Extremal Rays in the Hermitian Eigenvalue Problem for Arbitrary Types

Abstract

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of G " SLpnq . There is a polyhedral cone (the "eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups G, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of G.We give a description of the extremal rays of the eigencones for arbitrary semisimple groups G by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face, and the remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize [Bel17] which handled the case of G " SLpnq.

On the Saturation Conjecture for Spin(2n)

“…We introduce a computationally feasible method (based on the polynomial realization of [BGG73]) for calculating cup products in the singular (or deformed) cohomology of any G{P , and we indicate some pseudocode for implementing this method on a computer in order to find the desired inequalities. This can also be used to find extremal rays of the cone from the formulas of [BK18]. The method lends itself to (partial) parallelization, and it was with the crucial aid of the parallel-capable supercomputer Longleaf, maintained at the University of North Carolina, that we obtained the aforementioned results.…”

“…The D 5 calculation can be carried out once the inequalities have been generated. However, for the D 6 calculation we will need to generate the extremal rays of CpGq according to the formulas given in [BK18], which we recall here. Every extremal ray of CpGq lies on a facet F p w, P q (see [BK18, Lemma 5.4]).…”

“…However, we found it is not computationally feasible to use Normaliz to find the Hilbert basis from these inequalities instead of from the rays mentioned above. That is to say, we use the formulas (5) and (7) of [BK18] in a crucial way.…”

“…At least in these low-rank situations, then, it seems the combination of Normaliz's primal algorithm on generators coming from formulas (5) and (7) of [BK18] is most efficient in determining the Hilbert basis of CpGq.…”

“…For part (a), we are able to produce the defining inequalities for CpGq and use software to deduce a generating set from these; this is exactly the [KKM09] approach. For part (b), the inequalities are too many in number, and accordingly we find a (redundant) set of extremal rays for CpGq based on the formulas of [BK18]; we then use software to deduce the minimal generating set from these rays.…”