Parabolic Lusztig varieties and chromatic symmetric functions

TL;DR

This paper explores the geometric and algebraic structures underlying chromatic symmetric functions and Kazhdan--Lusztig elements, aiming to develop recursive formulas and address conjectures in algebraic combinatorics.

Contribution

It characterizes local systems in the cohomology of certain varieties, linking them to subgroup representations and proposing a recursive approach to compute related characters.

Findings

Local systems correspond to subgroup representations of S_n

A recursive formula for characters and chromatic symmetric functions is proposed

Connections are established with the Grojnowski--Haiman hybrid basis

Abstract

The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$ (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of certain subvarieties of the flag variety. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of $S_n$. An explicit characterization of such representations would provide a recursive formula for the computation of such characters/chromatic symmetric functions, which could settle Haiman's conjecture about the positivity of the monomial characters of Kazhdan--Lusztig elements and Stanley--Stembridge conjecture about $e$-positivity of chromatic symmetric function of indifference graphs. We also find a connection between the character of certain homology groups of subvarieties of the partial flag varieties and the Grojnowski--Haiman hybrid basis of the Hecke algebra.