Matrix representations of the twisted virtual braid group and its extensions

2025/06/04 by Mohamad N. Nasser, Nasser, Mohamad N., Vaibhav Keshari +3 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2506.03806

Abstract

This paper classifies complex local representations of the twisted virtual braid group, TVB₂, into GL₃(ℂ). It shows that such representations fall into eight types, all of which are unfaithful and reducible to a two-dimensional representation. Further reducibility to a one-dimensional representation is analyzed for specific types. The paper also examines complex homogeneous local representations of TVB_n into GL_n+1(ℂ) for n ≥ 3, identifying seven unfaithful types. Additionally, complex local representations of the singular twisted virtual braid group, STVB₂, into M₃(ℂ) are classified into thirteen unfaithful types. Finally, the paper demonstrates that not all complex local extensions of TVB₂ representations to STVB₂ conform to a Φ-type extension.

Citations

• On extensions of the standard representation of the braid group to the singular braid group
• The monoid structure of singular twisted virtual braids
• Classification of Homogeneous Local Representations of the Singular Braid Monoid
• On the Faithfulness of a Family of Representations of the Singular Braid Monoid SM_n
• Extensions of braid group representations to the monoid of singular braids
• Twisted virtual braids and twisted links
• Parameters for which the Lawrence-Krammer representation is reducible
• Virtual Braids
• Virtual braids
• Braid Groups are Linear
• THE SINGULAR BRAID MONOID EMBEDS IN A GROUP

Cited by

• Twin groups representations

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• Virtual and universal braid groups, their quotients and representations
• Representations of flat virtual braids by automorphisms of free group
• Twisted Virtual Braid Group