Permutation representations on Schubert varieties

Details Permutation representations on Schubert varieties Permutation representations on Schubert varieties

2006-04-26

arxiv.org/abs/math/0604578v2

Mathematics

Representation Theory

20 pages, v2: minor revisions

Scientific paper

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.

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