Generalised Cartan invariants of symmetric groups

TL;DR

This paper confirms a conjecture about the invariant factors of the Cartan matrix in a generalized modular representation theory of symmetric groups, extending previous work to a broader algebraic context.

Contribution

It proves a more precise blockwise conjecture, validating the combinatorial description of Cartan invariants in the generalized setting.

Findings

Confirmed the conjecture on invariant factors of the Cartan matrix.

Validated the combinatorial description for the generalized theory.

Extended the understanding of modular representation theory beyond prime moduli.

Abstract

K\"ulshammer, Olsson, and Robinson developed an l-analogue of modular representation theory of symmetric groups where l is not necessarily a prime. They gave a conjectural combinatorial description for invariant factors of the Cartan matrix in this context. We confirm their conjecture by proving a more precise blockwise conjecture due to Bessenrodt and Hill.