Equivariant Schubert Calculus

2007/03/15 by Letterio Gatto, Gatto, Letterio, Taíse Santiago +2
Mathematics · #14F43 #14N15 #15A75 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG #msc:14F43 #msc:14N15 #msc:15A75

paper · pdf · doi:10.48550/arxiv.math/0703445

15 pages, no figures, part of the doctoral thesis of the second author

arxiv created 2007/03/15 · openalex publication_date 2007/03/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let T be a torus acting on \CCⁿ in such a way that, for all 1≤ k≤ n, the induced action on the grassmannian G(k,n) has only isolated fixed points. This paper proposes a natural, elementary, explicit description of the corresponding T-equivariant Schubert calculus. In a suitable natural basis of the T-equivariant cohomology, seen as a module over the T-equivariant cohomology of a point, it is formally the same as the ordinary cohomology of a grassmann bundle. The main result, useful for computational purposes, is that the T-equivariant cohomology of G(k,n) can be realized as the quotient of a ring generated by derivations on the exterior algebra of a free module of rank n over the T-equivariant cohomology of a point.

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