Geometric representations of the permutation group using
Hessenberg varieties
Abstract: Geometric representation theorists study group actions by
constructing representations on geometric objects, usually on the
cohomology of an algebraic variety. The classical example of a
geometric representation is Springer's representation, which
constructs representations of the permutation group on the cohomology
of a family of varieties now known as Springer fibers. Amazingly, the
top-dimensional cohomology is an irreducible representation of the
symmetric group, and every irreducible representation can be recovered
uniquely in this way.
In this talk, we'll describe a family of algebraic varieties called
Hessenberg varieties that naturally generalize Springer fibers. We'll
show that we can construct geometric representations of the
permutation group using Hessenberg varieties, too. At the end, we
will describe how these representations appear in conjectures of
Shareshian-Wachs and Stembridge in combinatorial representation