Inequalities for Moment Cones of Finite-Dimensional Representations

TL;DR

This paper provides a comprehensive description of moment cones for finite-dimensional representations of compact Lie groups, introducing new inequalities and geometric interpretations relevant to physics and algebra.

Contribution

It generalizes Horn's inequalities, introduces new inequalities for quantum marginal problems, and offers a geometric perspective on tensor product invariants.

Findings

General description of moment cones via linear inequalities

New inequalities for quantum marginal problems

Geometric interpretation of tensor product invariants

Abstract

We give a general description of the moment cone associated with an arbitrary finite-dimensional unitary representation of a compact, connected Lie group in terms of finitely many linear inequalities. Our method is based on combining differential-geometric arguments with a variant of Ressayre's notion of a dominant pair. As applications, we obtain generalizations of Horn's inequalities to arbitrary representations, new inequalities for the one-body quantum marginal problem in physics, which concerns the asymptotic support of the Kronecker coefficients of the symmetric group, and a geometric interpretation of the Howe-Lee-Tan-Willenbring invariants for the tensor product algebra.