The $q$-Schur category and polynomial tilting modules for quantum $\mathrm{GL}_{n}$

Jonathan Brundan

Vol. 336 (2025), No. 1-2, 63–112

DOI: 10.2140/pjm.2025.336.63

Abstract

The $q$-Schur category is a $\mathbb{Z}[q,q^{-1}]$-linear monoidal category closely related to the $q$-Schur algebra. We explain how to construct it from coordinate algebras of quantum ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$ for all $n\ge 0$. Then we use Donkin’s work on Ringel duality for $q$-Schur algebras to make precise the relationship between the $q$-Schur category and a $\mathbb{Z}[q,q^{-1}]$-form for the $U_q{\mathfrak{𝔤}\mathfrak{𝔩}}_n$-web category of Cautis, Kamnitzer and Morrison. We construct explicit integral bases for morphism spaces in the latter category, and extend the Cautis–Kamnitzer–Morrison theorem to polynomial representations of quantum ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$ at a root of unity over a field of any characteristic.

Keywords

$q$-Schur algebra, tilting module, monoidal category

Mathematical Subject Classification

Primary: 17B10

Milestones

Received: 30 June 2024

Revised: 30 August 2024

Accepted: 1 September 2024

Published: 26 May 2025