Let Q be a bipartite quiver, V a real representation of Q, and σ an integral
weight of Q orthogonal to the dimension vector of V. Guided by quiver invariant theoretic considerations,
we introduce the Brascamp-Lieb operator associated to (V,σ) and study its capacity, denoted by D(V,σ).
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 D(V,σ) is equivalent to the σ-semi-stability of V. We also find a character
formula for D(V,σ) 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 σ 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 find necessary
and sufficient conditions for the existence and uniqueness of gaussian extremals for (V,σ). Finally, we explain how to
find the gaussian extremals of a gaussian-extremisable datum (V,σ) using the algebraic variety associated to (V,σ).
Last updated: 12/28/2020, 24 pages.