Folding with Soergel bimodules

TL;DR

This paper introduces folding in the context of equivariant categories and demonstrates that quasi-split Hecke algebras with unequal parameters are categorified by equivariant Soergel bimodules, supported by detailed computations.

Contribution

It provides an elementary introduction to folding and equivariant categories, and confirms the categorification of certain Hecke algebras via equivariant Soergel bimodules.

Findings

Quasi-split Hecke algebras with unequal parameters are categorified by equivariant Soergel bimodules in most cases.

Provides computational methods for working with equivariant categories and Grothendieck groups.

Offers an accessible explanation of the technical aspects of folding and categorification.

Abstract

We give a gentle introduction to the concept of folding. That is, we provide an elementary discussion of equivariant categories, their weighted Grothendieck groups, and the technical aspects of computing with them. We then perform the computations required to confirm that quasi-split Hecke algebras with unequal parameters are categorified by equivariant Soergel bimodules, in almost every case.