On unitriangular basic sets for symmetric and alternating groups

TL;DR

This paper investigates the existence of unitriangular basic sets in the modular representation theory of symmetric and alternating groups, providing new labeling methods for simple modules and highlighting cases where such sets do not exist.

Contribution

It introduces new approaches to construct unitriangular basic sets for symmetric groups and their Hecke algebras, and demonstrates their non-existence in certain cases for alternating groups.

Findings

New methods to obtain basic sets for symmetric groups and Hecke algebras.

Existence of basic sets is confirmed for symmetric groups but not always for alternating groups.

Explicit counterexamples in characteristic 3 show non-existence for some alternating groups.

Abstract

We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the decomposition matrices, that is, the existence of a unitriangular basic set. We study several ways to obtain such sets in the general case of a symmetric algebra. We apply our results to the symmetric groups and to their Hecke algebras and thus obtain new ways to label the simple modules for these objects. Finally, we show that these sets do not always exist in the case of the alternating groups by studying two explicit cases in characteristic 3.