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Symmetric Lorentzian Polynomials

Tracy Chin, Daniel Qin

TL;DR

This work extends the theory of Lorentzian polynomials to the realm of symmetric polynomials and symmetric functions, establishing that the space of Lorentzian symmetric polynomials forms a closed Euclidean ball and providing a constant-time reduction scheme for testing Lorentzianity. It delivers explicit semialgebraic descriptions for degrees up to six and develops a Hessian-based framework (via Haynsworth-type block matrices) that reduces Lorentzianity to a finite set of linear-algebraic conditions. The paper applies these tools to simplify proofs of Lorentzianity for chromatic symmetric functions and two-column Schur functions, and reveals inherent tensions by showing certain natural symmetric operations do not preserve Lorentzianity. Collectively, the results advance a practical, combinatorially tractable approach to Lorentzian symmetric structures and lay groundwork for further exploration of symmetry-constrained Lorentzianity in algebraic combinatorics.

Abstract

We study the class of Lorentzian symmetric polynomials and Lorentzian symmetric functions, which are defined to be symmetric functions for which every truncation of variables is Lorentzian. Similar to the space of Lorentzian polynomials, we show that the space of Lorentzian symmetric polynomials is homeomorphic to a closed Euclidean ball. Our main result is a reduction scheme that significantly reduces the complexity of testing for Lorentzianity. Using this method, we provide explicit semialgebraic descriptions of the spaces of Lorentzian symmetric polynomials and functions for degrees up to six. These techniques can also be applied to simplify the proofs to known cases of Lorentzian symmetric functions. We conclude by showing that some natural symmetric operators fail to preserve Lorentzianity which in turn highlights an inherent tension between symmetry in variables and the Lorentzian property.

Symmetric Lorentzian Polynomials

TL;DR

This work extends the theory of Lorentzian polynomials to the realm of symmetric polynomials and symmetric functions, establishing that the space of Lorentzian symmetric polynomials forms a closed Euclidean ball and providing a constant-time reduction scheme for testing Lorentzianity. It delivers explicit semialgebraic descriptions for degrees up to six and develops a Hessian-based framework (via Haynsworth-type block matrices) that reduces Lorentzianity to a finite set of linear-algebraic conditions. The paper applies these tools to simplify proofs of Lorentzianity for chromatic symmetric functions and two-column Schur functions, and reveals inherent tensions by showing certain natural symmetric operations do not preserve Lorentzianity. Collectively, the results advance a practical, combinatorially tractable approach to Lorentzian symmetric structures and lay groundwork for further exploration of symmetry-constrained Lorentzianity in algebraic combinatorics.

Abstract

We study the class of Lorentzian symmetric polynomials and Lorentzian symmetric functions, which are defined to be symmetric functions for which every truncation of variables is Lorentzian. Similar to the space of Lorentzian polynomials, we show that the space of Lorentzian symmetric polynomials is homeomorphic to a closed Euclidean ball. Our main result is a reduction scheme that significantly reduces the complexity of testing for Lorentzianity. Using this method, we provide explicit semialgebraic descriptions of the spaces of Lorentzian symmetric polynomials and functions for degrees up to six. These techniques can also be applied to simplify the proofs to known cases of Lorentzian symmetric functions. We conclude by showing that some natural symmetric operators fail to preserve Lorentzianity which in turn highlights an inherent tension between symmetry in variables and the Lorentzian property.

Paper Structure

This paper contains 23 sections, 32 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Background and Notation
  4. Lorentzian Polynomials
  5. Symmetric Polynomials
  6. Partitions
  7. General Theory on Lorentzian Symmetric Polynomials
  8. Symmetric M-Convex Sets
  9. Topology of the Space of Lorentzian symmetric Polynomials
  10. Hessians of Symmetric Polynomials
  11. Applying Reductions for Small Degrees
  12. Symmetric Quadratics
  13. Symmetric Cubics
  14. Symmetric Quartics
  15. Symmetric Quintics
  16. ...and 8 more sections

Key Result

Proposition 2.3

Let $A \in {\mathbb R}^{n\times n}_{\geq 0}$ be a symmetric matrix with nonnegative entries. Then $A$ has at most one positive eigenvalue if and only if for all nonempty subsets $S \subseteq [n]$, $(-1)^{|S| - 1}A_S \geq 0$, where $A_S = \det(A[S])$ denotes the principal minor with rows and columns

Figures (1)

  • Figure 1: Regions defined by the conditions of \ref{['thm:sym-cubic-lc-iff']} for $n = 2, 5$ and as $n\to\infty$, intersected with the hyperplane $a+b+c = 1$. Note that each region is homeomorphic to a closed Euclidean ball, as shown in \ref{['thm:proj sym lor=ball']}.

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Definition 3.1
  • Proposition 3.2
  • ...and 64 more