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

TL;DR

This paper explores the affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces, explicitly realizing the automorphism group and providing formulas for their product actions.

Contribution

It generalizes previous results to explicitly embed the fundamental group of automorphisms into the invertible elements of the quantum cohomology ring, with explicit formulas and new algebraic insights.

Findings

Explicit embedding of $ ext{Aut}(X)$ into $QH^*_T(X)_{loc}^ imes$

Formulas for the product by these invertible elements

Generalization of Magyar's formula and use of Peterson's result

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 arXiv:math/0609796 and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in arXiv:dg-ga/9511011. 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 arXiv:0705.3826. It also uses Peterson's unpublished result -- recently proved by Lam and Shimozono in arXiv:0705.1386 -- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.