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.

Efficient Algorithms for Tensor Scaling, Quantum Marginals, and Moment Polytopes

“…We emphasize again the amazing, surprising and non-trivial fact that ∆pX q is a rational convex polytope [NM84, Kir84b, Kir84a, Bri87] -known as the moment polytope or Kirwan polytope of X . This means that ∆pX q can in principle be given in terms of finitely many affine inequalities in eigenvalues of the one-body marginals [Kly06,Res10,VW17]. In particular, the preceding applies to X " PpV q, so we can rephrase Problem 1.1 as follows: Given p P P`pn 1 , .…”

“…Before that, only low-dimensional special cases were known [HSS03,Bra03,Fra02]. Further developments include the minimal complete description from [Res10] and the cohomology-free variant [VW17]. Yet, all these descriptions in terms of We note that the closely related Littlewood-Richardson coefficients (which capture the same problem for the representations of the general linear group) satisfy the so called saturation property: cpλ, µ, νq ą 0 iff cpsλ, sµ, sνq ą 0 [KT99].…”

“…inequalities are largely computationally infeasible; explicit descriptions are known up to formats 3ˆ3ˆ9 [Kly06] and 4ˆ4ˆ4 [VW17], and when all dimensions are two [HSS03]. Problems 1.3 and 1.4 can in principle be approached using classical computational invariant theory (e.g., [Stu08,DK15,WDGC13]), based on the invariant-theoretic description of ∆pX q and degree bounds (we recall both in Section 2).…”

On the asymptotics of Kronecker coefficients

“…For example Klyachko in [10] suggested to use the results of Berenstein and Sjamaar, and applied them in small cases. This method has recently been improved in [29]. This approach has two important limitations.…”

Implications of pinned occupation numbers for natural orbital expansions. II: rigorous derivation and extension to non-fermionic systems

“…The solution of one-body quantum marginal problem is obtained mutatis mutandis to the case of fermions as all the essential mathematical structures are also present for distinguishable particles, i.e. ( )   is a Kähler manifold and μ arises as the momentum map of the local unitary action on the space of states (see [5,6,[11][12][13][14][15][16][17][18][19][20][21][22][23] for more examples of the usage of geometric techniques in quantum information). A similar observation applies to bosons.…”