Mathematical Research Letters

Volume 16 (2009)

Number 1

Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces

Pages: 7 – 21

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n1.a2

Authors

Pierre-Emmanuel Chaput (Laboratoire de Mathématiques Jean Leray)

Laurent Manivel (Université de Grenoble)

Nicolas Perrin (Université Pierre et Marie Curie)

Abstract

Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of \cite{cmp2} and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in \cite{seidel}. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\times$. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar \cite{Magyar}. It also uses Peterson’s unpublished result \cite{Peterson} –- recently proved by Lam and Shimozono in \cite{Lam-Shi} –- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.

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