Matrix representations of the twisted virtual braid group and its extensions

This paper classifies complex local representations of the twisted virtual braid group, $\mathrm{TVB}_2$, into $\mathrm{GL}_3(\mathbb{C})$. 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 $\mathrm{TVB}_n$ into $\mathrm{GL}_{n+1}(\mathbb{C})$ for $n \geq 3$, identifying seven unfaithful types. Additionally, complex local representations of the singular twisted virtual braid group, $\mathrm{STVB}_2$, into $M_3(\mathbb{C})$ are classified into thirteen unfaithful types. Finally, the paper demonstrates that not all complex local extensions of $\mathrm{TVB}_2$ representations to $\mathrm{STVB}_2$ conform to a $\Phi$-type extension.