Permutation representations on Schubert varieties

• Autores: Julianna S. Tymoczko
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 130, Nº 5, 2008, págs. 1171-1194
• Idioma: inglés
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• Resumen

  • This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over ${\Bbb C}$ and over ${\Bbb C}[t_1, t_2,\ldots,t_n]$. 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 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.