The Capacity of Quiver Representations and Brascamp–Lieb Constants

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