On the Identity Problem for Unitriangular Matrices of Dimension Four

TL;DR

This paper proves that determining whether a finitely generated sub-semigroup of 4x4 unitriangular integer matrices contains the identity matrix can be decided efficiently in polynomial time, along with related reachability problems.

Contribution

It establishes the polynomial-time decidability of the Identity Problem and subset reachability problems for sub-semigroups of 4x4 unitriangular integer matrices.

Findings

Identity Problem is decidable in polynomial time for $ ext{UT}(4, bZ)$

Polynomial-time algorithms for subset reachability problems

Advances understanding of matrix semigroup decision problems

Abstract

We show that the Identity Problem is decidable in polynomial time for finitely generated sub-semigroups of the group $\mathsf{UT}(4, \mathbb{Z})$ of $4 \times 4$ unitriangular integer matrices. As a byproduct of our proof, we also show the polynomial-time decidability of several subset reachability problems in $\mathsf{UT}(4, \mathbb{Z})$.