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On the stabilizer of the graph of linear functions over finite fields

  • Valentino Smaldore , Corrado Zanella and Ferdinando Zullo EMAIL logo
Published/Copyright: January 13, 2025
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 6

Abstract

In this paper we will study the action of 𝔽 q n 2 × 2 on the graph of an 𝔽 q -linear function of 𝔽 q n to itself. In particular, we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also give some examples where this is not the case. We will also connect such a stabilizer to the right idealizer of the rank-metric code defined by the linear function, and give some structural results in the case where the polynomials are partially scattered.

Keywords: Linearized polynomial; scattered polynomial; rank-metric code
MSC 2020: 11T06

Communicated by Philipp Habegger

Funding statement: This research was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). The research of the first and third authors was supported by the INdAM - GNSAGA project Tensors over finite fields and their applications, number E53C23001670001. The research of the third author was supported by the “VALERE: VAnviteLli pEr la RicErca” project of the University of Campania “Luigi Vanvitelli” and by the COMBINE from “VALERE: VAnviteLli pEr la RicErca” project of the University of Campania “Luigi Vanvitelli”.

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Received: 2023-10-06
Revised: 2024-11-12
Published Online: 2025-01-13
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
https://doi.org/10.1515/forum-2023-0353
Keywords for this article
Linearized polynomial; scattered polynomial; rank-metric code

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
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