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

Pierre-Emmanuel Chaput (1) , Laurent Manivel (2) , Nicolas Perrin (3)

1 Géométrie
2 CRM - Centre de Recherches Mathématiques [Montréal]
3 HCM - Hausdorff Center for Mathematics

Let X be a rational homogeneous space and let QH*(X)×loc 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 π1(Aut(X))→QH∗(X)×loc 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. 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 It also uses Peterson's unpublished result -- recently proved by Lam and Shimozono in arXiv:0705.1386 -- on the comparison between the equivariant homology ring of the affine Grassmannian and the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.

Domaines: Géométrie algébrique [math.AG]

Pierre-Emmanuel Chaput, Laurent Manivel, Nicolas Perrin. Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces. Mathematical Research Letters, 2009, 16 (1), pp.7-21. ⟨hal-01256776⟩