The University of Michigan Combinatorics Seminar
Winter 2007
January 26, 4:10-5:00, 3866 East Hall



Divided difference operators in the
equivariant cohomology of Grassmannians

Julianna Tymoczko

University of Michigan


Abstract

Divided difference operators were introduced by Bernstein, Gelfand, and Gelfand to find explicit maps between algebraic constructions of the cohomology of G/P and geometric constructions. A third, more combinatorial construction of the (equivariant) cohomology of G/P was discovered independently by Kostant and Kumar and, in more generality, by Goresky, Kottwitz, and MacPherson (GKM). This method builds the equivariant cohomology of G/P by applying an algebraic algorithm to a combinatorial graph based on the geometry of G/P. We show how to construct divided difference operators using certain automorphisms of this combinatorial graph. One application is a generalization of Billey's formula for the localizations of equivariant Schubert classes to arbitrary G/P; other applications will be discussed as time permits.