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Sign patterns of principal minors of real symmetric matrices

Tobias Boege, Jesse Selover, Maksym Zubkov

TL;DR

This work studies sign patterns arising from the signs of all principal minors of real symmetric matrices, encoding them as admissible patterns tied to uniform Lagrangian matroids. It develops a robust combinatorial framework of minors, duality, and hyperoctahedral symmetry, and analyzes the topology of representation spaces PR_N(s) across representable and admissible patterns. The authors prove that almost all admissible sign patterns are non-representable, establish asymptotic growth rates via a 3-coloring correspondence, and provide detailed results for PR_3 and small n, including a (likely) minimal forbidden minor for representability. They also explore the complexity of representability testing and the topology of spaces in leading-principal-minor scenarios, outlining open questions and computational challenges.

Abstract

We analyze a combinatorial rule satisfied by the signs of principal minors of a real symmetric matrix. The sign patterns satisfying this rule are equivalent to uniform oriented Lagrangian matroids. We first discuss their structure and symmetries and then study their asymptotics, proving that almost all of them are not representable by real symmetric matrices. We offer several conjectures and experimental results concerning representable sign patterns and the topology of their representation spaces.

Sign patterns of principal minors of real symmetric matrices

TL;DR

This work studies sign patterns arising from the signs of all principal minors of real symmetric matrices, encoding them as admissible patterns tied to uniform Lagrangian matroids. It develops a robust combinatorial framework of minors, duality, and hyperoctahedral symmetry, and analyzes the topology of representation spaces PR_N(s) across representable and admissible patterns. The authors prove that almost all admissible sign patterns are non-representable, establish asymptotic growth rates via a 3-coloring correspondence, and provide detailed results for PR_3 and small n, including a (likely) minimal forbidden minor for representability. They also explore the complexity of representability testing and the topology of spaces in leading-principal-minor scenarios, outlining open questions and computational challenges.

Abstract

We analyze a combinatorial rule satisfied by the signs of principal minors of a real symmetric matrix. The sign patterns satisfying this rule are equivalent to uniform oriented Lagrangian matroids. We first discuss their structure and symmetries and then study their asymptotics, proving that almost all of them are not representable by real symmetric matrices. We offer several conjectures and experimental results concerning representable sign patterns and the topology of their representation spaces.
Paper Structure (17 sections, 17 theorems, 37 equations, 2 figures, 2 tables)

This paper contains 17 sections, 17 theorems, 37 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Minors, duality and symmetry
  3. Minors and duality
  4. Negation and the hyperoctahedral symmetry
  5. Actions on representations and topology
  6. Case study of PR_3
  7. Enumeration and asymptotics
  8. Enumerating sign patterns
  9. Asymptotics of admissible sign patterns
  10. Asymptotics of representable sign patterns
  11. Representability and topology
  12. Representability for n <= 5
  13. Topology for reducible sign patterns
  14. Further remarks
  15. Complexity of representability testing
  16. ...and 2 more sections

Key Result

Lemma 1

For any $\Sigma \in \mathrm{Sym}_N$, $i \neq j$ and $K \subseteq N \setminus ij$, the following polynomial identity holds: Consequently $[\mathop{\mathrm{signs}}\nolimits_\Sigma(iK) \neq \mathop{\mathrm{signs}}\nolimits_\Sigma(jK)] \implies [\mathop{\mathrm{signs}}\nolimits_\Sigma(K) \neq \mathop{\mathrm{signs}}\nolimits_\Sigma(ijK)]$.

Figures (2)

  • Figure 1: The arrangement of hypersurfaces $\setsepchar{:} \ignoreemptyitems \readlist*\mylist{K:\Sigma} \mylist[2] [\mylist[1]] = 0$ whose complement is $\widehat{\mathrm{PR}}_3$ as well as the pieces $\widehat{\mathrm{PR}}_3(s)$ for the $\mathfrak{B}_3$ representatives $s$ from \ref{['tab:B3']}, together in the middle and and separately (on different scales) on the right.
  • Figure 2: An admissible sign pattern $s_*$ of size 5 for which we have no representation, depicted as a coloring of the lattice of subsets of $\{1, \ldots, 5\}$. Positive signs are in orange, and negative in blue.

Theorems & Definitions (53)

  • Lemma
  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • ...and 43 more