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Chromatic quasisymmetric functions and regular semisimple Hessenberg varieties

John Shareshian, Washington University in St. Louis
Wednesday, October 3, 2018 - 3:30pm
PDL C-401
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Richard Stanley introduced the chromatic symmetric function of a graph, which contains at least as much information about the graph as the well-studied chromatic polynomial.  In joint work with Michelle Wachs, we introduced a refinement of Stanley's chromatic symmetric function, called the chromatic quasisymmetric function.  When G is the incomparability graph of a unit interval order, the chromatic quasisymmetric function of G turns out to be a symmetric function, and is Schur positive.  This means that there is a representation of the symmetric group associated to G.  Also associated to G is a subvariety of the type A flag variety, called a regular semisimple Hessenberg variety.  As described by Julianna Tymoczko, there is a nice representation of the symmetric group on the cohomology of this variety.  Wachs and I conjectured that, after a sign twist, the two representations just described are the same.  This was proved by Patrick Brosnan and Tim Chow.  I will discuss this story and some recent developments.

Note: You can find links to relevant background material below.

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