Parametrizations of $k$-Nonnegative Matrices: Cluster Algebras and $k$-Positivity Tests

TL;DR

This paper explores efficient methods for testing $k$-positivity in matrices using cluster algebra structures, reducing the computational complexity from testing many minors to a minimal set of $n^2$ minors.

Contribution

It characterizes the sub-cluster algebras corresponding to $k$-positivity tests and provides methods to transition between these tests, offering a combinatorial perspective.

Findings

Minimal $k$-positivity tests require testing only $n^2$ minors.

Sub-cluster algebras encode different $k$-positivity tests.

An alternative combinatorial description of many tests is provided.

Abstract

A $k$-positive matrix is a matrix where all minors of order $k$ or less are positive. Computing all such minors to test for $k$-positivity is inefficient, as there are $\sum_{\ell=1}^k \binom{n}{\ell}^2$ of them in an $n\times n$ matrix. However, there are minimal $k$-positivity tests which only require testing $n^2$ minors. These minimal tests can be related by series of exchanges, and form a family of sub-cluster algebras of the cluster algebra of total positivity tests. We give a description of the sub-cluster algebras that give $k$-positivity tests, ways to move between them, and an alternative combinatorial description of many of the tests.