For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes the group of invertible stochastic matrices, i.e., the set of matrices (with entries of either sign) whose row sums are unity. Also, write $\langle R \rangle \equiv \pi$ for $R \in \langle \pi \rangle$. $\langle \pi \rangle$ is a $(n-1)^2$-dimensional Lie group under matrix multiplication. Its Lie algebra $\mathfrak{lie}(\langle \pi \rangle)$ has a basis of the form
$$e_{(j,k)}^{\langle \pi \rangle} := e_{(j,k)} - \frac{\pi_j}{\pi_n}e_{(n,k)}, \quad 1 \le j, k \le n-1$$
where $e_{(j,k)} = e_j(e_k^* - e_n^*)$ and we use typical notation for the standard basis of $\mathbb{R}^n$. A block matrix decomposition shows that $\mathfrak{lie}(\langle \pi \rangle) \cong \mathfrak{gl}(n-1)$, and it is easy to show that
$$\exp Ce_{(j,k)}^{\langle \pi \rangle} = I + \frac{e^{C(\pi_j/\pi_n + \delta_{jk})} - 1}{\pi_j/\pi_n + \delta_{jk}} e_{(j,k)}^{\langle \pi \rangle}.$$
In particular, this object seems pretty tractable (although I haven't bothered to think about the Haar measure).
My question is, has it been used in Bayesian statistics (or elsewhere)? It seems naturally suited for such an application.
