On Invariant Conjugate Symmetric Statistical Structures on the Space of Zero-Mean Multivariate Normal Distributions

By the results of Furuhata–Inoguchi–Kobayashi [Inf. Geom. (2021)] and Kobayashi–Ohno [Osaka Math. J. (2025)], the Amari–Chentsov $\alpha$-connections on the space $\mathcal{N}$ of all $n$-variate normal distributions are uniquely characterized by the invariance under the transitive action of the affine transformation group among all conjugate symmetric statistical connections with respect to the Fisher metric. In this paper, we investigate the Amari–Chentsov $\alpha$-connections on the submanifold ${\mathcal{N}}_0$ consisting of zero-mean $n$-variate normal distributions. It is known that ${\mathcal{N}}_0$ admits a natural transitive action of the general linear group $GL(n,\mathbb{R})$. We establish a one-to-one correspondence between the set of $GL(n,\mathbb{R})$-invariant conjugate symmetric statistical connections on ${\mathcal{N}}_0$ with respect to the Fisher metric and the space of homogeneous cubic real symmetric polynomials in $n$ variables. As a consequence, if $n\ge 2$, we show that the Amari–Chentsov $\alpha$-connections on ${\mathcal{N}}_0$ are not uniquely characterized by the invariance under the $GL(n,\mathbb{R})$-action among all conjugate symmetric statistical connections with respect to the Fisher metric. Furthermore, we show that any invariant statistical structure on a Riemannian symmetric space is necessarily conjugate symmetric.