Rank-metric separation in irreducible representations of finite groups

2025/12/22 by Zeev Dvir, Dvir, Zeev
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #graph theory and CDMA systems

paper · doi:10.48550/arxiv.2512.19638

openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

Abstract

We give a general lower bound on the rank of matrices of the form ρ(h) - I with ρ: G → GL(\mathbb Fⁿ) 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.

Citations

• A Near-Cubic Lower Bound for 3-Query Locally Decodable Codes from Semirandom CSP Refutation
• Average dimension of fixed point spaces with applications
• Generation of finite almost simple groups by conjugates

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