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 ${\sum }_{\ell =1}^k{\left(\begin{array}{c}n\\ \ell \end{array}\right)}^2$ 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.

中文翻译：

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k负矩阵的参数化：簇代数和k阳性检验

甲ķ阳性矩阵是一个矩阵，其中订单的所有未成年人ķ或更少是积极的。计算所有这样的未成年人以测试k阳性是无效的，因为${\sum }_{\ell =\mathrm{1个}}^{ķ}{（\begin{array}{c}ñ\\ \ell \end{array}）}^2$ 他们在 $ñ\times ñ$矩阵。但是，只有极少数的k阳性测试仅需要测试${ñ}^2$未成年人。这些最小测试可以通过一系列交换来关联，并形成总阳性测试的簇代数的子簇代数族。我们给出子集群代数的描述，这些子集群给出k正性测试，在它们之间移动的方式以及许多测试的替代组合描述。