On self-extensions of irreducible modules over symmetric groups

TL;DR

This paper investigates a long-standing conjecture about the absence of non-trivial self-extensions of irreducible modules over symmetric groups in odd characteristic, providing partial positive results.

Contribution

It offers new partial results supporting the conjecture that irreducible modules over symmetric groups have no non-trivial self-extensions in characteristic not equal to 2.

Findings

Partial positive results confirming the conjecture

Evidence supporting the non-existence of self-extensions in certain cases

Progress towards resolving a decades-old conjecture

Abstract

A conjecture going back to the eighties claims that there are no non-trivial self-extensions of irreducible modules over symmetric groups if the characteristic of the ground field is not equal to $2$. We obtain some partial positive results on this conjecture.