The weighted fusion category algebra and the q-Schur algebra for \mathrm{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.

8 pages, 2 figures, to appear in Journal of Algebra