On Word Representations and Embeddings in Complex Matrices

Bell, PC, Kenison, G, Niskanen, R ORCID: 0000-0002-2210-1481, Potapov, I and Semukhin, P ORCID: 0000-0002-7547-6391 On Word Representations and Embeddings in Complex Matrices. In: Lecture Notes in Computer Science (LNCS) Series . (Developments in Language Theory 2026, 30th - 3rd July 2026, Rouen, France). (Accepted)

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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 2×2 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 2×2 matrix semigroups.

Item Type:	Conference or Workshop Item (Paper)
Subjects:	Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Divisions:	Computer Science and Mathematics
Publisher:	Springer-Verlag
Date of acceptance:	10 April 2026
Date Deposited:	20 Apr 2026 14:39
Last Modified:	20 Apr 2026 14:39
URI:	https://researchonline.ljmu.ac.uk/id/eprint/28411

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