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I'm having a surprisingly hard time finding references for some facts about $GL_k$-equivariant cohomology of the space of $k\times n$ matrices. Specifically, I believe the following things to be true:

  1. The ring $H^*_{GL_k}(\mathrm{Mat}_{k\times n})$ is isomorphic to the ring $\Lambda_k$ of symmetric polynomials in $k$ variables.
  2. Under this isomorphism, the $GL_k$-equivariant cohomology class of a Schubert variety $\Omega_\lambda$ in $\mathrm{Mat}_{k\times n}$ maps to the Schur polynomial $s_\lambda$.
  3. If we restrict to the subvariety $\mathrm{Mat}_{k\times n}^\circ$ of full-rank $k\times n$ matrices, the corresponding pullback map $$H^*_{GL_k}(\mathrm{Mat}_{k\times n})\to H^*_{GL_k}(\mathrm{Mat}_{k\times n}^\circ)=H^*(Gr(k,n))$$ is the one that kills the Schur polynomials corresponding to partitions $\lambda$ with $\lambda_k>n-k$.

Does anyone know either a reference or a simple argument for these? I think (1) should follow simply from the fact that $\mathrm{Mat}_{k\times n}$ is contractible, and that the isomorphism is even canonical. I also think that (1) and (2) together should imply (3) without too much effort, but I'm not quite comfortable enough with these ideas to fill in all the details.

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The action of $GL_k$ on $Mat_{k\times n}$ is linear. Therefore, the scaling retraction of $Mat_{k\times n}$ to $\{0\}$ is $GL_k$-equivariant. It follows that the restriction map $$H^*_{GL_k}(Mat_{k\times n})\rightarrow H^*_{GL_k}(\{0\})$$ is an isomorphism. That you obtain an isomorphism $H^*_{GL_k}(Mat_{k\times n})\cong\Lambda_k$ then follows from the general fact that $$H_G^*(pt)\cong H_T^*(pt)^W,$$ where $T\subseteq G$ is a maximal torus and $W$ is the Weyl group (and also that $H_T^*(pt)$ is a polynomial ring in $\dim(T)$ indeterminates). I hope this helps answer your question.

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  • $\begingroup$ This is helpful; this seems to be (1) on the nose. Do you know of a reference for the isomorphism $H^*_G(pt)\cong H^*_T(pt)^W$? $\endgroup$
    – Nicolas Ford
    Commented Oct 17, 2013 at 0:22
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    $\begingroup$ I would suggest Brion's article Equivariant Cohomology and Equivariant Intersection Theory. $\endgroup$
    – Peter Crooks
    Commented Oct 17, 2013 at 1:25
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    $\begingroup$ In fact $H_G^*(X) = H_{N(T)}^*(X) = H_T^*(X)^W$ (with rational coefficients) for any space $X$, not just $X=pt$. This comes from Leray-Hirsch applied to the bundle $G/N(T) \to (EG \times X)/N(T) \to (EG \times X)/G$, and the fact that $G/N(T)$ has trivial rational cohomology. For the latter, one shows first that $H^*(G/T)$ is the regular representation of $N(T)/T$, so the invariants are $1$-dimensional. $\endgroup$
    – Allen Knutson
    Commented Oct 17, 2013 at 16:29

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