Insights on the Homogeneous 3-local representations of the Twin Groups
Keywords:
Twin Groups, Braid Groups, Irreducibility, FaithfulnessAbstract
We provide a complete classification of the homogeneous 3-local representations of the twin group T_n, the virtual twin group VT_n, and the welded twin group WT_n, for all n\geq 4. Beyond this classification, we examine the main characteristics of these representations, particularly their irreducibility and faithfulness. More deeply, we show that all such representations are reducible, and most of them are unfaithful. Also, we find necessary and sufficient conditions of the first two types of the classified representations of T_n to be irreducible in the case n=4. The obtained results provide insights into the algebraic structure of these three groups.
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