Representations of unitary groups and free convolution
The main result of this paper is the following Theorem. Let \(S\) be a finite set, \(\{n_m\mid m\in \mathbb{N}\}\) a sequence such that \(n_m\to \infty\). Let for each \(m\geq 0\), \(R_m(s)\), \(s\in S\), be a family of independent isotypic representations of \(U(n_m)\) (i.e. the representations \(\widetilde R_m(s)\) are irreducible), and let \(\varepsilon_m\), \(m\geq 0\), be a sequence in \(\mathbb{R}_+\) such that: (i) for every \(r\in \mathbb{N}\), \(\varepsilon_m(n_m)^r\to 0\); (ii) there is a constant \(c\) such that \(\varepsilon_m|M^m(s)|\leq c\) for all \(m\in \mathbb{N}\), where \(M^m(s)\) is the highest weight of \(R_m(s)\); (iii) there exist probability distributions \(\alpha(s)\) such that \({\mathcal M}^{\varepsilon_m}(R_m(s))\to \alpha(s)\). Let furthermore \(D(t, m)_{t\in T}\) be a family of \(n_m\times n_m\) diagonal matrices which have a limit distribution as \(m\to \infty\) and such that \(\sup_m |D(t, m)|< \infty\) for all \(t\in T\). We consider them as elements of \(M_{n_m}(\mathbb{C})\otimes {\mathcal L}(v)\) by tensoring with \(\text{Id}_v\). Then \((C(R_m(s))_{s\in S}, \{D(t, m)_{t\in T}\})\) is an asymptotically free family of sets of random variables.
scientific article; zbMATH DE number 785366