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PreprintarXiv licencearXiv

Range-compatible homomorphisms on Hermitian matrices

  • ClΓ©ment de Seguins Pazzis
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Abstract

Let 𝔻 be a division ring with an involution x ↦ x⋆, and n β‰₯ 2 be an integer. Denote by β„‹n(𝔻) the set of all n-by-n Hermitian matrices with entries in 𝔻, and by π’œβ„‹n(𝔻) the set of all matrices Aβˆ’A⋆ with A ∈ β„³n(𝔻). Here, we give a complete solution to the following problem: Determine all group homomorphisms from β„‹n(𝔻) to 𝔻n (respectively, from π’œβ„‹n(𝔻) to 𝔻n unless (βˆ’)⋆ is the identity) that take every matrix to a right linear combination of its columns. The solution to this problem was already known when (βˆ’)⋆ is the identity, and the novelty here lies in the generalization to arbitrary involutions, and in particular in the noncommutative case. These results are to be used in a subsequent article on subspaces of Hermitian matrices of bounded rank, and on large spaces of diagonalisable matrices.

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Cite

BibTeXRIS
BibTeX
@misc{pazzis2026rangecompatible,
  title = {{Range-compatible homomorphisms on Hermitian matrices}},
  author = {ClΓ©ment de Seguins Pazzis},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20363},
  eprint = {2609.20363},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20363},
}
RIS
TY  - UNPB
TI  - Range-compatible homomorphisms on Hermitian matrices
AU  - ClΓ©ment de Seguins Pazzis
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20363
UR  - https://arxiv.org/abs/2609.20363
AB  - Let 𝔻 be a division ring with an involution x ↦ x^⋆, and n β‰₯ 2 be an integer. Denote by β„‹_n(𝔻) the set of all n-by-n Hermitian matrices with entries in 𝔻, and by π’œβ„‹_n(𝔻) the set of all matrices Aβˆ’A^⋆ with A ∈ β„³_n(𝔻). Here, we give a complete solution to the following problem: Determine all group homomorphisms from β„‹_n(𝔻) to 𝔻^n (respectively, from π’œβ„‹_n(𝔻) to 𝔻^n unless (βˆ’)^⋆ is the identity) that take every matrix to a right linear combination of its columns. The solution to this problem was already known when (βˆ’)^⋆ is the identity, and the novelty here lies in the generalization to arbitrary involutions, and in particular in the noncommutative case. These results are to be used in a subsequent article on subspaces of Hermitian matrices of bounded rank, and on large spaces of diagonalisable matrices.
ER  -