What question did this study set out to answer?

This work aims to establish a lower bound on the rank of specific matrices derived from irreducible representations of finite groups.

June 15, 2026Open Access

Rank-metric separation in irreducible representations of finite groups

Key Points

This work aims to establish a lower bound on the rank of specific matrices derived from irreducible representations of finite groups.
Developed a general lower bound for rank using matrices of the form ρ ( h ) − I with ρ : G → GL ( F n ).
Utilized a strengthening of Efremenko's reduction relating low rank matrices to locally decodable codes (LDCs).
Applied results from known 2-query LDCs to derive the rank bound.
Established a new lower bound for the rank of matrices related to irreducible representations.
Demonstrated the connection between matrix rank and properties of locally decodable codes.
Provided a framework for further explorations in the intersection of group theory and coding theory.

Abstract

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

Rank-metric separation in irreducible representations of finite groups

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Abstract

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