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

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