Rank-metric separation in irreducible representations of finite groups

TL;DR

This paper establishes a lower bound on the rank of matrices derived from irreducible representations of finite groups, linking representation theory with locally decodable codes to derive new algebraic bounds.

Contribution

It introduces a novel connection between matrix rank bounds in group representations and locally decodable codes, extending previous reduction techniques.

Findings

Provides a general lower bound on matrix rank in irreducible representations

Links representation theory with error-correcting codes to derive bounds

Utilizes known results on 2-query LDCs to establish the rank bound

Abstract

We give a general lower bound on the rank of matrices of the form $\rho(h) - I$ with $\rho : G \rightarrow GL({\mathbb F}^n)$ an irreducible representation of a finite group $G$. The main tool in the proof is a (strengthening) of a reduction due to Efremenko from low rank matrices spanned by a few images of $\rho$ to Locally Decodable Codes (LDCs), which are a special kind of error correcting codes. We then apply the known results on 2-query LDCs to derive our rank bound.