Projective modules for the symmetric group and Young's seminormal form

TL;DR

This paper explores the representation theory of symmetric groups in positive characteristic, proposing a conjectural description of projective modules and verifying it for small cases using an algorithm inspired by Khovanov-Lauda-Rouquier algebras.

Contribution

It introduces a conjectural framework for projective modules in modular representation theory and provides an algorithm to verify these conjectures for small symmetric groups.

Findings

Conjectural description of projective covers for certain partitions.

Verification of the conjecture for n ≤ 15 cases.

Connection between ladder partitions and module structure.

Abstract

We study the representation theory of the symmetric group $S_n$ in positive characteristic $p$. Using features of the LLT-algorithm we give a conjectural description of the projective cover $P(\lambda)$ of the simple module $D(\lambda)$ where $\lambda$ is a $p$-restricted partition such that all ladders of the corresponding ladder partition are of order less than $p$. Inspired by the recent theory of Khovanov-Lauda-Rouquier algebras we explain an algorithm that allows us to verify this conjectural description for $n \leq 15$, at least.