# Combinatorics-Tymoczko's dot action on the cohomology of Hessenberg varieties

Speaker: Patrick Brosnan, University of Maryland

Date and time: Tuesday, February 13, 1–2pm

Place: Rome 771

Title: Tymoczko's dot action on the cohomology of Hessenberg varieties

Abstract: Hessenberg varieties are certain subvarieties of flag varieties. For GLn, they are defined in terms of an nxn matrix S along with a certain function f from {1,...,n} to itself, which we call a Hessenberg function. When all of the eigenvalues of the matrix are distinct, the associated Hessenberg variety is smooth and the associated variety B(f,s) comes equipped with an action of the symmetric group first studied by Tymoczko, who called it the "dot action." In a theoretical sense, Tymoczko's action is computable in terms of a combinatorial object called the moment graph associated to the Hessenberg variety. But the complexity of actually carrying out such a computation (even using a computer) is rather daunting. I will explain joint work with Tim Chow proving a conjecture of Shareshian and Wachs, which gives a more or less closed form expression for the action in terms of a chromatic symmetric function associated to a certain graph. This function is a generalization of Stanley's chromatic symmetric function introduced by Shareshian and Wachs, who were partially motivated by conjectures of Stanley and Stembridge on the e-positivity of Stanley's chromatic symmetric function. In my talk I will explain some of this, and I will also say a few words about Guay-Paquet's independent proof of the Shareshian–Wachs conjecture (along with a generalization of some of his arguments from the case of GLn to other groups).