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

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 ${\sum }_{\ell =1}^k{\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)}^2$ 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.

arXiv comment21 pages, 11 figures. Improved diagrams, fixed typos

References in corpus (3)

• Introduction to Cluster Algebras. Chapters 1-3
• Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples
• Double Rim Hook Cluster Algebras

Cited by in corpus (2)

• Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull
• 1324- and 2143-avoiding Kazhdan-Lusztig immanants and k-positivity