Rank-metric separation in irreducible representations of finite groups

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.