SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI SPACES

LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

1. Introduction

In this paper we approach the theory of Schur and Schubert polynomials from the the point of view of generalized Thom polynomials. The starting point is that we realize Schur and Schubert polynomials—of Chern classes—as first obstructions of certain fibre bundles (in the spirit of Stiefel, Whitney and Steenrod who defined Chern classes as first obstructions). This realization makes Schur and Schubert polynomials a special case of the generalized Thom polynomials. Applying a general method to compute them as Thom polynomials we obtain new definitions for these polynomials. Along the way we also redefine the double Schubert polynomials and define “double Schur polynomials” which are also first obstructions. We show how these results are related to the structure of the cohomology ring of the Grassmannian and the flag manifold. The authors hope that this Thom polynomial technique may turn to be useful in the study of the cohomology ring structure of various moduli spaces, see section 6 for a discussion.

We will define Schur and Schubert polynomials as unique solutions of linear equations. This might seem somewhat implicit however it allows us to give closed formulas for the double Schur polynomials (Theorem 5.1) and to deduce the recursion formula of Lascoux and Schützenberger [11] for double Schubert polynomials (Theorem 4.2).

For geometrically defined fibre bundles—on which we mean fibre bundles associated to a principal G-bundle where G is a Lie group—first obstructions are usually called Thom polynomials (see [3] on this subject). The word polynomial is justified since the first obstruction of the universal bundle is an element in H^{∗}(BG) which is a polynomial ring, at least rationally.

Supported by OTKA D29234 (first author), OTKA T029759 (second author)

Keywords: Thom polynomials, global singularity theory, Schur and Schubert polynomials, double Schubert polynomials, cohomology of quotients AMS Subject classification 14N10, 57R45.

2 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

Calculating Thom polynomials has a long history. It was René Thom who initiated their study in the case of singularities of smooth maps. The major tool for calculating them was the method of resolution (see [1] for an account of the method and results).

Works of V. Vassiliev [15] and M. Kazarian [7], [9] clarified the connection of Thom polynomials with the underlying symmetry groups. Their works also show that the so called degeneracy loci formulas in algebraic geometry are also Thom polynomials for group actions (see [5] for many examples). In particular in [2] double Schubert polynomials are described as degeneracy loci formulas for certain quiver representations. In [4] we showed how to describe and calculate quiver type formulas as Thom polynomials for group actions. In this paper however we use a more “economic” group action. In Remark 6.9 we compare the two methods.

Based on works of A. Szűcs ([14]) the second author introduced a different method (called the method of restriction equations) to calculate Thom polynomials for singularities of smooth maps ([12]). In [3] we generalized the method to the case of Thom polynomials for group actions. It turned out that though the idea is quite simple the method of restriction equations is very powerful. We reproduced and improved earlier results in several directions ([3]). In all these applications we start from a representation of G with the property that G has only finitely many orbits (at least up to a given codimension). In the final section of this paper we make an attempt to generalize the definitions and methods to the case where continuous families of orbits occur—the case of “moduli”. We also propose that via Thom polynomials we can calculate the cohomology rings of certain moduli spaces.

The idea of the paper was born from a question of Tamás Hausel who drew our attention to the problem of calculating cohomology rings of moduli spaces.

2. Thom polynomials for group actions

In this section we give a short introduction to the theory of Thom polynomials for group actions. They were defined by M. Kazarian in [9]. The approach given here is somewhat different since we concentrate on complex representations, see [3] for a detailed introduction along these lines.

Given a complex representation ρ : G → GL(V) and a principal G-bundle P → M we can look for obstruction of having a section of the V -bundle E = P ×_{G} V associated to this

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 3

representation avoiding a certain orbit η (or more generally a G-invariant subset of V). Of course the zero section avoids any orbit different from the zero orbit but this is pathological: we want obstructions for a generic section. In effect we want to avoid the closure of η. The obstruction we will deal with is the cohomology class represented by η̄(s) ⊂ M for a generic section s. Thus this class is an obstruction for having a section in the complement V \ η̄. Let us call this class the Thom polynomial Tp(η)(E).

In [3] we explain that Tp(η)(E) is equal to the first obstruction class (in the homotopy theoretic sense) of the fibre bundle P ×_{G} (V \ η̄). By naturality it is enough to look at the universal V -bundle BV = EG ×_{G} V (where B refers to the Borel construction), and the value of the Thom polynomial here: let Tp(η) denote Tp(η)(BV) ∈ H^{∗}(BG) = H_{G}^{∗}(pt). This characteristic class Tp(η) can be thought as the G-equivariant Poincaré dual of η̄ (see [3]).

One of the early examples is the case what Stiefel, Whitney and Steenrod studied (see e.g. [13]) to define Chern classes:

Example 2.1. Let V = Hom(ℂ^{n}, ℂ^{n+k}) and let G = GL(n + k) act on V by composition. We define a G-invariant subset of V:

Σ_{1}(k) := {v ∈ V : dim_{ℂ}(ker v) = 1}.

Then Tp(Σ_{1}(k)) is the first obstruction of the bundle with fibre V \ Σ̄_{1}(k). This set can be identified with the set of n-frames in ℂ^{n+k} i.e. the complex Stiefel manifold St(n, n + k).

So given a vector bundle E = P ×_{GL(n+k)} ℂ^{n+k}, the cohomology class Tp(Σ_{1}(k)) is the first obstruction for having n linearly independent sections of E. In other words if we have n sections s_{1}, …, s_{n} in generic position then Tp(Σ_{1}(k)) is the cohomology class represented by the subset of the base where s_{1}, …, s_{n} are not linearly independent (notice that s_{1}, …, s_{n} defines a section of Hom(ℂ^{n}, E) = P ×_{GL(n+k)} V).

Theorem 2.2 (Stiefel, Whitney and Steenrod). Tp(Σ_{1}(k)) = c_{k+1}.

3. Calculation of Thom polynomials via the method of restriction equations

The G-equivariant Poincaré dual has similar properties as the ordinary Poincaré dual. Since Tp(η) is the G-equivariant Poincaré dual of η̄ we get:

4 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

Proposition 3.1 ([3]). Suppose that θ and η are orbits of ρ and j_{θ}^{∗} : H_{G}^{∗}(V) → H_{G}^{∗}(θ) is induced by the inclusion θ ⊂ V.

Then

j_{θ}^{∗} Tp(η) = \begin{cases}e(ν_{η}) & if θ = η \\ 0 & if η ⊄ θ̄, \end{cases}

where ν_{η} is the normal bundle of η in V and e denotes the equivariant Euler class.

Remark 3.2. ν_{η} is a G-equivariant bundle so it has an equivariant Euler class in H_{G}^{∗}(η). A simple calculation shows (see e.g. [3]) that H_{G}^{∗}(η) ≅ H_{G_{η}}^{∗}(pt) and e(ν_{η}) is equal to the Euler class of the representation of G_{η} on a normal space of η. The Euler class of a representation—using the description of H_{G_{η}}^{∗}(pt) as symmetric polynomials—is simply the product of all weights.

In certain cases Tp(η) is the unique solution of these equations. In fact even less equations are enough to determine Tp(η):

Theorem 3.3 ([3]). Let ρ : G → GL(V) be a linear representation on a complex vector space V with finitely many orbits. Suppose that for every orbit η we have e(ν_{η}) ≠ 0. Then the restriction equations

j_{θ}^{∗} Tp(η) = \begin{cases}e(ν_{η}) & if θ = η  ‘principal equation’ \\ 0 & if θ ≠ η, codim θ ≤ codim η  ‘homogeneous equations’ \end{cases}

have a unique solution.

4. Schubert polynomials

Let the group G := _{}(n) ×_{}(n)—where _{}(n) = {upper triangular n by n matrices} and (n) = {lower triangular n by n matrices}—act on the vector space V := Hom(ℂ^{n}, ℂ^{n}) by (Y, X) · M := XMY^{−1}. The orbits of this action are in a one-to-one correspondence with the rook-arrangements on an n by n chessboard (c.f. Bruhat-form of matrices). A rook-arrangement is a number of rooks placed on a chessboard with the property that no two are in one row or column. Associated to such a rook-arrangement we consider the n × n matrix where 1’s stand for rooks and 0’s otherwise. We will encode such a rook-arrangement by a permutation π ∈ S_{m} for some m ≥ n as follows (and as is in [6, p.9.]). Expand the matrix (to the right and down)

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 5

to an m × m matrix and add 1’s to rows which do not have any, starting from the top down, putting a 1 in the left-most column outside the n × n matrix that does not yet have a 1 in it. Then set π(i) = j if in the i^{th} row the 1 is in the j^{th} column. For example for the rookarrangement ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) the permutation π ∈ S_{3} is 2, 3, 1 or π = 2, 3, 1, 4, 5, 6 in S_{6}. The ambiguity of which S_{m} is considered π in will not effect our reasoning or formulas.

Let M_{π} denote the n×n rook-arrangement matrix encoded by the permutation π ∈ S_{m≥n} and let Σ_{π} be its orbit. Easy computation (matrix multiplication) shows that the tangent space to Σ_{π} at M_{π} can be obtained as follows. Let T_{π} be the set of boxes in the matrix which are either directly below or right of any 1 in M_{π} (including where the 1’s are). The set of the remaining boxes will be denoted by N_{π}. (Observe that N_{π} does not depend on n only on π ∈ S_{m}.) Then the mentioned tangent space is ℂ^{T_{π}}, so for a normal slice to its orbit at M_{π} we can take ℂ^{N_{π}}. This also gives us the codimension of the orbit Σ_{π}:

codim Σ_{π} = |N_{π}| = |{(i, j)|there are 0’s directly above and to the left of (i, j) in M_{π}}| =

= |{(i, j)|π(i) > j, π(l) ≠ j for all l = 1, 2, …, i − 1}| =
= |{(i, j)|π(i) > j, π^{−1}(j) > i}| = l(π),

where the length l(π) of π is the number of inversions in π, i.e. the number of i < j such that π(i) > π(j).

According to the general theory now we need to determine the maximal compact symmetry group of M_{π} together with its representation on an invariant normal slice at M_{π} to Σ_{π}. This is also an easy computation. For the sake of simplicity in our formulas, instead of considering the actual maximal compact symmetry group we will map a group onto the maximal compact symmetry group. Here diag(a, b, …) means the diagonal matrix with a, b, … in the diagonal.

Proposition 4.1. The homomorphism

G_{π} := {diag(α_{1}, α_{2}, …, α_{m}) : |α_{i}| = 1} → {( diag(α_{1}, α_{2}, …, α_{n}), diag(α_{π(1)}, α_{π(2)}, …, α_{π(n)}))}

maps G_{π} ≅ U(1)^{m} onto the maximal compact symmetry group of M_{π}.

Fortunately enough the normal slice ℂ^{N_{π}} is invariant under the action of G_{π}, so now we have the two ingredients of the restriction equations:

6 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

• the map \xymatrix{H^{∗}(BG) \ar[d]^{≅} \ar[r]^{j_{π}^{∗}} & H^{∗}(BG_{π}) \ar[d]^{≅} \\ ℤ[y_{1}, …, y_{n}, x_{1}, …, x_{n}] & ℤ[a_{1}, …, a_{m}]}

is given by y_{i} ↦ a_{i}, x_{i} ↦ a_{π(i)}.

• the Euler class of the representation of G_{π} on ℂ^{N_{π}} is

e_{π} = ∏_{(i,j)∈N_{π}} (j_{π}^{∗}(x_{i}) − j_{π}^{∗}(y_{j})) = ∏_{(i,j)∈N_{π}}(a_{π(i)} − a_{j}). (1)

Observe that this latter Euler class is never zero (since (i, j) ∈ N_{π} implies π(i) > j), so according to Theorem 3.3 the restriction equations are uniquely solvable, giving the Thom polynomial Tp(π) := Tp(Σ_{π}) associated to the orbit Σ_{π}. If we consider a permutation π ∈ S_{n} in a larger permutation group S_{m} (where the permutation acts by identity on the extra m − n elements) then the equations do not change, so Tp does not change either. That is, the map Tp is really defined on S_{∞} := ∪S_{i} (here S_{i} ⊂ S_{j} is meant if i < j).

These polynomials have already been studied in geometry and named double Schubert polynomials. So our approach can be regarded as a new way to compute double Schubert polynomials as solutions of (large) linear equation systems.

In enumerative geometry the standard procedure is to compute double Schubert polynomials by recursion. In the rest of this section we will show that these recursion formulas occur naturally in our approach.

Theorem 4.2. The recursion formulas of Lascoux and Schützenberger ([11]) hold for Tp(π)’s, i.e.

1. Tp(n, n − 1, …, 2, 1) = ∏_{i+j<n}(x_{i} − y_{j});

2. if for some i the inequality π(i) > π(i+1) holds and ρ = π ·s_{i}, where s_{i} is the transposition

(i, i + 1), then

Tp(ρ) = ∂_{i} Tp(π) := \frac{Tp(π) − s_{i}(Tp(π))}{x_{i} − x_{i+1}},

where s_{i}(Tp(π)) is obtained from Tp(π) by interchanging x_{i} and x_{i+1}.

Proof. Let π_{n} be the permutation n, n − 1, …, 2, 1, and let E_{i,j} be the matrix which is 1 at the (i, j) position and 0 otherwise. Matrix multiplication shows that any matrix in the orbit

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 7

of M_{π_{n}} has zeros in the (i, j) entries if i + j < n. It means that for such (i, j)’s E_{i,j} does not belong to the closure of Σ_{π_{n}}, so the homogeneous equation j_{orbit of E_{i,j}}^{∗}(Tp(π_{n})) = 0 holds. This j^{∗} assigns different indeterminates to x_{k}’s and y_{l}’s (k, l = 1, …, n) except x_{i} and y_{j} are both mapped to a_{j}. This means that Tp(π_{n}) must be divisible by (x_{i} − y_{j}), and so in effect it must be a constant times ∏_{i+j<n}(x_{i}−y_{j}). The constant is calculated to be 1 using the principal equation j_{π_{n}}^{∗}(Tp(π_{n})) = ∏_{i+j<n}(a_{n+1−i} − a_{j}).

Now we turn to the proof of (2). For this let us study the matrices M_{π} and M_{ρ}. On the figure below we show the parts in which they differ:

π(i + 1) ⋯ π(i)  ρ(i) ⋯ ρ(i + 1)

\begin{matrix}i \\ i + 1 \end{matrix}  1 1  \begin{matrix}i \\ i + 1 \end{matrix}  1 1

M_{π}  M_{ρ}

Here the boxes correspond to N_{π} and N_{ρ} respectively. This shows that codim(Σ_{ρ}) is one less than codim(Σ_{π}). Also we can keep track of how the corresponding Euler classes change: e_{π} = e_{ρ} · (a_{π(i)} − a_{π(i+1)}) (c.f. formula (1)).

To prove the statement we need to prove the principal and the homogeneous equations (corresponding to ρ) for ∂_{i} Tp(π). The principal one is easily verified as follows

j_{ρ}^{∗}(∂_{i} Tp(π)) = \frac{j_{ρ}^{∗}(Tp(π)) − j_{ρ}^{∗}(s_{i} Tp(π))}{a_{ρ(i)} − a_{ρ(i+1)}} = \frac{0 − j_{π}^{∗}(Tp(π))}{a_{π(i+1)} − a_{π(i)}} = \frac{−e(π)}{a_{π(i+1)} − a_{π(i)}} = e(ρ).

For the homogeneous ones let τ ≠ ρ be a permutation with length not greater than that of ρ. Then we have that codim(Σ_{τ}) ≤ codim(Σ_{ρ}) = codim(Σ_{π})−1, so codim(Σ_{τs_{i}}) = codim(Σ_{τ})−1 ≤ codim(Σ_{π}) and τs_{i} ≠ π. Then

j_{τ}^{∗}(∂_{i} Tp(π)) = \frac{j_{τ}^{∗}(Tp(π)) − j_{τ}^{∗}(s_{i} Tp(π))}{a_{ρ(i)} − a_{ρ(i+1)}} = \frac{0 − j_{τs_{i}}^{∗}(Tp(π))}{a_{ρ(i)} − a_{ρ(i+1)}} = \frac{0 − 0}{a_{ρ(i)} − a_{ρ(i+1)}} = 0.

This proves the theorem.

8 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

Remark 4.3. There were some awkward points in our reasonings—e.g. a matrix of size n × n was encoded by a permutation in S_{m} greater than n (and m was not even fixed) or that we did not describe the maximal compact symmetry group of M_{π} explicitly but as an image of a homomorphism. These could have been avoided by considering the ‘infinite’ case from the very beginning: infinitely big triangular matrices act on the space of infinitely big matrices. Then the orbits correspond exactly to elements in S_{∞}. And the described homomorphism to the symmetry group is an isomorphism. However doing that we would face other technical obstacles, such as: the described symmetry group would not be compact and there would be infinitely many equations for infinitely many indeterminates, etc. Although these problems can be overcome they would make the reasoning even more technical.

Remark 4.4. To obtain the orbits Σ_{π} we could have started with a smaller group action. If we act (e.g.) from the left by unipotent triangular matrices—i.e. ones having 1 in the diagonal— then the orbits are the same, and we expect the ‘single’ Schubert polynomials as solutions. However carrying out the calculation one finds that in this case the Euler classes are 0, so our method does not work: the solutions of the restriction equations are not unique. This phenomenon, i.e. ‘the addition of extra symmetries makes the method work’ was noticed by M. Kazarian too in [8].

5. Schur polynomials

Let d ≤ n, and let the group G := GL(d) ×_{}(n) act on the vector space V := Hom(ℂ^{d}, ℂ^{n}) by (Y, X) · M := XMY^{−1}. The action of GL(d) in the source geometrically means that two linear maps are in one orbit if their image is the same. The action of the triangular matrices (= the automorphism group of a complete flag) geometrically means that two linear subspaces are equivalent if they have the same intersection structure with the complete flag. So the orbits of this action are exactly the so called Schubert cells of the Grassmannian Gr_{r}(ℂ^{n}) for r ≤ d. Encoding a Schubert cell by a partition as usual we have that orbits are in a 1-to-1 correspondence with the set

{(r, λ′) | r = 0, …, d; λ′ ⊂ (n − r)^{r} a partition}

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 9

It will be convenient to index the orbits not with such a pair (r, λ′) but with a single partition, so we embed the above set in

{λ ⊂ n^{d}}

by attaching a d − r by n − r rectangle above λ′, i.e.

λ_{i}(r, λ′) = \begin{cases}n − r & if i ≤ d − r \\ λ′_{i−(d−r)} & if i > d − r. \end{cases}

We will also associate a ‘strict partition’ μ to λ by μ_{i} := λ_{d+1−i} + i. Then

1 ≤ μ_{1} < μ_{2} < … < μ_{d} ≤ n + d.

Let the orbit corresponding to λ be called Σ_{λ}. A representative of this is the n by d matrix M_{λ} having 1 at (μ_{i}, i) (if μ_{i} ≤ n) and 0 everywhere else. We should not forget about the (μ_{i}, i) entries when this does not fit into the n by d matrix. We imagine them outside M_{λ}. Observe that they form a d − r by d − r identity matrix below M_{λ} as in Figure 1.

r  d − r

μ_{1} 1
⋮
μ_{2}  1
⋮  ...
μ_{i}  1
⋮

1

...

1

Figure 1. The matrix M_{λ}

10 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

Computing the image of the Lie algebra of G in the tangent space of V at M_{λ} (i.e. carrying out matrix multiplications) we find that the tangent space of Σ_{λ} at M_{λ} is ℂ^{T_{λ}} where T_{λ} is the set of boxes which are directly below or in the row of an entry 1 in M_{λ}. Denoting the set of the remaining boxes by N_{λ} it is clear that ℂ^{N_{λ}} is a normal slice to Σ_{λ} at M_{λ}. As a byproduct we have that

codim Σ_{λ} = |N_{λ}| = ∑ |N_{λ} ∩ i^{th} column| = λ_{d} + … + λ_{2} + λ_{1} = |λ|.

Now we need to determine the maximal compact symmetry group of M_{λ} together with its representation on an invariant normal slice at M_{λ} to Σ_{λ}. Easy computation (e.g. using the symmetry group calculations for GL(n) × GL(n) as in [3] and restricting to G) shows that

G_{λ} := {( diag(α_{μ_{1}}, α_{μ_{2}}, …, α_{μ_{r}}) ⊕ A, diag(α_{1}, α_{2}, …, α_{n})) : |α_{i}| = 1, A ∈ U(d − r)}
≅ U(1)^{n} × U(d − r)

is the maximal compact symmetry group. It keeps ℂ^{N_{λ}} invariant and its maximal torus acts on ℂ^{N_{λ}} with the representation

⨁_{(i,j)∈N_{λ}} ε_{i} ⊗ \overline{ε_{μ_{j}}},

where ε_{k} is the standard 1-dimensional representation of the k^{th} U(1) and ε_{μ_{j}} for j > r correspond to the maximal torus of the U(d − r) part. From these we can compute

• the map \xymatrix{H^{∗}(BG) \ar[d]^{≅} \ar[r]^{j_{λ}^{∗}} & H^{∗}(BG_{λ}) \ar[d]^{≅} \\ ℤ[x_{1}, …, x_{n}, y_{1}, …, y_{d}]^{S_{d}} & ℤ[a_{1}, …, a_{n}, a_{n+1}, …, a_{n+d−r}]^{S_{d−r}}}

is given by x_{i} ↦ a_{i}, y_{i} ↦ a_{μ_{i}}.

• the Euler class of the representation of G_{λ} on ℂ^{N_{λ}} is

e_{λ} = ∏_{(i,j)∈N_{λ}}(a_{i} − a_{μ_{j}}).

Observe that this latter is never zero, so according to Theorem 3.3 the restriction equations are uniquely solvable, giving the Thom polynomial Tp(λ) := Tp(Σ_{λ}) associated to the orbit Σ_{λ}. Variants of these polynomials have been extensively studied in geometry, and by analogy

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 11

with them let us call Tp(λ) the “double Schur polynomial” associated to λ. Our approach (the solution of the restriction equations) is thus a way of computing double Schur polynomials.

In the rest of this section we show a closed form for Tp(λ).

Theorem 5.1.

Tp(λ) = Δ_{λ}(c^{(d)}, c^{(d−1)}, …, c^{(1)}),

where c^{(j)} = c_{0}^{(j)} + c_{1}^{(j)}t + c_{2}^{(j)}t^{2} + … is the Taylor expansion of

\frac{∏_{i<μ_{j}}(1 + x_{i}t)}{∏_{i=1}^{d}(1 + y_{i}t)},

and Δ_{λ} is the determinant of (c_{λ_{i}+j−i}^{(d+1−i)})_{d×d}.

Proof. First we show that the principal equation holds, i.e. the j_{λ}^{∗} image of Δ_{λ}(c^{(d)}, c^{(d−1)}, …, c^{(1)}) is equal to the Euler class e(λ). We have

j_{λ}^{∗}(Δ_{λ}(c^{(d)}, c^{(d−1)}, …, c^{(1)})) = Δ_{λ}(\frac{∏_{N_{d}}(1 + a_{j}t)}{(1 + a_{μ_{d}}t)}, \frac{∏_{N_{d−1}}(1 + a_{j}t)}{(1 + a_{μ_{d}}t)(1 + a_{μd−1}t)}, …, \frac{∏_{N_{1}}(1 + a_{j}t)}{∏_{j=1}^{d}(1 + a_{μ_{j}}t)}),

where N_{i} = {j < μ_{i}, j ≠ μ_{l}∀l}. (Let us remark that N_{λ} = ⋃ N_{i} × {i}.) In this latter determinant carry out the following column transformations: add a_{μ_{d}} times the j − 1^{st} column to the j^{th} one for j = d, d − 1, …, 2 (in this order). Then—keeping in mind that

\frac{∏_{N_{i}}(1 + a_{j}t)}{∏_{j=i}^{d}(1 + a_{μ_{j}}t)}|_{v} + a_{μ_{d}} \frac{∏_{N_{i}}(1 + a_{j}t)}{∏_{j=i}^{d}(1 + a_{μ_{j}}t)}|_{v−1} = \frac{(1 + a_{μ_{d}}t) ∏_{N_{i}}(1 + a_{j}t)}{∏_{j=i}^{d}(1 + a_{μ_{j}}t)}|_{v} = \frac{∏_{N_{i}}(1 + a_{j}t)}{∏_{j=i}^{d−1}(1 + a_{μ_{j}}t)}|_{v},

we get the matrix

(\begin{matrix}\frac{∏_{N_{d}}(1+a_{j}t)}{(1+a_{μ_{d}}t)} |_{λ_{1}} & 0 \\ ∗ \end{matrix}) ,

where the bottom right d − 1 by d − 1 matrix is exactly the one whose determinant is

Δ_{λ\{λ_{1}}}(c^{(d−1)}, …, c^{(1)}).

The top left entry is exactly the resultant of the enumerator and the denominator, so it is ∏_{N_{d}}(a_{j} − a_{μ_{d}}). So either going on with similar column transformations or by induction we get

∏_{i=1}^{d} ∏_{j∈N_{i}}(a_{j} − a_{μ_{i}}),

which is e(λ), what we wanted to prove.

12 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

Now we prove the homogeneous equations, that is if codim Σ_{λ} = |λ| ≥ codim Σ_{η′} = |λ′|, λ ≠ λ′ then applying j_{λ′}^{∗} to the formula in the theorem yields 0. Let the strict partition assigned to λ and λ′ be μ and μ′ respectively. We will use the following index sets

T_{0} := {μ′_{l}|μ′_{l} < μ_{1}}  S_{0} := {j < μ_{1}} \ T_{0}
T_{1} := {μ′_{l}|μ_{1} ≤ μ′_{l} < μ_{2}} S_{1} := {μ_{1} ≤ j < μ_{2}} \ T_{1}
⋮ ⋮
T_{d} := {μ′_{l}|μ_{d} ≤ μ′_{l}}  S_{d} := {μ_{d} ≤ j} \ T_{d}

Applying j_{λ′}^{∗} to the formula in the theorem we get

Δ_{λ}(\frac{S_{0}T_{0}S_{1}T_{1} ⋯ S_{d−1}T_{d−1}}{T_{0}T_{1} ⋯ T_{d}} , \frac{S_{0}T_{0}S_{1}T_{1} ⋯ S_{d−2}T_{d−2}}{T_{0}T_{1} ⋯ T_{d}} , …, \frac{S_{0}T_{0}}{T_{0}T_{1} ⋯ T_{d}} ) =

Δ_{λ}(\frac{S_{0}S_{1} ⋯ S_{d−1}}{T_{d}} , \frac{S_{0}S_{1} ⋯ S_{d−2}}{T_{d−1}T_{d}} , …, \frac{S_{0}}{T_{1}T_{2} ⋯ T_{d}} ),

where for simplicity instead of ∏_{j∈J}(1 + a_{j}t) we only wrote J in the formulas.

Now we are going to apply column operations as earlier: adding a_{i} times the v−1^{st} column to the v^{th} one. First do it with the greatest index element of T_{d} and v = d, d−1, …, 2. Then with the next a_{i} in T_{d} and v = n, n−1, …, 3, and so on. Keeping our eyes on how the entries change (see in the first part of the proof) wee see that in the first row there will be no denominators after the |T_{d}|^{th} element. The degree of the enumerator will be ∑_{i=0}^{d−1} |S_{i}| = λ_{1} − 1 + |T_{d}|. After the |T_{d}|^{th} element we have to take the (at least) λ_{1} + |T_{d}|^{th} coefficient. So, we see that after the |T_{d}|^{th} element we actually get 0.

Now forget the first |T_{d}| column, and consider only the rest. Here the a_{i}’s from T_{d} already disappeared. Now do the same column operations as before, but now with the elements of |T_{d−1}|. As above we get that after the |T_{d}| + |T_{d−1}|^{th} element (in the original matrix) in the second row we get polynomials with degree ∑_{i=0}^{d−2} |S_{i}| = λ_{2} − 2 + |T_{d}| + |T_{d−1}|. After the |T_{d}| + |T_{d−1}|^{th} element we have to take at least the λ_{2} − 1 + |T_{d}| + |T_{d−1}|^{th} coefficients, which are thus all zero.

Carrying out this procedure with all rows, we arrive at a matrix which is zero in the i^{th} row after the ∑_{j=d−i+1}^{d} |T_{j}|^{th} element. In other words, a (reflected) Young diagram is covered from a matrix (i.e. 0’s stand there), and we would like to conclude that the determinant of the matrix is 0. We can certainly conclude so—in the case (and only in the case) when this Young

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 13

diagram covers at least one box of the diagonal. That is, the determinant is 0 unless |T_{d}| > 0 and |T_{d}| + |T_{d−1}| > 1, …. But if these all hold then the definitions of the index sets T_{i} gives that either λ = λ′ or |λ′| > |λ|, which we wanted to prove.

Remark 5.2. The ordinary Schur polynomials can be obtained from our double Schur polynomials by substituting 0 for all x_{i}. This can be seen either from the definition (only unipotent triangular matrices act in the target space) or from the concrete form of these polynomials. Interestingly enough the ordinary Schur polynomials can not be obtained directly by our method (c.f. Remark 4.4) only via the double ones.

6. Cohomology of moduli spaces

The computation of the cohomology ring of various quotients spaces is a hot area in numerous branches of geometry and topology, see e.g. [10]. In this section we show how the theory of Thom polynomials for group actions can be applied to this problem; and we also show how this approach works in the case of the Grassmannians.

6.1. Obstructions as relations. The idea is very simple: Suppose that X is a G-space. Then we have a map j_{X}^{∗} : H_{G}^{∗}(pt) → H_{G}^{∗}(X) induced by the G-equivariant map X → pt. In [3] we called the kernel O_{X} of this map the obstruction ideal of X since the elements of O_{X} are the G-characteristic classes which are obstructions for having a section of an X-bundle associated to a principal G-bundle. If in addition the map j_{X}^{∗} is surjective, then the generators of H_{G}^{∗}(pt) are generators for the ring H_{G}^{∗}(X), too and the elements of O_{X} are the relations for H_{G}^{∗}(X). If the G-action is free, then H_{G}^{∗}(X) ≅ H^{∗}(X/G) i.e. we calculate the ordinary cohomology ring of the space X/G. (In this case X itself is a principal G-bundle and elements of O_{X} are the vanishing G-characteristic classes of X.)

6.2. Using Thom polynomials. If there is a geometric realization of the map X → pt i.e. a G-equivariant embedding i_{X} : X → V into a contractible G-space V —in other words X is an invariant subset of a representation of G—then we can use Thom polynomial methods to generate O_{X}: It is easy to see that Thom polynomials of orbits outside of X are in O_{X}. However we have to extend the definition for the case where continuous families of orbits occur.

6.3. Thom polynomials for moduli.

Definition 6.1. A G-invariant stratification Ξ of V is called a Vassiliev stratification if for every S ∈ Ξ the projection p : S → S/G = M is a fibre bundle over the manifold M.

If M has positive dimension we call it moduli space. Suppose now that p : S → S/G = M is a complex algebraic map and the manifold M admits a complex algebraic cell stratification C = {C_{i}} (i.e. the cells C_{i} are complex submanifolds and their closures are complex algebraic varieties). Since in this case the G-invariant subsets p^{−1}(C̄_{i}) ⊂ V are also complex algebraic varieties they have Thom polynomials ([3]) and we can define a homomorphism:

Definition 6.2. The Thom polynomial map Tp_{M} : H^{∗}(M) → H_{G}^{∗+d}(pt) is defined by

Tp_{M}([C_{i}]) := Tp (p^{−1}(C_{i})),

where [C_{i}] denotes the cohomology class defined by the cell C_{i} in the cellular cohomology of the cell decomposition C.

Remark 6.3. In general M doesn’t have a complex cell decomposition. Then take any cell decomposition C = {C_{i}} which is a stratification of M at the same time. Extend this stratification into a strict Vassiliev stratification Ξ of V (i.e. we additionally assume that for every S ∈ Ξ the manifold S/G is contractible). Then we have a map from the group of cellular cochains C^{∗}(C) to E_{1}^{0,∗+d} of the Kazarian spectral sequence of Ξ (i.e. the cohomology spectral sequence of the induced filtration, see [9]). If this map is a cochain map (see Example 6.5 for a counterexample) then it induces a map H^{∗}(M) → E_{2}^{0,∗+d}. Composing with the edge homomorphism we can define Tp_{M}.

Examples 6.4. In the following cases the Thom polynomial map can be defined:

(i) Let V = Hom(ℂ^{n}, ℂ^{n}) as in Section 4 but restrict the action to the subgroup _{}(n). The orbit of a map φ ∈ V is determined by the image of the standard flag. So a natural stratification of V is V = ⋃ S_{λ} where S_{λ} = {φ ∈ V : dim φ(ℂ^{i}) = λ_{i}}. The moduli space S_{λ}/_{}(n) is a partial flag manifold Fl_{λ}(ℂ^{n}). In particular the moduli space corresponding to the open stratum is the full flag manifold Fl(ℂ^{n}).

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 15

We can see that the orbits of _{}(n) ×_{}(n) correspond to the Schubert cells of Fl_{λ}(ℂ^{n}) so not only they give a complex algebraic cell stratification but we can calculate their Thom polynomials by restricting the _{}(n) ×_{}(n) Thom polynomials to H_{(n)}^{∗}(pt). In other words we substitute y_{i} = 0 into the _{}(n) ×_{}(n) Thom polynomials.

This method can be generalized: Whenever we can enlarge the the symmetry group such that the moduli space becomes the union of finitely many orbits of the larger group we have a chance to calculate Tp_{M}. In an ideal situation the orbit decomposition defines a complex algebraic cell stratification of M.

(ii) Let V = Hom(ℂ^{d}, ℂ^{n}) as in Section 5 but restrict the action to the subgroup GL(d). The orbit of a map φ ∈ V is determined by the image Im(φ). So a natural stratification of V is V = ⋃ Σ_{i} where Σ_{i} = {φ ∈ V : dim ker(φ) = i}. The moduli space Σ_{i}/GL(d) ≅ Gr_{d−i}(ℂ^{n}). Again the Schubert cells give a complex algebraic cell stratification and the Thom polynomial maps can be calculated by restricting the GL(d) × _{}(n) Thom polynomials calculated in Section 4.

(iii) Generalizing case (i) let V = V (Q) be a quiver representation of A_{n}-type (see [4]) and forget the action on the last vertex space ℂ^{d_{n}}. Then the moduli spaces are partial flag manifolds as in (i) but the moduli space corresponding to the open stratum is also a partial flag manifold. Now if we enlarge the symmetry group by letting _{}(d_{n}) act on the last vertex space ℂ^{d_{n}} then we get a situation analogous to case (i). (Notice that if we forget the action on the last vertex space of a representation finite quiver different from A_{n} then the moduli spaces are not compact and don’t admit complex algebraic cell stratifications.)

Example 6.5. This example shows how bad the situation can be even in a very simple case. Let G = GL(1) acting on V = ℂ^{2} by ρ(α) = ( \begin{matrix}α \\ & α^{−1} \end{matrix} ). There are three exceptional orbits: the zero orbit, η_{x} = {(x, y) : x ≠ 0, y = 0} and η_{y} = {(x, y) : y ≠ 0, x = 0}. And there are the orbits η_{c} = {(x, y) : xy = c} for c ≠ 0. So a Vassiliev stratification can be 0, η_{x}, η_{y}, S = ∪η_{c}. The moduli space S/G ≅ ℂ^{×} has a cell stratification C_{0} = ℂ \ [0, ∞), C_{1} = (0, ∞). Though C_{1} is a cocycle in the cellular cochain complex and [C_{1}] generates H^{1}(M) ≅ ℤ but δ_{1}p^{−1}(C_{1}) = η_{x} − η_{y} in the Vassiliev complex (E_{1}^{0,∗}, δ_{1}). So it cannot have a Thom polynomial. (It is easy to construct a transversal section s of the trivial ℂ^{2}-bundle over the two-torus T^{2} such that [\overline{s^{−1}(C_{1})}] is not zero in H^{1}(T^{2}). On the other hand for a trivial bundle any Thom polynomial has to be zero.)

16 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

It might be tempting to ask what is the Thom polynomial of a circle around 0 in M, but it represents 0 ∈ H^{1}(M) so its Thom polynomial has to be zero by trivial reasons.

6.4. Surjectivity. As we mentioned at the beginning of the section, surjectivity of the map j_{X}^{∗} : H_{G}^{∗}(pt) → H_{G}^{∗}(X) is an essential step in calculating H_{G}^{∗}(X). Using the Thom polynomial map we can prove surjectivity for the zero codimensional moduli space in cases of Examples 6.4:

Proposition 6.6. Suppose that for an open and G-invariant X ⊂ V the G-action on X is free and M = X/G admits a complex algebraic cell stratification. Then the map j_{X}^{∗} : H_{G}^{∗}(pt) → H_{G}^{∗}(X) ≅ H^{∗}(M) is surjective.

Proof. E = X ×_{G} V has a tautological section which is transversal since X is an open subset of V (notice that transversality will fail for higher codimensional moduli). It shows that Tp_{M}(α)(E) = j_{X}^{∗}(Tp_{M}(α)) = α, i. e. Tp_{M} is a left inverse to j_{X}^{∗}.

Remark 6.7. The proof of Proposition 6.6 also shows that in the Grassmannian Gr_{n}(ℂ^{n+k}) the Poincaré dual [σ_{λ}] of the Schubert variety σ_{λ} is equal to Δ_{λ}(c_{1}, …, c_{n}) where c_{i} are the Chern classes of the universal ℂ^{n}-bundle over Gr_{n}(ℂ^{n+k}), so we gave an independent proof of the classical Schubert calculus. Similarly the single Schubert polynomials Schubert_{π}(x_{1}, …, x_{n})— when we substitute y_{i} = 0 into the double Schubert polynomials—expresses the Poincaré duals of the Schubert varieties σ_{π} of the flag manifold Fl(ℂ^{n}) in terms of x_{i} = c_{1}(L_{i}) where L_{i} is the i^{th} tautological line bundle over the flag manifold.

6.5. Cohomology of the Grassmannian. According to Section 5 we can calculate the Thom polynomial maps for Example 6.4.(ii):

Tp_{M_{r}}(σ_{λ′}) = Δ_{λ(r,λ′)}|_{{x_{i}=0}}.

As we mentioned, all Thom polynomials coming from the higher codimensional moduli are relations for H^{∗}(M_{0}). In fact, Thom polynomials of M_{1} already generate the ideal of relations O(M_{0}):

Theorem 6.8. For any 0 < d ≤ n

H^{∗}(M_{0}) ≅ Im Tp_{M_{0}} / Im Tp_{M_{1}} .

SCHUR AND SCHUBERT POLYNOMIALS AS THOM POLYNOMIALS—COHOMOLOGY OF MODULI 17

Proof. There are well known descriptions of the ideal I for which H^{∗}(M_{0}) = ℂ[c_{1}, …, c_{d}]/I, see e.g. [6, p.27.]. One such is I = ⊕_{λ⊄(n−d)^{d}}ℤΔ_{λ}(1/c). Another is I = (Δ_{(n−d+i)}(1/c)|i = 1, …, d) (ideal generators). What we have to prove is that I is also equal to J := (Δ_{λ}(1/c)|λ ⊂ (n − d + 1)^{d}, λ_{1} = n − d + 1). According to the first description of I clearly J ⊂ I. To prove the converse observe that Δ_{(n−d+i)}(1/c) = Δ_{1^{n−d+i}}(c) and in the latter determinant carry out the Lagrange expansion corresponding to the first i − 1 columns. In each term we get a polynomial times an (n − d + 1) × (n − d + 1) determinant, and this latter determinant is verified to be of the form Δ_{λ^{T}} (c) = Δ_{λ}(1/c), with λ ⊂ (n − d + 1), λ_{1} = n − d + 1. This proves J ⊃ I, so our theorem. [Remark that an even finer analysis of this proof gives an even more economical ideal generator system for I, namely Δ_{(n−d+1,1^{i})}(1/c), i = 0, …, d − 1.]

Remark 6.9. In [2] there is a different description of the double Schubert polynomials. They use the quiver representation G(Q) = X_{i=1}^{n} GL(i) × X_{i=n}^{1} GL(i) acting on

V (Q) = ⨁_{i=1}^{n−1} Hom(ℂ^{i}, ℂ^{i+1}) ⊕ Hom(ℂ^{n}, ℂ^{n}) ⊕ ⨁_{i=n−1}^{1} Hom(ℂ^{i+1}, ℂ^{i}).

Looking at the map V = Hom(ℂ^{n}, ℂ^{n}) → V (Q) we can see that the the two methods should give the same result.

Remark 6.10. It would be interesting to find analogues of Theorem 6.8 for the flag manifold (Example 6.4 (i)), for partial flags (Example 6.4(iii)), for analogues where GL(n) is replaced by different complex simple groups and for real Grassmannians and flag manifolds.

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[2] A. Buch and W. Fulton. Chern class formulas for quiver varieties. Inv. Math., (135):665–687, 1999.

[3] L. Fehér and R. Rimányi. Computation of Thom polynomials for group actions.

www.cs.elte.hu/analysis/rimanyi/cikkek.

[4] L. Fehér and R. Rimányi. Thom polynomials for quivers. www.cs.elte.hu/analysis/rimanyi/cikkek.

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[6] W. Fulton and P. Pragacz. Schubert Varieties and Degeneracy Loci. Springer, 1989.

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18 LÁSZLÓ FEHÉR AND RICHÁRD RIMÁNYI

[8] M. Kazarian. Thom polynomials for Lagrange, Legendre and critical point singularities. Isaac Newton Institute for Math. Sci. preprint, 2000.

[9] M. É. Kazarian. Characteristic classes of singularity theory. In V. I. Arnold et al., editors, The Arnold-Gelfand mathematical seminars: geometry and singularity theory, pages 325–340, 1997.

[10] F. Kirwan. Cohomology of quotients in symplectic and algebraic geometry. Number 31 in Mathematical Notes. Princeton University Press, 1984.

[11] Lascoux and Schützenberger. Polynômes de Schubert. C. R. Acad. Sci. Paris, (294):447–450, 1982.

[12] R. Rimányi. Thom polynomials, symmetries and incidences of singularities. Inv. Math., (143):499–521, 2001.

[13] E. Stiefel. Richtungsfelder und Fernparallelismus in Mannigfaltigkeiten. Comm. Math. Helv., (8):3–51, 1936.

[14] A. Szűcs. Analogue of the Thom space for mapping with singularity of type Σ^{1}. Math. Sb. (N. S.), 108 (150)(3):438–456, 1979.

[15] V. A. Vassiliev. Lagrange and Legendre Characteristic Classes. Gordon and Breach, 1988.

Rényi Institute, Reáltanoda u. 13-15, 1053, Hungary

E-mail address: lfeher@math-inst.hu

Department of Analysis, ELTE TTK, Múzeum krt. 6-8., Budapest 1088, Hungary

E-mail address: rimanyi@cs.elte.hu