On Tensor Products of Simple Modules for Simple Groups

TL;DR

This paper investigates the algebraic properties of simple modules across various finite groups, revealing that most natural modules are non-algebraic in groups of Lie type, while certain modules in symmetric and sporadic groups are algebraic.

Contribution

It provides new results on the algebraicity of modules for simple groups, especially in groups of Lie type, symmetric groups, and sporadic groups, expanding understanding of their module structures.

Findings

Natural modules in most Lie type groups are non-algebraic.

Simple modules in specific blocks of symmetric groups are algebraic for p ≤ 5.

All simple modules in analyzed sporadic groups are algebraic for various primes.

Abstract

In an attempt to get some information on the multiplicative structure of the Green ring we study algebraic modules for simple groups, and associated groups such as quasisimple and almost-simple groups. We prove that, for almost all groups of Lie type in defining characteristic, the natural module is non-algebraic. For alternating and symmetric groups, we prove that the simple modules in $p$-blocks with defect groups of order $p^2$ are algebraic, for $p\leq 5$. Finally, we analyze nine sporadic groups, finding that all simple modules are algebraic for various primes and sporadic groups