Geometry and algorithms for upper triangular tropical matrix identities

TL;DR

This paper introduces geometric methods and algorithms to verify and construct identities in upper triangular tropical matrix semigroups, linking algebraic identities to lattice polytope signatures and providing new minimal identities.

Contribution

It develops polyhedral algorithms for identity verification in $ ext{UT}_n$, proves a structural theorem for $ ext{UT}_2$, and constructs minimal identities for $ ext{UT}_3$, challenging existing conjectures.

Findings

Algorithms achieve optimal complexity in some cases.

A structural theorem enables enumeration of identities for $ ext{UT}_2$.

Constructed minimal identities for $ ext{UT}_3$ and counterexamples to Izhakian's conjecture.

Abstract

We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed $m$-letter alphabet) that form identities in the semigroup $\ut{n}$ of $n\times n$ upper triangular tropical matrices. In the case $n=2$ these identities are precisely those satisfied by the bicyclic monoid, whilst in the case $n=3$ they form a subset of the identities which hold in the plactic monoid of rank $3$. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for $\ut{n}$ if and only if their signatures are equal. Our algorithms are thus based on polyhedral computations and achieve optimal complexity in some cases. For $n=m=2$ we prove a Structural Theorem, which allows us to quickly enumerate the pairs of words of fixed length which form identities for $\ut{2}$. This allows us to recover a short proof of Adjan's theorem on minimal length identities for the bicyclic monoid, and to construct minimal length identities for $\ut{3}$, providing counterexamples to a conjecture of Izhakian in this case. We conclude with six conjectures at the intersection of semigroup theory, probability and combinatorics, obtained through analysing the outputs of our algorithms.