Positive bases and invariant cones for symmetric group representations

Abstract

The irreducible representations of the symmetric group over characteristic 0 have been classified since the seminal work of Frobenius, Schur and Young. When representing elements as matrices, we need to pick a basis.

Various bases are well understood such as Young’s seminormal basis. However, the relationships between other natural bases such as the Kazhdan—Lusztig, Springer or p-canonical bases remain subtle. Such bases and their base change are related to various fundamental questions and conjectures in algebraic geometry and geometric representation theory.

In this talk, we explore a common framework for comparing these bases. In particular, in all the bases above the matrices of the operators 1+s for a simple reflection s have only non-negative entries. Hence, we can view these bases as spanning a proper cone that is invariant under the operators 1+s.

We study the moduli space of all such bases which are related by a unitriangular base change (relative to a fixed ordered basis) over the real numbers. It turns out that this moduli space is a semi-algebraic compact set. 

Among the basic constructions, we will see various properties of this moduli space and how the bases above sit inside it. At the end, I will briefly mention similar considerations for Lie algebra representations and some open questions.

This talk is based on joint work with Geordie Williamson.