R-cyclic families of matrices in free probability
Alexandru Nica, Dimitri Shlyakhtenko, and Roland Speicher
We introduce the concept of "R-cyclic family" of matrices with entries in a non-commutative probability space; the definition consists in asking that only the "cyclic" non-crossing cumulants of the entries of the matrices are allowed to be non-zero.
Let $A_{1}, \ldots , A_{s}$ be an R-cyclic family of $d \times d$ matrices over a non-commutative probability space $({\mathcal A}, \varphi)$. We prove a convolution-type formula for the explicit computation of the joint distribution of $A_{1}, \ldots , A_{s}$ (considered in $M_{d} ( {\mathcal A} )$ with the natural state), in terms of the joint distribution (considered in the original space $({\mathcal A}, \varphi)$) of the entries of the $s$ matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula.
Moreover, let $A_{1}, \ldots , A_{s}$ be a family of $d \times d$ matrices over a non-commutative probability space $({\mathcal A}, \varphi)$, let ${\mathcal D} \subset M_{d} ( {\mathcal A} )$ denote the algebra of scalar diagonal matrices, and let ${\mathcal C}$ be the subalgebra of $M_{d} ( {\mathcal A} )$ generated by $\{ A_{1}, \ldots , A_{s} \} \cup {\mathcal D}$. We prove that the R-cyclicity of $A_{1}, \ldots , A_{s}$ is equivalent to a property of ${\mathcal C}$ – namely that ${\mathcal C}$ is free from $M_{d} ( {\mathbf C} )$, with amalgamation over ${\mathcal D}$.