Chromatic quasisymmetric functions

By John Shareshian, Michelle L. Wachs. 2014

251 citations recorded by the source corpus.

We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the $q$-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the $e$-positivity conjecture of Stanley and Stembridge for incomparability graphs of $(3+1)$-free posets.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9bf99e42283267ccc3d23a8b8fc85914388808e5

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In Algebra and Number Theory

1. Victor Reiner
2. Hiraku Abe
3. Megumi Harada
4. Darij Grinberg
5. M. Masuda
6. T. Horiguchi
7. Martha Precup
8. T. Panov

In Discrete Mathematics and Combinatorics

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. Victor Reiner
6. Darij Grinberg
7. Martha Precup
8. T. Panov

In Geometry and Topology

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. Victor Reiner
6. Darij Grinberg
7. Martha Precup
8. T. Panov

Citations in the corpus

• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, methodology): “o’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 defined from the Hessenberg function h [43, 44]. (Details are in Section 11.) When both sides of their conjectured equality are expanded in”
• Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties (2015, unknown): “Theorem 33 (Tymoczko) Let H ( m , s ) be a regular Hessenberg variety and let the partition λ encode the sizes of the Jordan blocks of s .”
• The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture (2017, background): “isabelian(cf. Definition 3.1). Beforeproceedingwetakeamomenttoexplainhowaproof of Conjecture 2.8 implies the classical Stanley-Stembridge conjecture. Since this story has been recorded elsewhere (e.g. [3,22]) we will be brief. We begin with a statement of the original Stanley-Stembridge conjecture. A”
• Hopf Algebras in Combinatorics (2014, background): “See Shareshian/Wachs [192] for more general criteria for such functions to be symmetric, as well as deeper results.”