Groups of singular alternating sign matrices

TL;DR

This paper explores the structure of groups formed entirely by singular alternating sign matrices (ASMs), demonstrating that any finite group can be represented within this framework and analyzing properties like size, rank, and order.

Contribution

It introduces constructions of groups of singular ASMs and proves that every finite group is isomorphic to such a group, with a singular idempotent ASM as the identity.

Findings

Every finite group is isomorphic to a group of singular ASMs

Singular idempotent ASM can serve as the identity element

Relationships between size, rank, and orders of singular ASMs are characterized

Abstract

We investigate multiplicative groups consisting entirely of singular alternating sign matrices (ASMs), and present several constructions of such groups. It is shown that every finite group is isomorphic to a group of singular ASMs, with a singular idempotent ASM as its identity element. The relationship between the size, the rank, and the possible multiplicative orders of singular ASMs is explored.