SL2 tilting modules in the mixed case

TL;DR

This paper explores the structure of tilting modules for SL2 in the mixed case using Temperley-Lieb calculus, generalizing several classical settings and providing explicit descriptions of character formulas, fusion rules, and categorical presentations.

Contribution

It introduces a unified approach to tilting modules in the mixed case, extending known theories and explicitly describing their categorical and monoidal structures.

Findings

Character formulas for tilting modules

Explicit fusion rules and Jones-Wenzl projectors

Presentation of the category via quiver with relations

Abstract

Using the non-semisimple Temperley-Lieb calculus, we study the additive and monoidal structure of the category of tilting modules for $\mathrm{SL}_{2}$ in the mixed case. This simultaneously generalizes the semisimple situation, the case of the complex quantum group at a root of unity, and the algebraic group case in positive characteristic. We describe character formulas and give a presentation of the category of tilting modules as an additive category via a quiver with relations. Turning to the monoidal structure, we describe fusion rules and obtain an explicit recursive description of the appropriate analog of Jones-Wenzl projectors. We also discuss certain theta values, the tensor ideals, mixed Verlinde quotients and the non-degeneracy of the braiding.