GIG random matrices and a Yang-Baxter extension of the Matsumoto-Yor property

Sasada and Uozumi, \cite{SasUoz2024}, identified independence preserving $[2:2]$ quadrirational parametric Yang-Baxter maps, see \eqref{YBEQ}, on $(0,\infty)$. In particular, the map denoted there by $H_{III,B}^{(α,β)}$, see \eqref{CS}, was connected to the independence preserving property of the GIG distributions on $(0,\infty)$. Remarkably, the property appears also naturally in probabilistic integrable models of discrete Korteweg de Vries type, as observed by Croydon and Sasada, \cite{CroSas2020}. In the case of $(α,β)=(1,0)$ the independence reduces to the classical Matsumoto-Yor property, \cite{MatYor2001}. In \cite{LetWes2024} we proposed an extension of $H_{III,B}^{(α,β)}$ to a map on the cone of symmetric positive definite matrices of a fixed dimension, showing that such extended map preserves independence of GIG random matrices. In the present paper we prove two results: (i) the matrix GIG distributions are characterized by the independence property governed by this map; (ii) the matrix variate extension of $H_{III,B}^{(α,β)}$ we use, is a parametric Yang-Baxter map.