Rank-metric separation in irreducible representations of finite groups

TL;DR

This paper establishes a fundamental link between the rank of deviations from the identity in irreducible finite-group representations and locally decodable codes (LDCs). By extending Efremenko’s reduction, it shows that if a small-rank combination of representation matrices exists, one can construct a linear q-LDC with parameters tied to the rank, thereby transferring lower bounds from LDC theory to representation-theoretic rank bounds. The authors prove a concrete bound for 2-LDCs over arbitrary fields, yielding a quantifiable lower bound on rank(ρ(h)−I) in terms of n and |G|, and discuss tightness with explicit group constructions. They also analyze the limitations for q≥3 LDCs, discuss current bounds, and pose conjectures linking LDC lower bounds to rank guarantees in representations, highlighting a deep interdisciplinary connection between coding theory and finite group representations.

Abstract

We give a general lower bound on the rank of matrices of the form $ρ(h) - I$ with $ρ: G \rightarrow GL({\mathbb F}^n)$ 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 $ρ$ 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.

Theorems & Definitions (18)

• Theorem 1.1
• Definition 1.2: $q$-Matching
• Definition 1.3: Linear $(q,\delta)$-LDC
• Definition 1.4: Special $(2,\delta)$-LDC
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• proof : Proof of Theorem \ref{['thm-main2LDCsep']}
• Claim 2.1
• proof