Quantum cohomology of partial flag manifolds - MathOverflow

Quantum cohomology of partial flag manifolds

2010-11-02T12:41:27Z

Is there a place in the literature where the quantum differential equation (or even just quantum cohomology algebra) of partial flag manifolds $G/P$ is computed for arbitrary semi-simple $G$ and arbitrary parabolic $P$? I actually think that I know one way to formulate (and prove) the answer but I was sure that this was well-known and to my surprise I couldn't find the reference for the general case (the case when $P$ is a Borel subgroup is well-known and there is a lot of literature for other parabolics in the case when $G$ is a classical group but again I couldn't find a treatment of the general case). For the quantum cohomology algebra many papers mention a result of Peterson (which I think coincides with what I want when one takes the appropriate limit going from quantum $D$-module to quantum cohomology algebra) which describes it, but I was unable to find a published proof of this result. Is it written anywhere?

Answer by Sheikraisinrollbank for Quantum cohomology of partial flag manifolds

2010-11-02T13:49:34Z

Edited in light of clarification made by OP in comments to his question:

Yes, the result you want is proved by Lam and Shimozono; it is Theorem 10.16 of their paper arXiv:0705.1386. Their theorem (which is followed by a proof) identifies a localization of $QH^T(G/P)$ with a localization of a quotient of the torus equivariant homology of the affine Grassmannian; specialization gives the earlier non-equivariant unpublished result of Peterson.

The Lam/Shimozono result depends on an earlier calculation (in arXiv:math/0501213) by Mihalcea of the equivariant quantum product of a Schubert class by a divisor class; this rule should already suffice to determine the QDE.

arxiv:1007.1683 by Leung and Li is the state of the art in relations between $QH(G/P)$ and $QH(G/B)$, as far as I am aware. See in particular Theorem 1.4 (which however restricts to the case P/B equal to a flag variety).

Answer by Erik Insko for Quantum cohomology of partial flag manifolds

2010-11-02T16:32:03Z

Among many other nice results, the paper "Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties" by Konstanze Rietsch contains a proof of Peterson's result. It's available at arXiv:math/0112024. The result appears as Theorem 4.2.

I believe Peterson's theorem says that if one takes the opposite Schubert cell $B_{-} w_P B/B$ and intersects that with what is now called the Peterson variety, then the coordinate ring of that space is the quantum cohomology of $G/P$.

Section 2 of Harada and Tymoczko's paper "A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties" has a concise description of the Peterson variety. This paper is available on the arxiv at arXiv:0908.3517.