Representation Theory of $0$-Schur Algebras and Related Categories

Jensen, Su, and Yang described the projective indecomposable modules of the $0$-Schur algebra $\mathbf{S}_0(n,r)$ using its geometric realization. In this paper, the simple modules of $\mathbf{S}_0(n,r)$ are identified by computing the tops of the projective indecomposable modules. Furthermore, functorial relations among the module categories $\mathbf{H}_r(0)$\textsf{-mod}, $\mathbf{S}_0(n,r)$\textsf{-mod}, and $U_0(\mathfrak{gl}_n)$\textsf{-mod} are examined, where $\mathbf{H}_r(0)$ denotes the $0$-Hecke algebra and $U_0(\mathfrak{gl}_n)$ denotes the degenerate quantum group.