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

Abstract

Let $X$ be a rational homogeneous space and let $Q{H}^{\ast }(X{)}_{loc}^{\times }$ 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 ${\pi }_1(\mathrm{Aut}(X))\to Q{H}^{\ast }(X{)}_{loc}^{\times }$ described in arXiv:dg-ga/9511011. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $Q{H}_T^{\ast }(X{)}_{loc}^{\times }$. 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 It also uses Peterson's unpublished result -- recently proved by Lam and Shimozono in arXiv:0705.1386 -- on the comparison between the equivariant homology ring of the affine Grassmannian and the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.

15 pages

References in corpus (2)

• Quantum cohomology of minuscule homogeneous spaces II : Hidden symmetries
• Notes on Schubert classes of a loop group