The sparsity of character tables over finite reductive groups and its additive analogue

TL;DR

This paper investigates the proportion of zero entries in character tables of finite reductive groups, establishing asymptotic bounds and an additive analogue, revealing how sparsity evolves with group size and rank.

Contribution

It provides the first asymptotic bounds on zero entries in character tables and introduces an additive analogue for reductive Lie algebras.

Findings

Proportion of zeros approaches one as semisimple rank increases.

Asymptotic lower bounds for fixed groups over larger fields.

Additive analogue results for reductive Lie algebras.

Abstract

We consider the proportion of zero entries in the character table of a sequence of reductive groups over a finite field. We prove an asymptotic lower bound when the reductive group is fixed and the size of the finite field increases. Furthermore, we prove that when considering a sequence of reductive groups with increasing semisimple rank, the proportion is asymptotically one. We also establish an additive analogue of this phenomenon in the context of a fixed reductive Lie algebra.