Explicit calculations in an infinitesimal singular block of $S{L}_N$

Abstract

Let $G=S{L}_{n+1}$ be defined over an algebraically closed field of characteristic $p>2$. For each $n\ge 1$ there exists a singular block in the category of $G_1$-modules which contains precisely $n+1$ irreducible modules. We are interested in the lift of this block to the category of $G_{1}T$-modules. Imposing only mild assumptions on $p$, we will perform a number of calculations in this setting, including a complete determination of the Loewy series for the baby Verma modules and all possible extensions between the irreducible modules. In the case where $p$ is extremely large, we will also explicitly compute the Loewy series for the indecomposable projective modules.

29 pages, made a number of changes due to referee comments