13 research outputs found
Power sum expansion of chromatic quasisymmetric functions
The chromatic quasisymmetric function of a graph was introduced by Shareshian
and Wachs as a refinement of Stanley's chromatic symmetric function. An
explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing
the chromatic quasisymmetric function of the incomparability graph of a natural
unit interval order in terms of power sum symmetric functions, is proven. The
proof uses a formula of Roichman for the irreducible characters of the
symmetric group.Comment: Final version, 9 pages; comments by a referee incorporate
Involutions and the Gelfand character
The Gelfand representation of is the multiplicity-free direct
sum of the irreducible representations of . In this paper, we
use a result of Adin, Postnikov, and Roichman to find a recursive generating
function for the Gelfand character. In order to find this generating function,
we investigate descents of so-called -unimodal involutions
Splitting the cohomology of Hessenberg varieties and e-positivity of chromatic symmetric functions
For each indifference graph, there is an associated regular semisimple
Hessenberg variety, whose cohomology recovers the chromatic symmetric function
of the graph. The decomposition theorem applied to the forgetful map from the
regular semisimple Hessenberg variety to the projective space describes the
cohomology of the Hessenberg variety as a sum of smaller pieces. We give a
combinatorial description of the Frobenius character of each piece. This
provides a generalization of the symmetric functions attached to Stanley's
local h-polynomials of the permutahedral variety to any Hessenberg variety.
As a consequence, we can prove that the coefficient of , where
is any partition of length 2, in the e-expansion of the chromatic
symmetric function of any indifference graph is non-negative.Comment: 23 page
Chromatic quasisymmetric functions
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
-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
-positivity conjecture of Stanley and Stembridge for incomparability graphs
of -free posets.Comment: 57 pages; final version, to appear in Advances in Mat