Freeness for the G-circulant Decomposition of the Partial Transpose of Random Matrices

@misc{arizmendi2026freeness,
  title = {{Freeness for the $G$-circulant Decomposition of the Partial Transpose of Random Matrices}},
  author = {Octavio Arizmendi and Julian Zazueta-Obeso},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09648},
  eprint = {2609.09648},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09648},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Freeness for the G-circulant Decomposition of the Partial Transpose of Random Matrices
AU  - Octavio Arizmendi
AU  - Julian Zazueta-Obeso
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09648
UR  - https://arxiv.org/abs/2609.09648
AB  - We introduce a left G-circulant decomposition for matrices indexed by an arbitrary finite group G, extending the diagonal decomposition associated with cyclic groups. We show that, when A∈ M_|G|(𝒜) is free from M_|G|(ℂ), the components arising from the left G-circulant decomposition of A^t form a free family, with the components associated with inverse pairs forming R-diagonal pairs. We also describe the distributions of these components in terms of the distribution of A. Our results recover the cyclic case and show that different group structures of the same order may lead to different free decompositions of the same matrix.
ER  -