The capacity of quiver representations and the Anantharam-Jog-Nair inequality

The Anantharam-Jog-Nair inequality [AJN22] in Information Theory provides a unifying approach to the information-theoretic form of the Brascamp-Lieb inequality [CCE09] and the Entropy Power inequality [ZF93]. In this paper, we use methods from Quiver Invariant Theory [CD21] to study Anantharam-Jog-Nair inequalities with integral exponents. For such an inequality, we first view its input datum as a quiver datum and show that the best constant that occurs in the Anantharam-Jog-Nair inequality is of the form $-{1\over 2}\log (\mathbf{cap}(V,\sigma))$ where $\mathbf{cap}(V, \sigma)$ is the capacity of a quiver datum $(V, \sigma)$ of a complete bipartite quiver. The general tools developed in [CD21], when applied to complete bipartite quivers, yield necessary and sufficient conditions for: (1) the finiteness of the Anantharam-Jog-Nair best constants; and (2) the existence of Gaussian extremizers. These results recover some of the main results in [AJN22] and [ACZ22]. In addition, we characterize gaussian-extremizable data in terms of semi-simple data, and provide a general character formula for the Anatharam-Jog-Nair constants. Furthermore, our quiver invariant theoretic methods lead to necessary and sufficient conditions for the uniqueness of Gaussian extremizers. This answers the third and last question left unanswered in [AJN22].