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Statistics of entries of matrices from the general linear group over a finite field

  • Ofir Gorodetsky
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Abstract

Let GLn(𝔽q) be the group of n× n invertible matrices over a finite field 𝔽q. How many matrices in GLn(𝔽q) are there with a subset of their entries prescribed? Motivated by the random matrix theory of classical compact groups, we address this question and various variants, with a focus on asymptotic results that are valid as n tends to infinity. Our methods are Fourier-analytic, with the main input being an exponential sum over GLn(𝔽q), evaluated in 1955 by Hodges via elementary methods. Our results turn out to be related to investigations carried out in other fields, including coding theory, universality for p-adic matrices and algebraic combinatorics.

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Cite

BibTeXRIS
BibTeX
@misc{gorodetsky2026statistics,
  title = {{Statistics of entries of matrices from the general linear group over a finite field}},
  author = {Ofir Gorodetsky},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29305},
  eprint = {2610.29305},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29305},
}
RIS
TY  - UNPB
TI  - Statistics of entries of matrices from the general linear group over a finite field
AU  - Ofir Gorodetsky
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29305
UR  - https://arxiv.org/abs/2610.29305
AB  - Let GL_n(𝔽_q) be the group of n× n invertible matrices over a finite field 𝔽_q. How many matrices in GL_n(𝔽_q) are there with a subset of their entries prescribed? Motivated by the random matrix theory of classical compact groups, we address this question and various variants, with a focus on asymptotic results that are valid as n tends to infinity. Our methods are Fourier-analytic, with the main input being an exponential sum over GL_n(𝔽_q), evaluated in 1955 by Hodges via elementary methods. Our results turn out to be related to investigations carried out in other fields, including coding theory, universality for p-adic matrices and algebraic combinatorics.
ER  -