The weighted fusion category algebra and the q-Schur algebra for \mathrm{GL}_2(q)

TL;DR

This paper establishes a connection between the weighted fusion category algebra of the principal 2-block of GL_2(q) and the q-Schur algebra, revealing a new algebraic quotient relationship and a bijection of module classes.

Contribution

It demonstrates that the weighted fusion category algebra is a quotient of the q-Schur algebra by its socle, providing new insights into the structure of modules for GL_2(q).

Findings

Weighted fusion category algebra is a quotient of the q-Schur algebra.

Establishes a bijection between simple modules and conjugacy classes of weights.

Provides a new algebraic relationship for the principal 2-block of GL_2(q).

Abstract

We show that the weighted fusion category algebra of the principal 2-block $b_0$ of $\mathrm{GL}_2(q)$ is the quotient of the $q$-Schur algebra $\mathcal{S}_2(q)$ by its socle, for $q$ an odd prime power. As a consequence, we get a canonical bijection between the set of isomorphism classes of simple $k\mathrm{GL}_2(q)b_0$-modules and the set of conjugacy classes of $b_0$-weights in this case.