Determinantally equivalent nonzero functions
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 F a field, and suppose that K,Q:2 are two functions such that for any n∈N and x1,x2,…,xn∈, the determinants of matrices (K(xi,xj))1≤ i,j≤ n and (Q(xi,xj))1≤ i,j≤ 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)-1f(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 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.