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Determinantally equivalent nonzero functions

Harry Sapranidis Mantelos

TL;DR

The paper addresses when two determinantally equivalent kernels $K$ and $Q$ must be related by simple transformations. It shows that in the general non-symmetric setting such a classification fails without extra non-vanishing-cycle hypotheses, and provides a natural, minimal set of conditions under which $Q$ can be obtained from $K$ by conjugation and transposition only. The authors present an elementary, graph-theoretic proof that relies on cycle identities for $3$- and $4$-cycles, avoiding heavy linear-algebraic machinery, and confirm the result in finite $\Lambda$ via a four-element case analysis. The results refine the understanding of determinantally equivalent kernels, with implications for discrete determinantal point processes and related matrix problems, by isolating the transformations that preserve determinant structure under non-symmetric settings.

Abstract

We study the problem raised in [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples. The problem is plainly stated as follows: Let $Λ$ be a set and $\mathbb{F}$ a field, and suppose that $K,Q:Λ^2\to\mathbb{F}$ are two functions such that for any $n\in\mathbb{N}$ and $x_1,x_2,\ldots,x_n\inΛ$, the determinants of matrices $(K(x_i,x_j))_{1\leq i,j\leq n}$ and $(Q(x_i,x_j))_{1\leq i,j\leq n}$ agree. What are all the possible transformations that transform $Q$ into $K$? In [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] the following two were conjectured: $(Tf)(x,y)=f(y,x)$; and $(Tf)(x,y)=g(x)g(y)^{-1}f(x,y)$ for some nowhere-zero function $g$. In the same paper, this conjectured classification is verified in the case of symmetric functions $K$ and $Q$. By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed. By taking $Λ$ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23--64]. Hence, our paper presents a simple non-linear-algebraic proof that uses only some elementary combinatorics and three simple algebraic identities involving $3$-cycles and $4$-cycles.

Determinantally equivalent nonzero functions

TL;DR

The paper addresses when two determinantally equivalent kernels and must be related by simple transformations. It shows that in the general non-symmetric setting such a classification fails without extra non-vanishing-cycle hypotheses, and provides a natural, minimal set of conditions under which can be obtained from by conjugation and transposition only. The authors present an elementary, graph-theoretic proof that relies on cycle identities for - and -cycles, avoiding heavy linear-algebraic machinery, and confirm the result in finite via a four-element case analysis. The results refine the understanding of determinantally equivalent kernels, with implications for discrete determinantal point processes and related matrix problems, by isolating the transformations that preserve determinant structure under non-symmetric settings.

Abstract

We study the problem raised in [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples. The problem is plainly stated as follows: Let be a set and a field, and suppose that are two functions such that for any and , the determinants of matrices and agree. What are all the possible transformations that transform into ? In [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] the following two were conjectured: ; and for some nowhere-zero function . In the same paper, this conjectured classification is verified in the case of symmetric functions and . By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed. By taking finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23--64]. Hence, our paper presents a simple non-linear-algebraic proof that uses only some elementary combinatorics and three simple algebraic identities involving -cycles and -cycles.
Paper Structure (6 sections, 15 theorems, 40 equations)

This paper contains 6 sections, 15 theorems, 40 equations.

Table of Contents

  1. Introduction and main result
  2. Counterexamples to the conjecture
  3. Terminology and notations
  4. Proof of the symmetric case
  5. Cycles on $4$ vertices
  6. Proof of the main result

Key Result

Theorem 1.1

Let $\Lambda$ be a set, $\mathbb{F}$ a field, and let $K,Q:\Lambda^2\to\mathbb{F}$ be two (not necessarily symmetric) nowhere-zero functions, except possibly on the set $\{(x,x):x\in\Lambda\}$. Suppose further that for every pairwise distinct $x,y,z,w\in\Lambda$, If $K$ and $Q$ are determinantally equivalent, then $Q$ can be transformed into $K$ through only conjugation and transposition transfor

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 4.1: Modified Theorem 1.5 from equiv_symm_kernels_for_dpps
  • Proposition 4.2
  • Proposition 4.3
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Lemma 6.1
  • Lemma 6.2
  • Proposition 6.3
  • ...and 5 more