Explicit calculations in an infinitesimal singular block of $SL_N$

TL;DR

This paper performs explicit calculations in a specific singular block of the algebraic group SL_{n+1} over a field of characteristic p, determining Loewy series and extensions in the category of G_1T-modules.

Contribution

It provides the first detailed calculations of Loewy series and extensions in the lift of an infinitesimal singular block of SL_N to G_1T-modules, including explicit projective module structures for large p.

Findings

Complete Loewy series for baby Verma modules

All extension groups between irreducible modules determined

Loewy series for indecomposable projective modules computed for large p

Abstract

Let $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$ 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 $G_1T$-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.