then C(ρ) contains no sequences that begin with a. Thus we assume that a appears exactly once in ρ. In this case, we must have ρ k = a for all k < (v) and ρ (v) = a. Indeed, if ρ k = a for some k < (v), then we can write w = z 1 z 2 where z 1 = s ρ 1 · · · s ρ k and z 2 ∈ W a . Now write z 1 = v u for some v ∈ W a and u ∈ W a . But then is a parabolic decomposition of w with (v ) ≤ k < (v) which contradicts the uniqueness of parabolic decompositions. The argument of the previous paragraph implies (ρ 1 , . . . , ρ (v) ) ∈ R(v). Now let β ∈ C(ρ). If β 1 ≥ a, then condition that β j ≤ ρ j for all j implies β i = a for all 1 ≤ i ≤ (v) as desired.
Let γ = (γ 1 , γ 2 , . . . , γ n−1 ) denote the Lehmer code of a permutation w. Then we may consider the reduced word for w determined by γ as in Lemma 5.7, given by: It is easy to see that x γ is the monomial corresponding to the sequence β 0 and hence Theorem 5.10 implies any Schubert polynomial can be written as The next proposition shows that x γ is the leading term of S w with respect to the lexicographical total ordering on monomials.
Proof. By Theorem 5.10, each monomial in the expansion of S w (x) corresponds to a compatible sequence of some reduced word of w. Let ρ ∈ R(w) and β = (β 1 , . . . , β ) ∈ C(ρ) with corresponding monomial x δ = x β 1 · · · x β . Using this notation, the composition δ has the property that δ 1 is equal to the number of 1's in β, δ 2 is equal to the number of 2's in β, and so on.
What remains to be shown is that if γ = δ, then ρ = ρ 0 and hence the monomial x γ only appears once in the expansion of S w (x) given in (5.2). Since compatible sequences are weakly increasing, γ = δ implies that β 0 = β. This further implies that ρ k ≥ j for all k ≥ 1 + γ 0 + γ 1 + · · · + γ j−1 (with the convention that γ 0 = 0). The only reduced word of w that satisfies this property is ρ 0 ; this follows easily from the uniqueness of parabolic decompositions.
We can now prove our main theorem, which shows that the images of the Schubert polynomials corresponding to elements from W (α) form a basis of H * (B α ).
Proof of Theorem 5.9. Let w ∈ W (α). Then there exists a unique Υ ∈ RSCT n (α) for which w = u Υ . Let γ denote the Lehmer code of w, which is also the inversion vector of Υ. By Proposition 5.13, we can write S w (x) = x γ + δ c δ x δ where the sum is over compositions δ and x γ < x δ for all c δ = 0. We now have that π 0 (S w (x)) = π 0 (x γ ) + δ c δ π 0 (x δ ).

Υ<Ω
g Ω x Ω for some coefficients g Ω . This equation implies that the transition matrix from the set {π 0 (S w (x)) | w ∈ W (α)} to the basis {x Υ | Υ ∈ RSCT n (α)} of A 0 is invertible. In fact, it is upper triangular with 1's on the diagonal. This proves the theorem.
Observe that x 1 x 3 , x 1 x 2 ∈ Sp (2,2) while the monomial x 2 x 3 is not an element of Sp (2,2) . This example shows that although the structure constants are nonnegative in many examples, there is typically some cancellation to take into account. The answer to Question 5.16 is known to be 'yes' in the special case that α is one of (n), (n − 1, 1), or (1, 1, . . . , 1). Note that when α = (1, 1, . . . , 1), the Springer fiber B α is the full flag variety, and we obtain a positive answer to Question 5.16 using the formula from Theorem 5.10.

Connections with the geometry of Springer fibers
It is well known that the Schubert polynomial S w (x) is a polynomial representative for the fundamental cohomology class of the Schubert variety B w := BwB/B. It is therefore natural to ask if the polynomials φ * 0 (S w (x)) represent a fundamental cohomology class of a subvariety in the Springer fiber B λ . Unfortunately, due the fact that Springer fibers are usually singular, the classical notion of a fundamental cohomology class of a subvariety using Poincaré duality is not defined. However, the notion of a fundamental homology class of a subvariety is well defined (see [6] or [15,Appendix B]).
We briefly recall the connections between homology classes and Schubert polynomials for the flag variety B, which is smooth. Recall that the inclusion φ : B λ → B induces a surjective map φ * 0 : H * (B) → H * (B λ ). Since we do not consider equivariant cohomology in this section, we denote will denote φ * 0 by just φ * . We now give a geometric interpretation of the classes φ * (σ w ) ∈ H * (B λ ), which are represented by the polynomials φ * (S w (x)) = π 0 (S w (x)) in A 0 . Note that the following proposition is true for any subvariety X of the flag variety with inclusion map φ : X → B (not just Springer fibers).  for generic g ∈ G.
Corollary 6.2. Let g ∈ G be generic and suppose that w∈W c w φ * (σ w ) = 0 in H * (B λ ) for some coefficients c w ∈ C. Then Note that φ * (σ w ) can be computed explicitly by expanding its polynomial representative π 0 (S w (x)) in terms of the Springer monomial basis using Theorem 4.5. Hence Corollary 6.2 gives a combinatorially sufficient condition to determine if the homology classes [B λ ∩ gB w ] = 0 for generic g ∈ G.