Multiple Horn problems for planar networks and invertible matrices

Abstract

The multiplicative multiple Horn problem is asking to determine possible singular values of the combinations $AB, BC$ and $ABC$ for a triple of invertible matrices $A,B,C$ with given singular values. There are similar problems for eigenvalues of sums of Hermitian matrices (the additive problem), and for maximal weights of multi-paths in concatenations of planar networks (the tropical problem). For the planar network multiple Horn problem, we establish necessary conditions, and we conjecture that for large enough networks they are also sufficient. These conditions are given by the trace equalities and rhombus inequalities (familiar from the hive description of the classical Horn problem), and by the new set of tetrahedron equalities. Furthermore, if one imposes Gelfand-Zeitlin conditions on weights of planar networks, tetrahedron equalities turn into the octahedron recurrence from the theory of crystals. We give a geometric interpretation of our results in terms of positive varieties with potential. In this approach, rhombus inequalities follow from the inequality $\Phi^t \leqslant 0$ for the tropicalized potential, and tetrahedron equalities are obtained as tropicalization of certain Pl\"ucker relations. For the multiplicative problem, we introduce a scaling parameter $s$, and we show that for $s$ large enough (corresponding to exponentially large/small singular values) the Duistermaat-Heckman measure associated to the multiplicative problem concentrates in a small neighborhood of the octahedron recurrence locus.