Combinatorics Seminar: “The Geometry and Combinatorics of Certain Hessenberg Varieties”

Abstract: It is known that the regular semisimple Hessenberg variety associated to the Hessenberg function h(i)=i+1, for i=1,..., n-1, is isomorphic to the permutohedral variety. The first goal in the talk is to make this isomorphism concrete, by constructing a natural isomorphism between the two varieties. In the process, we introduce a sequence of varieties which we call the prepermutohedral varieties. These varieties have some interesting geometric and combinatorial properties; and are closely related to the regular semisimple Hessenberg varieties associated to the Hessenberg function represented by h_k= (2,3, ... , k+1, n,...,n)$, k=1,..., n-3. If time allows, I will also describe the cohomology ring of these Hessenberg varieties and the dot representation of the permutation groups on them.

Host: Nathan Lesnevich