Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Spring 2017

Speaker:  John Shareshian, Washington University
Title: Chromatic quasisymmetric functions and regular semisimple Hessenberg varieties
Time: 2:30 PM, Monday, May 1st, 2017
Place:  Malott 206

Abstract: We call a weakly increasing sequence $m=(m_1,\ldots,m_n)$ of positive integers a $\textit{Hessenberg vector}$ if $i \leq m_i \leq n$ for all $i$. Given a Hessenberg vector $m$ and an $n \times n$ matrix $s$, we define a graph $G_m$ and a variety $X(m,s)$ as follows: the graph $G_m$ has vertex set $\{1,\ldots,n\}$ and edges all $ij$ such that $i< j \leq m_i$. The $\textit{Hessenberg variety}$ $V(m,s)$ is the subvariety of the flag variety in ${\mathbb C}^n$ consisting of all flags $0< V_1< \ldots< V_n={\mathbb C}^n$ such that $sV_i \leq V_{m_i}$ for all $i$. We consider only the case where $s$ is diagonalizable with $n$ distinct eigenvalues, that is, a regular semisimple matrix.

Certain classes of Hessenberg varieties have been of interest to geometers for some time. The study of regular semisimple Hessenberg varieties was initiated in papers DeMari-Shayman and DeMari-Procesi-Shayman. Tymoczko observed that the action of $C_{GL_n({\mathbb C})}(s)$ on $V(m,s)$ allows one to apply the theory of torus actions developed by Goresky-Kottwitz-MacPherson, from which one obtains a representation of the symmetric group $S_n$ on each cohomology group of $V(m,s)$. The cohomology of $V(m,s)$ is concentrated in even degrees.

Stanley defined, for any finite graph $G$, the chromatic symmetric function $X_G({\mathbf x})$ in variables $x_1,x_2,\ldots$. In their study of a problem about immanants raised by Stembridge, Stanley-Stembridge conjectured that if $G$ is the incomparability graph of a $3+1$-free poset $P$, then $X_G({\mathbf x})$ is a non-negative integer combination of elementary symmetric functions (an $\textit{$e$-positive}$ symmetric function). Guay-Paquet showed that to prove the Stanley-Stembridge conjecture, it suffices to show that $X_G({\mathbf x})$ is $e$-positive when $P$ is both $3+1$-free and $2+2$-free, in which case $G$ is isomorphic with $G_m$ for some Hessenberg vector $m$.

In joint work with Wachs, we introduced a graded version $X_G({\mathbf x};t)$ of Stanley's chromatic symmetric function. The power series $X_G({\mathbf x};t)$ is not symmetric in general, but is a quasisymmetric function. One can consider $X({\mathbf x};t)$ to be a polynomial in $t$ with coefficients in the ring of quasisymmetric functions.

We showed that if $m$ is a Hessenberg vector, then $X_{G_m}({\mathbf x};t)$ is symmetric. In fact, our conjecture that the Frobenius characteristic of the representation of $S_n$ on $H^{2j}(X(m,s))$ is the coefficient of $t^j$ in $\omega(X_{G_m}({\mathbf x};t))$ was proved by Brosnan-Chow. A different proof was given by Guay-Paquet. (Here $\omega$ is the involution on the ring of symmetric functions mapping elementary symmetric functions to complete homogeneous symmetric functions.) As we have determined the decomposition of $X_{G_m}({\mathbf x};t)$ into Schur functions (generalizing a result of Gasharov for $X_{G_m}({\mathbf x})$), and Athanasiaids has proved a conjecture of ours giving a formula for the decomposition of $X_{G_m}({\mathbf x};t)$ into power sum symmetric functions, the Brosnan-Chow theorem tells us the irreducible decomposition and character values of the representation of $S_n$ on the cohomology of $X(m,s)$, thus answering a question of Tymoczko. The Brosnan-Chow theorem also makes possible a geometric approach to proving the Stanley-Stembridge conjecture, which has yet to succeed.

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