2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141904We 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 AlgebraRepresentation Theory20C20; 20J05text