The capacity of quiver representations and Brascamp-Lieb constants

Abstract

Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp-Lieb operator $T_{V,\sigma}$ associated to $(V,\sigma)$ and study its capacity, denoted by $\mathbf{D}_Q(V, \sigma)$. When $Q$ is the $m$-subspace quiver, the capacity of quiver data is intimately related to the Brascamp-Lieb constants that occur in the $m$-multilinear Brascamp-Lieb inequality in analysis. We show that the positivity of $\mathbf{D}_Q(V, \sigma)$ is equivalent to the $\sigma$-semi-stability of $V$. We also find a character formula for $\mathbf{D}_Q(V, \sigma)$ whenever it is positive. Our main tool is a quiver version of a celebrated result of Kempf-Ness on closed orbits in invariant theory. This result leads us to consider certain real algebraic varieties that carry information relevant to our main objects of study. It allows us to express the capacity of quiver data in terms of the character induced by $\sigma$ and sample points of the varieties involved. Furthermore, we use this character formula to prove a factorization of the capacity of quiver data. We also show that the existence of gaussian extremals for $(V, \sigma)$ is equivalent to $V$ being $\sigma$-polystable, and that the uniqueness of gaussian extremals implies that $V$ is $\sigma$-stable. Finally, we explain how to find the gaussian extremals of a gaussian-extremisable datum $(V, \sigma)$ using the algebraic variety associated to $(V,\sigma)$.