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

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 $A\in M_{\mathrm{\mid }G\mathrm{\mid }}(\mathcal{A})$ is free from $M_{\mathrm{\mid }G\mathrm{\mid }}(\mathbb{C})$, 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.