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.


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.

  • $\begingroup$ 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 Oct 17 '13 at 0:22
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    $\begingroup$ I would suggest Brion's article Equivariant Cohomology and Equivariant Intersection Theory. $\endgroup$ – Peter Crooks Oct 17 '13 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 Oct 17 '13 at 16:29

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