Matrix representations of the twisted virtual braid group and its extensions

Abstract

This paper classifies complex local representations of the twisted virtual braid group, $TV{B}_2$, into ${\mathrm{GL}}_3(\mathbb{C})$. 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 $TV{B}_n$ into ${\mathrm{GL}}_{n+1}(\mathbb{C})$ for $n\ge 3$, identifying seven unfaithful types. Additionally, complex local representations of the singular twisted virtual braid group, $STV{B}_2$, into ${\mathrm{M}}_3(\mathbb{C})$ are classified into thirteen unfaithful types. Finally, the paper demonstrates that not all complex local extensions of $TV{B}_2$ representations to $STV{B}_2$ conform to a $\hat{I}¦$-type extension.