Algebra and Combinatorics Seminar

Date: September 22, 2017

Time: 3:00PM - 4:00PM

Location: BLOC 117

Speaker: Patrick Brosnan , University of Maryland

Title: Hessenberg varieties and a conjecture of Shareshian---Wachs

Abstract: I will explain joint work with Tim Chow proving a conjecture of Shareshian---Wachs which relates a combinatorial object, the so-called chromatic symmetric function of a certain graph, to a certain action of the symmetric on the cohomology of a Hessenberg variety first studied by J. Tymoczko. I should mention that, shortly after Chow and I posted our proof to the ArXiv, a completely independent proof relying on a map of Hopf algebras and the theorem of Aguiar---Bergeron---Sottile was posted by M. Guay-Paquet. The Hessenberg varieties in the title are certain smooth subvarieties of the the complete flag variety studied first by the applied mathematicians de Mari and Shayman. They were later generalized by de Mari, Procesi and Shayman to a setting where the general linear group is replaced with an arbitrary reductive group. In this case, Tymoczko's dot action becomes a representation of the Weyl group, and it is an interesting problem to determine this representation. I will present some results in this direction. In particular, I will explain a restriction formula that generalizes Guay-Paquet's proof that his Hopf algebra map respects comultiplication.