For si-stable and almost-si-stable condition sets, the dot-action module H_C decomposes as a direct sum of a fixed subspace and a multiply-shifted copy, realizing the modular relation at the representation level.
An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson
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This paper provides an introduction to equivariant cohomology and homology using the approach of Goresky, Kottwitz, and MacPherson. When a group G acts suitably on a variety X, the equivariant cohomology of X can be computed using the combinatorial data of a skeleton of G-orbits on X. We give both a geometric definition and the traditional definition of equivariant cohomology. We include a discussion of the moment map and an algorithm for finding a set of generators for the equivariant cohomology of X. Many examples and explicit calculations are provided.
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Divided difference operators for Hessenberg representations
For si-stable and almost-si-stable condition sets, the dot-action module H_C decomposes as a direct sum of a fixed subspace and a multiply-shifted copy, realizing the modular relation at the representation level.