Chromatic quasisymmetric functions and Hessenberg varieties

By John Shareshian, Michelle L. Wachs. 2011

87 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/57b6a1c8e6d33d903205e6e98b74632e72cba36a

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, methodology): “As briefly discussed above, Shareshian and Wachs conjectured a precise relationship between the (Frobenius characteristic of) Tymoczko’s Sn-representation on the cohomology group of a regular semisimple Hessenberg variety Hess(S, h) and the chromatic quasisymmetric function XG(x, t) of a graph G def”
Recent Trends in Quasisymmetric Functions (2018, background): “Shareshian andWachs recently proposed a new approach to the Stanley-Stembridge e-positivity conjecture in the form of a refinement of Stanley’s chromatic symmetric functions.”
Hopf Algebras in Combinatorics (2014, background): “208A recent refinement for incomparability graphs of posets which are both (3 + 1)- and (2 + 2)-free, also known as unit interval orders is discussed by Shareshian and Wachs [131].”