# Counting characters in linear group actions

## TL;DR

This paper establishes upper bounds on the sizes of specific subsets of irreducible characters in finite group modules, which could aid in addressing the non-coprime $k(GV)$-problem in representation theory.

## Contribution

It provides new upper bounds for subsets of irreducible characters in finite group modules, advancing understanding in the representation theory of finite groups.

## Key findings

- Derived upper bounds for subsets of irreducible characters
- Potential applications to the non-coprime $k(GV)$-problem
- Enhanced tools for analyzing group actions on modules

## Abstract

Let $G$ be a finite group and $V$ be a finite $G$--module. We present upper bounds for the cardinalities of certain subsets of $\Irr(GV)$, such as the set of those $\chi\in\Irr(GV)$ such that, for a fixed $v\in V$, the restriction of $\chi$ to $<v>$ is not a multiple of the regular character of $<v>$. These results might be useful in attacking the non--coprime $k(GV)$--problem.