Factorizations of $k$-Nonnegative Matrices

TL;DR

This paper characterizes the structure of $k$-nonnegative matrices for specific cases, providing generators, relations, and a cell decomposition that generalizes Bruhat cells from totally nonnegative matrices.

Contribution

It introduces a parametrized set of generators and relations for $k$-nonnegative matrices when $k=n-1$ and $k=n-2$, and describes their cell decompositions.

Findings

Cells are homeomorphic to open balls.

The cell structure forms a Bruhat-like CW-complex.

Provides a topological understanding of $k$-nonnegative matrices.

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 $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to 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 the cells form a Bruhat-like CW-complex.