On the classification and irreducibility of 2-local representations of the twin group T_n

2025/08/20 by Taher I. Mayassi, Mayassi, Taher I., Mohamad N. Nasser +1 · 1 citation
Mathematics · #20F36 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2508.14505

openalex publication_date 2025/08/20 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We investigate the homogeneous 2-local representations of the twin group T_n for all integers n\geqslant 2. A complete classification is obtained, yielding three distinct families of representations. We show that each of these families is reducible by explicitly constructing one-dimensional invariant subspaces, with particular emphasis on the first family, namely ξ₁: T_n → GL_n(ℂ). Passing to the corresponding quotients, we construct a reduced representation of ξ₁, namely ξ₁: T_n → GL_n-1(ℂ). The core of the paper is that we establish, through a precise criterion, a necessary and sufficient condition for the irreducibility of the representation ξ₁.

Citations

• On representations of the multi-virtual braid group M_kVB_n and the multi-welded braid group M_kWB_n
• Twin groups representations
• Local Representations of the Flat Virtual Braid Group
• Classification of Homogeneous Local Representations of the Singular Braid Monoid
• Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups

Cited by

• Algebraic and topological aspects of the singular twin group and its representations

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