A module isomorphism between and
We give an explicit (new) morphism of modules between and and prove (the known result) that the two modules are isomorphic. Our map identifies submodules of the cohomology of the flag variety that are isomorphic to each of and . With this identification, the map is simply the product within the ring . We use this map in two ways. First we describe module bases for that are different from traditional Schubert classes and from each other. Second we analyze a -representation on via restriction to subgroups . In particular we show that the character of the Springer representation on is a multiple of the restricted representation of on .
In this paper we construct a large family of distinct bases for the equivariant cohomology of the generalized flag variety. To do this we give an explicit formula for the Leray-Hirsch isomorphism for the fibration of flag varieties . The Leray-Hirsch theorem says
but does not provide an explicit map. In fact the procedure in the Leray-Hirsch theorem sends classes through series of quotients, isomorphisms, and identifications in a spectral sequence, so explicitly writing the output is challenging. Indeed, a prori the output of the Leray-Hirsch map is a basis for the cohomology of the total space as a module over the cohomology of the base space. We bypass these subtleties by choosing explicit bases of Schubert classes for each cohomology module as a submodule of and proving the isomorphism directly.
Moreover we solve this problem in equivariant rather than ordinary cohomology. The equivariant cohomology of a variety is an enhanced version of the ordinary cohomology ring that records information about an underlying group action on the variety. Certain computational tools can make equivariant cohomology easier to construct than ordinary cohomology, as well as permitting us to recover ordinary cohomology. (Knutson and Tao’s work computing the structure constants of the equivariant and ordinary cohomology ring of is one example of this principle [Knutson-Tao].) These computational tools can be used in many important cases, including generalized flag varieties and partial flag varieties both with the left-multiplication action of the torus .
We consider a presentation of the equivariant cohomology due to Kostant and Kumar [KostantKumar] and use it to construct a module isomorphism between the tensor product and all treated as modules over . The map naturally descends to a module isomorphism on the ordinary cohomology. The fact that these modules are isomorphic is not new [Dou04, Theorem 2.1]. But in Schubert calculus people want very explicit answers—even down to specific numbers and elementary combinatorial formulas. The main result of this paper is a pleasingly elementary identification of classes inside that realize Leray-Hirsch and provide a useful tool for fields like Schubert calculus (see Theorem LABEL:thm:_distinct_bases).
To construct our map, we identify each factor in the tensor product with a submodule of . The bilinear module isomorphism is multiplication of classes inside the ring . More precisely our main theorem states:
Identify and isomorphically with submodules of as described in Section 2.1. Then the multiplication map
induces a bilinear isomorphism of modules
Guillemin-Sabatini-Zara realize this isomorphism differently for a class of varieties called GKM spaces, proving a combinatorial version of the Leray-Hirsch theorem for fibrations of graphs associated to GKM spaces [GKMfiberbundles]. As an application, they show their results apply to the map in classical types [GKMfiberbundles, Section 5] and apply it to a number of specific Grassmannians and symplectic varieties [Balancedfiberbundles]. Establishing their hypotheses for in classical types takes longer than our direct proof for all types. Indeed in their calculations they use an interesting basis whose elements are stable under an action of the Weyl group. In a colloquial sense Guillemin-Sabatini-Zara’s basis is the “opposite” of our flow-up classes, which are closer to the classes constructed from Morse flows or Bialynicki-Birula decompositions.
We give two applications of our result in Section 4. First our explicit module isomorphism gives rise to a large collection of module bases of indexed by the parabolic subgroups . An immediate corollary of the isomorphism is that if is any module basis for and is any module basis for then the products form a module basis for .
Fix two distinct parabolic subgroups with . Let denote the product basis of obtained from the equivariant Schubert bases for and , and similarly for . Then the bases and are distinct.
Theorem LABEL:thm:_distinct_bases proves this result for connected and Corollary LABEL:cor:_disconnected_G generalizes the theorem to disconnected . One way to state the core problem of Schubert calculus is: analyze combinatorially and explicitly the cohomology ring of a generalized flag variety in terms of the basis of Schubert classes. Thus these parabolic bases allow us to optimize the choice of basis to make particular computations in Schubert calculus as simple as possible.
As another application we show how these bases can be used to analyze a well-known action of on called the Springer representation. In particular Theorem LABEL:thm:_kostant_kumar_character says that the character of the restricted action of on is the scalar multiple where is the character of the –representation on .
In Section 3 we prove Theorem 1.1 in the equivariant setting. Our proofs use what many call GKM theory, after Goresky-Kottwitz-MacPherson’s algebraic–combinatorial description of equivariant cohomology rings [GKMtheory]. The GKM presentation comes with an explicit formula for the Schubert classes that is due to Billey [Billey, Theorem 4] and Anderson-Jantzen-Soergel [A-J-S, Remark p. 298]. These tools permit an elegant combinatorial and linear-algebraic proof of Theorem 1.1. The result then descends to ordinary cohomology (see Corollary 3.7).
We are grateful to J. Matthew Douglass for showing us his work on both equivariant and ordinary cohomology isomorphism and inspiring this proof; to Alexander Yong for useful discussions; and to the anonymous referee for very helpful comments.
We denote by a complex reductive linear algebraic group and fix a Borel subgroup . We denote the maximal torus in by and the Weyl group associated to by . Let be any parabolic subgroup containing .
Let denote the subgroup of associated to . This is also a Weyl group, specifically the Weyl group of . For elements the length refers to the minimal number of simple reflections required to write as a word in the generators of . Let denote the subset of minimal-length coset representatives of . The following fact is so essential to our work that we state it explicitly here; many texts give proofs, including Björner-Brenti [Bjorner-Brenti, Lemma 2.4.3].
Every minimal-length word for each element ends in a simple reflection .
2.1. Restricting to fixed points
We use a presentation of torus-equivariant cohomology that is often referred to as GKM theory, after Goresky, Kottwitz, and MacPherson [GKMpaper], though key ideas are due to many others [AtiyahBott, Kirwan, Chang-Skjelbred] (see [GKMpaper, Section 1.7] for a fuller history). For suitable spaces the inclusion map of fixed points induces an injection on cohomology . Straightforward algebraic conditions determine the image of the injection explicitly [Chang-Skjelbred, Lemma 2.3], though we do not use them in this manuscript.
Through this map we think of equivariant classes as collections of polynomials in . We use functional notation to describe the elements meaning that for each we have .
GKM theory applies to varieties like , , and that have only even-dimensional ordinary cohomology [GKMpaper, Theorem 14.1(1)]. In fact each of , , and is a CW-complex whose cells are Schubert cells indexed by the elements of , , and respectively. The fixed point sets of , , and are also naturally isomorphic to , , and .
As modules over the equivariant cohomology of , , and each have a basis of (equivariant) Schubert classes that are again indexed by the elements of , , and respectively. The restrictions of each Schubert class to each fixed point are given explicitly by what we call Billey’s formula (see Section 2.2). The formula is the same in all three cases , , and . Thus the map that sends the Schubert class to the corresponding Schubert class is a module isomorphism onto its image, and similarly for . We identify the images of and in with the modules and themselves, so
For this inclusion is only a homomorphism of modules and not a homomorphism of rings.
The map that we consider is the ordinary product of classes inside .
2.2. Billey’s formula
This section describes an explicit combinatorial formula for evaluating the polynomial in . Anderson, Jantzen, and Soergel originally discovered this formula [A-J-S]; Billey independently found it as well [Billey, Theorem 4]. While proven originally for it also holds for [TymoczkoG/P, Theorem 7.1] and [Kum02, Corollary 11.3.14]. Fix a reduced word for and define Then
The polynomial has the following properties:
The polynomial does not depend on the choice of reduced word for [Billey, Theorem 4].
The polynomial is homogeneous of degree [Billey, Corollary 5.2].
The polynomial if and only if [KostantKumar, Proposition 4.24] .
For any we have .
Let have Weyl group and let and . The word is found as a subword of in the two places and .
3. Main Theorem
This section proves the main theorem of the paper. First in the equivariant setting, and then for ordinary cohomology, we prove that the module map from to induced by is a bilinear isomorphism of modules. We show this first in the equivariant case by proving that it takes a module basis for to a module basis for . The non-equivariant case follows from the equivariant case.
The set of Schubert class products is a linearly independent set over .
In this section we will prove Theorem 3.1 by arranging these products in the matrix
Our notational convention is to index rows by pairs and columns by pairs also in .
We begin by establishing an order on . The elements of both and are partially ordered by length; fix a total order on (respectively ) consistent with this partial order and extend this lexicographically to all of . For instance all rows and columns corresponding to pairs in come before any pair in .
For the remainder of this section we will consider the matrix to have rows and columns ordered as above. The proof of Theorem 3.1 is given in Section 3.2.
3.1. Key lemmas
We begin with two lemmas. The first will prove that given the above ordering of its rows and columns, the matrix is block upper-triangular. The second lemma will construct a matrix where is an invertible matrix and is known to have linearly independent rows and columns. The main theorem will then show how and are related.
The matrix is block upper-triangular.
Choose . Consider the entries of whose rows are indexed by pairs in and whose columns are indexed by pairs in . By construction this is a square block. Its entries are where range over all of . As established in Proposition 2.1, the last letter in every reduced word for is a simple reflection . Thus every reduced word for inside is contained in the prefix . Therefore . By Property 3 of Billey’s formula is non-zero if and only if in the Bruhat order. Therefore whenever and the entire block is zero. ∎
Consider the parabolic subgroup in the type Weyl group . The minimal coset representatives are . Let denote the collection . Then the blocks of the matrix are
This example treats pairs with the same length. Let be the parabolic subgroup . The elements of with length two are and . The restriction of the matrix to the diagonal block where and both have length two has form:
In the next lemma we show that the rows of the diagonal blocks of the matrix are linearly independent. It is not immediately obvious that the matrices in this lemma are in fact the diagonal blocks; that result is part of the main theorem.
Lemma 3.5 (Linear independence of diagonal blocks).
Fix . Assume that the elements of are ordered consistently with the partial order on length. Let be the matrix defined by
where . Define the matrix by for . Consider the algebra isomorphism induced from the action . Denote the image of under this action of by .
Then the rows of the matrix are linearly independent over .
Note that does not permute the rows or columns of . For computational clarity, the isomorphism is equivalent to the map if one passes to .
If then by construction . If then unless . Therefore is an upper-triangular matrix. The entries on the diagonal have the form . Since is a unit in the matrix is invertible.
We defined to be the matrix of Schubert classes in . The rows of are the Schubert class basis for so the rows and columns of the matrix are linearly independent. The function acts on by sending each to . This operation is invertible and so preserves linear independence of the matrix rows. Thus the new matrix also has linearly independent rows.
Since is invertible over and has linearly independent rows over the rows of the matrix product are also linearly independent over . ∎
3.2. Proof of Theorem 3.1
We now show that each of the diagonal blocks of identified in Lemma 3.2 is a scalar multiple of the matrix defined by Lemma 3.5. This proves that the rows of matrix are linearly independent and thus the collection of Schubert class products is linearly independent over .
Consider the matrix with rows and columns ordered lexicographically subordinate to the length partial order on and described above.
Partition the matrix into blocks according to the pairs . Lemma 3.2 proved that is block-upper-triangular with this partition. Now consider the blocks along the diagonal, namely the blocks of the form
for each . Proposition 3 of Billey’s formula guarantees that is non-zero so it suffices to consider the matrix . We will show that
where and are the matrices of Lemma 3.5. Multiplying matrices gives
We now show that for any the polynomial can be decomposed as the sum . Consider Billey’s formula for and group terms according to which part of is a subword of and which part is a subword of . More precisely:
The map induces a bilinear isomorphism of modules
For any pair the polynomial degree of the homogeneous class is just like that of . The map given by is a bijection [Bjorner-Brenti] and induces a bijection which preserves polynomial degree. Thus the set contains the correct number of elements of each polynomial degree to be a basis of .
By Theorem 3.1 the set is also linearly independent over . Thus it is a basis for . ∎
The equivariant isomorphism induces a similar isomorphism in ordinary cohomology, essentially by Koszul duality. We confirm this below, re-deriving the Leray-Hirsch isomorphism. (Note that not all equivariant results immediately descend to ordinary cohomology; see for instance Theorem LABEL:thm:_distinct_bases.)
The map induces a bilinear isomorphism of modules
Let be the augmentation ideal, namely . Recall that the ordinary cohomology is the quotient when satisfies certain conditions, for instance, if has no odd-dimensional ordinary cohomology [GKMpaper]. Consider the two projections
If and then so the kernels of the two projections agree. It follows that we have an isomorphism
The map is an isomorphism of -modules so it commutes with taking the quotient by the augmentation . Combining these results gives
or in other words . ∎
The parabolic basis is generally not the Schubert basis. In fact we will show that, with the exception of and , each of the bases is distinct not only from the Schubert basis but from any other parabolic basis as well. As with different bases of symmetric functions, this is a useful computational tool. As another application we compute the character of a particular Springer representation.
4.1. The parabolic basis
We begin with an example illustrating that the basis is not the Schubert basis.
We again use the example and . Four of the classes in are also Schubert classes: