Abstract
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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 497-551 |
| Number of pages | 55 |
| Journal | Advances in Mathematics |
| Volume | 295 |
| DOIs | |
| State | Published - Jun 4 2016 |
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Keywords
- E-Positivity
- Eulerian polynomials
- Hessenberg varieties
- Symmetric functions
- Unimodality
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Chromatic quasisymmetric functions. / Shareshian, John; Galloway, Michelle L.
In: Advances in Mathematics, Vol. 295, 04.06.2016, p. 497-551.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Chromatic quasisymmetric functions
AU - Shareshian, John
AU - Galloway, Michelle L
PY - 2016/6/4
Y1 - 2016/6/4
N2 - 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.
AB - 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.
KW - E-Positivity
KW - Eulerian polynomials
KW - Hessenberg varieties
KW - Symmetric functions
KW - Unimodality
UR - http://www.scopus.com/inward/record.url?scp=84962803749&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84962803749&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2015.12.018
DO - 10.1016/j.aim.2015.12.018
M3 - Article
AN - SCOPUS:84962803749
VL - 295
SP - 497
EP - 551
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -