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.
Chromatic quasisymmetric functions and Hessenberg varieties
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abstract
We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections between these topics.
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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.