Let $X=G/B$ , where $G=GL_n(\mathbb{C}^n)$ and $B$ be the upper triangular matrices. I am curoius about the structure of $H^*_T(G/B)$ which I consider as a $H_T^*(pt)$-module. If we just consider ordinary cohmology, we know that the Schubert classes $[X_w]$ form a $\mathbb{Z}$-linear basis , their product is a positive combination of the others and it is described by Schubert polynomial. In Knutson and Tao's paper, they provided a nice puzzle to calculate the equivariant cohomology of $G/P$, the grassmanian variety.

Do we have something similar happening in $H_T^*(G/B)$? To be more precise, do we have the following properites?

  1. $H_T^*(G/B)$ is graded module over $H_T^*(pt)$.
  2. We have a natural basis $[\tilde{X_w}]$ for this $H_T^*$-module.
  3. The forgetful functor from $H_T^*(-)$ to $H^*(-)$ maps $[\tilde{X_w}]$ to $[X_w]$.

If the above listed properties are true, is it possible to create a simlar puzzle as in Knutson and Tao's paper? What are the progress in this area?

  • $\begingroup$ You mean equivariant. $\endgroup$ – Qiaochu Yuan Oct 30 '15 at 3:02
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    $\begingroup$ Oh right. I am so drunk. $\endgroup$ – Ben Oct 30 '15 at 3:03
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    $\begingroup$ I am not specialist in this, but I think similar problems, if not exactly the same, are studies in the context of cohomology of quasi-toric manifolds. You may look at papers by Bahri, Buchstaber, Fred Cohen, Sam Gitler, Nige Ray, and others in this school. A good source is Ray's homepage ma.man.ac.uk/~nige/prepr.html In particular, the following paper also can be interesting journals.cambridge.org/… $\endgroup$ – user51223 Oct 30 '15 at 9:33
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    $\begingroup$ There is a puzzle rule for equivariant cohomology of the two step flag variety (I'm on a cell phone, but the paper is by Buch) and there's a conjecture for ordinary cohomology of the three step (also by Buch, but published in a paper by Vakil). There isn't a rule of any kind known even for ordinary cohomology of the complete flag variety, so who knows, but of course a puzzle rule is conceivable. $\endgroup$ – Matt Samuel Nov 1 '15 at 23:30
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    $\begingroup$ Properties 1-3 hold for any algebraic torus action on a smooth complex projective variety (say). $\endgroup$ – Allen Knutson Nov 3 '15 at 20:44

Points 1-3 are available for equivariant cohomology. As an economical link, this paper on the arXiv discusses three well-known available presentations for $T$-equivariant cohomology of the flag manifold $G/T$ and how to translate between them. It also contains the relevant literature references for all three presentations. (Although the reference is fairly new, the presentations relevant for the question are very classical, going back to work of Chevalley, Borel, Goresky-Kottwitz-MacPherson...)

Edit: as requested, I include some more details on points 1-3 (most of this can be found in the linked paper or the references therein). The presentation most relevant for the question is called Chevalley representation in this paper.

Equivariant cohomology is defined via the Borel construction $ET\times_T G/B$. There is a fibre bundle $G/B\to ET\times_TG/B\to BT$, where $BT$ is the classifying space of the torus $T$; this fibre bundle is associated to the universal $T$-bundle $T\to ET\to BT$ and the standard left action of $T$ on $G/B$. The projection $\pi:ET\times_TG/B\to BT$ induces a graded ring homomorphism $H^\ast_T(\ast)\to H^\ast_T(G/B)$. This shows (1).

For each Schubert variety $X_w$, there is an associated class $[\tilde{X}_w]$ in $H^\ast_T(G/B)$. Then the Chevalley presentation states that $H^\ast_T(G/B)$ is a free $H^\ast_T(\ast)$-module generated by the classes $[\tilde{X}_w]$. This is (2).

For (3), it is possible to recover ordinary cohomology from equivariant cohomology via $$ H^\ast(G/B)\cong H^\ast_T(G/B)\otimes_{H^\ast_T(\ast)}\mathbb{Q} $$ and under this identification, the Schubert classes in equivariant cohomology are mapped to the Schubert classes in ordinary cohomology as required. (Check the discussion of equivariant cohomology of equivariantly formal spaces in the paper of Goresky-Kottwitz-MacPherson.)

  • $\begingroup$ I am just a beginner in equivariant cohomology. Do you know where I can find a reference showing that the three properties hold? Thank you! $\endgroup$ – Ben Oct 30 '15 at 14:34

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