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# Journal of Combinatorics

## Volume 13 (2022)

### Number 2

### Factorizations of $k$-nonnegative matrices

Pages: 201 – 250

DOI: https://dx.doi.org/10.4310/JOC.2022.v13.n2.a2

#### Authors

#### Abstract

A matrix is $k$-*nonnegative* if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $(n-1)$-nonnegative $n \times n$ invertible matrices and $(n-2)$-nonnegative $n \times n$ unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a topological structure that neatly relates closure of cells to subwords of factorizations. In the case of $(n-2)$-nonnegative unitriangular matrices, we show that the link of the identity forms a Bruhat-like CW complex, as in the Bruhat decomposition of unitriangular totally nonnegative matrices. Unlike the totally nonnegative case, we show this CW complex is not regular.

#### Keywords

matrix semigroup, $k$-nonnegativity, total positivity, Bruhat cell, Bruhat order

#### 2010 Mathematics Subject Classification

Primary 15A23, 22A15. Secondary 15A24, 57Q05.

Received 14 October 2019

Accepted 10 February 2021

Published 30 March 2022