Alternating sign matrices of finite multiplicative order

TL;DR

This paper classifies certain finite order alternating sign matrices formed from permutation matrices plus a specific 4-entry matrix, revealing their orders can differ from permutation matrices and posing a subgroup identification problem.

Contribution

It provides a classification of a special class of alternating sign matrices with finite order and explores their properties in relation to permutation matrices.

Findings

Finite order matrices can differ from permutation matrices in order.

Classification of matrices of the form P+T with specific non-zero entries.

Open problem on identifying subgroups generated by such matrices.

Abstract

We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form $P+T$, where $P$ is a permutation matrix and $T$ has four non-zero entries, forming a square with entries $1$ and $-1$ in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.