On Word Representations and Embeddings in Complex Matrices

2026-04-16

AI Summary

This study addresses the problem of embedding word structures into low-dimensional complex matrix semigroups, aiming to overcome their inherent structural limitations. By integrating combinatorial word theory with representation-theoretic techniques from complex matrix groups such as SL(2, ℂ), the work proposes a novel approach that successfully constructs word representations for Euclidean Bianchi groups. This method transcends the constraints of conventional low-dimensional matrix embeddings and establishes the first symbolic word embedding framework tailored to 2×2 complex matrix semigroups. The resulting framework provides a rigorous theoretical foundation for the systematic analysis of fundamental decision problems—including membership and equivalence—within this algebraic setting.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Natural Language Processing: Lexical Semantics and Morphology

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• word embeddings
• matrix semigroups
• complex matrices
• Bianchi groups
• decision problems

Innovation

• word embeddings
• matrix semigroups
• complex matrices
• Bianchi groups
• symbolic representation