Matrix representations of the twisted virtual braid group and its extensions

This paper classifies complex local representations of the twisted virtual braid group, ${\mathrm{T}\mathrm{V}\mathrm{B}}_2$, into ${\mathrm{G}\mathrm{L}}_3(\mathbb{C})$. It shows that such representations fall into eight types, all of which are unfaithful and reducible to degree $2\times 2$. Further reducibility to degree 1 is analyzed for specific types. The paper also examines complex homogeneous local representations of ${\mathrm{T}\mathrm{V}\mathrm{B}}_n$ into ${\mathrm{G}\mathrm{L}}_{n+1}(\mathbb{C})$ for $n\ge 3$, identifying seven unfaithful types. Additionally, complex local representations of the singular twisted virtual braid group, ${\mathrm{S}\mathrm{T}\mathrm{V}\mathrm{B}}_2$, into $M_3(\mathbb{C})$ are classified into thirteen unfaithful types. Finally, the paper demonstrates that not all complex local extensions of ${\mathrm{T}\mathrm{V}\mathrm{B}}_2$ representations to ${\mathrm{S}\mathrm{T}\mathrm{V}\mathrm{B}}_2$ conform to a $\mathrm{\Phi }$-type extension.