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)$$\mathsf{\text{-mod}}$, ${\mathbf{S}}_0(n,r)$$\mathsf{\text{-mod}}$, and $U_0({\mathfrak{g}\mathfrak{l}}_n)$$\mathsf{\text{-mod}}$ are examined, where ${\mathbf{H}}_r(0)$ denotes the $0$-Hecke algebra and $U_0({\mathfrak{g}\mathfrak{l}}_n)$ denotes the degenerate quantum group.