Permutation Representations on Schubert Varieties

Authors

Julianna S. Tymoczko, University of IowaFollow

Document Type

Article

Publication Date

10-1-2008

Publication Title

American Journal of Mathematics

Abstract

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over ℂ and over ℂ[t1, t2, . . . , tn]. 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 of 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.

Volume

130

Issue

5

First Page

1171

Last Page

1194

DOI

10.1353/ajm.0.0018

ISSN

00029327

Comments

Peer reviewed accepted manuscript.

Recommended Citation

Tymoczko, Julianna S., "Permutation Representations on Schubert Varieties" (2008). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/108