Inequalities for Moment Cones of Finite-Dimensional Representations

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.

42 pages, to appear in Journal of Symplectic Geometry

References in corpus (6)

• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Quantum marginal problem and representations of the symmetric group
• Quantum state transformations and the Schubert calculus
• Multipartite Quantum States and their Marginals
• On the one-particle reduced density matrices of a pure three-qutrit quantum state
• A Basis for the GL_n Tensor Product Algebra

Cited by in corpus (16)

• Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes
• On vanishing of Kronecker coefficients
• Quantum Horn's lemma, finite heat baths, and the third law of thermodynamics
• Efficient algorithms for tensor scaling, quantum marginals and moment polytopes
• Generalized Pauli constraints in small atoms
• Locally Maximally Entangled States of Multipart Quantum Systems
• Membership in moment polytopes is in NP and coNP
• Computation of Dilated Kronecker Coefficients
• Permanent versus determinant, obstructions, and Kronecker coefficients
• The Horn inequalities from a geometric point of view
• Multiplicity of compact group representations and applications to Kronecker coefficients
• Vector partition functions and Kronecker coefficients
• Implications of pinned occupation numbers for natural orbital expansions. II: Rigorous derivation and extension to non-fermionic systems
• Ressayre's pairs in the Kähler setting
• Horn conditions for quiver subrepresentations and the moment map
• Refuting spectral compatibility of quantum marginals