Sejong Park

The weighted fusion category algebra and the $q$-Schur algebra for $\mathrm{GL}_2(q)$, to appear in Journal of Algebra



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.