ISM Discovery School: Geometry and Combinatorics of Hessenberg Varieties

June 6-10, 2022

There are relatively new developments connecting the geometry of Hessenberg varieties and symmetric functions and their associated combinatorics, which have shown that the (equivariant) geometry and topology of Hessenberg varieties are intimately connected with a deep unsolved problem in the theory of symmetric (and quasisymmetric) functions called the Stanley-Stembridge conjecture.

The subject of Hessenberg varieties lies in the fruitful intersection of algebraic geometry, combinatorics, and geometric representation theory. A fundamental contribution in this area, over a decade ago, was Julianna Tymoczko’s construction of an action of the symmetric group on the cohomology rings of regular semisimple Hessenberg varieties. Tymoczko’s action provided the first link between Hessenberg varieties and symmetric functions because representations of symmetric groups give rise to symmetric functions via the Frobenius characteristic map.

The second link was developed via the notion of the chromatic symmetric function of a graph, introduced by Richard Stanley in 1995 as a generalization of the classic chromatic polynomial of a graph. The Stanley-Stembridge conjecture concerns the structure of the chromatic symmetric functions of a special family of graphs; it states that the chromatic symmetric functions of these graphs are non-negative linear combinations of elementary symmetric functions. This conjecture is still open.

A close relationship between chromatic symmetric functions and Hessenberg varieties was discovered by John Shareshian and Michelle Wachs, who associated a graph with each regular semisimple Hessenberg variety and formulated a conjecture relating the chromatic symmetric function of the graph with the symmetric function associated with the Hessenberg variety via Tymoczko’s action. Their conjecture has since been proved, and there is reason to hope that further progress on the Stanley-Stembridge conjecture can be made by better understanding the relation between these two areas.

The Summer School is aimed at graduate students specializing in geometry or combinatorics, and the goal is to introduce both the theories of Hessenberg varieties and of symmetric functions in such a way that a student can have access to the exciting developments linking these areas.