Quantum invariant families of matrices in free probability

TL;DR

This paper characterizes families of infinite matrices of noncommutative random variables invariant under free quantum group conjugation, revealing a novel operator-valued R-cyclicity condition that contrasts classical exchangeability.

Contribution

It provides a complete operator-valued R-cyclicity characterization of invariant matrix families under free orthogonal and hyperoctahedral groups, extending free probability theory.

Findings

Invariant families characterized by R-cyclicity

Contrast with classical exchangeability results

New operator-valued invariance conditions

Abstract

We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued $R$-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.