2007/12/19 by Pierre–Emmanuel Chaput, Pierre-Emmanuel Chaput, Laurent Manivel +5
Mathematics · #14M15 #14N35 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:14M15 #msc:14N35
paper · pdf · doi:10.48550/arxiv.0712.3131
15 pages
arxiv created 2007/12/19 · openalex publication_date 2007/12/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(\rm 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 It also uses Peterson's unpublished result -- recently proved by Lam and Shimozono in arXiv:0705.1386 -- on the comparison between the equivariant homology ring of the affine Grassmannian and the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.