Journal of Symplectic Geometry

Volume 15 (2017)

Number 4

Inequalities for moment cones of finite-dimensional representations

Pages: 1209 – 1250



Michèle Vergne (Institut Mathématique de Jussieu, Université Paris 7 Diderot, Paris, France)

Michael Walter (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands; Institute for Theoretical Physics, Stanford University, Stanford, California, U.S.A.; and Institute for Theoretical Physics, ETH Zürich, Switzerland)


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.

Received 14 April 2015

Accepted 27 January 2016

Published 28 November 2017