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Multiple Horn problems for planar networks and invertible matrices

Anton Alekseev, Arkady Berenstein, Anfisa Gurenkova, Yanpeng Li

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 $Φ^t \leqslant 0$ for the tropicalized potential, and tetrahedron equalities are obtained as tropicalization of certain Plücker 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.

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 and for a triple of invertible matrices 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 for the tropicalized potential, and tetrahedron equalities are obtained as tropicalization of certain Plücker relations. For the multiplicative problem, we introduce a scaling parameter , and we show that for 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.

Paper Structure

This paper contains 25 sections, 46 theorems, 181 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Horn problems
  4. Three Horn problems
  5. Comparison of Horn problems
  6. Solution of the Horn problem
  7. Multiple Horn problems
  8. Three versions of the problem
  9. Main results
  10. Planar networks
  11. Preliminaries
  12. Tropical multiple Horn problem
  13. Results
  14. Proofs
  15. Positive functions and inequalities
  16. ...and 10 more sections

Key Result

Theorem 2.1

For all $s \in \mathbb{R}_{\neq 0}$, we have and for all $\lambda, \mu$ the corresponding Duistermaat-Heckman measures coincide: For all $\Pi_1, \Pi_2$, we have and for $\Pi_1, \Pi_2$ sufficiently large networks ( e.g standard networks) this inclusion is an equality.

Figures (20)

  • Figure 1: A tropical line: $\max\{\alpha, \beta, \gamma\}$.
  • Figure 2:
  • Figure 3: A planar network of rank 3. In red is a 2-multipath of weight $(b\cdot e)\cdot d$.
  • Figure 4: A concatenation of two planar networks. In red is a (1,1)-multipath of weight $b_1\cdot e_1\cdot a_2\cdot d_2\cdot d_1$.
  • Figure 5:
  • ...and 15 more figures

Theorems & Definitions (108)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: Theorems \ref{['theorem PN rhombi inequalities']} and \ref{['theorem PN tetrahedra equalities']}
  • Conjecture A
  • Theorem 2.7: Theorem \ref{['theorem PN octahedron recurrence']}
  • Theorem 2.8: Theorems \ref{['Thm:octrec']} and \ref{['thm:sumofmt3']}
  • Conjecture B
  • ...and 98 more