  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.
