On Word Representations and Embeddings in Complex Matrices

TL;DR

This paper explores embeddings of word structures into low-dimensional complex matrix semigroups, developing new techniques and representations, especially for Euclidean Bianchi groups, to analyze decision problems in matrix semigroups.

Contribution

It introduces novel methods for constructing word representations in 2x2 complex matrices and applies them to Euclidean Bianchi groups, advancing understanding of matrix semigroup embeddings.

Findings

Developed new techniques for word embeddings in 2x2 complex matrices.

Constructed representations for Euclidean Bianchi groups.

Provided a framework for analyzing decision problems in matrix semigroups.

Abstract

Embeddings of word structures into matrix semigroups provide a natural bridge between combinatorics on words and linear algebra. However, low-dimensional matrix semigroups impose strong structural restrictions on possible embeddings. Certain finitely generated groups admit faithful representations in SL(2, C) and other similar matrix groups. On the other hand, it is known that the product of two free semigroups on two generators cannot be embedded into the 2x2 complex matrices. In this paper we study embeddings of word structures into low-dimensional matrix semigroups over the complex numbers and develop new techniques for constructing word representations of the Euclidean Bianchi groups. These representations provide a symbolic framework and a natural first step towards analysing fundamental decision problems in 2x2 matrix semigroups.