Local Representations of the Flat Virtual Braid Group

We prove that any complex local representation of the flat virtual braid group, $FVB_2$, into $GL_2(\mathbb{C})$, has one of the types $\lambda_i: FVB_2 \rightarrow GL_2(\mathbb{C})$, $1\leq i\leq 12$. We find necessary and sufficient conditions that guarantee the irreducibility of representations of type $\lambda_i$, $1\leq i\leq 5$, and we prove that representations of type $\lambda_i$, $6\leq i\leq 12$, are reducible. Regarding faithfulness, we find necessary and sufficient conditions for representations of type $\lambda_6$ or $\lambda_7$ to be faithful. Moreover, we give sufficient conditions for representations of type $\lambda_1$, $\lambda_2$, or $\lambda_4$ to be unfaithful, and we show that representations of type $\lambda_i$, $i=3, 5, 8, 9, 10, 11, 12$ are unfaithful. We prove that any complex homogeneous local representations of the flat virtual braid group, $FVB_n$, into $GL_{n}(\mathbb{C})$, for $n\geq 2$, has one of the types $\gamma_i: FVB_n \rightarrow GL_n(\mathbb{C})$, $i=1, 2$. We then prove that representations of type $\gamma_1: FVB_n \rightarrow GL_n(\mathbb{C})$ are reducible for $n\geq 6$, while representations of type $\gamma_2: FVB_n \rightarrow GL_n(\mathbb{C})$ are irreducible if and only if $b\neq y$, for $n\geq 3$. Then, we show that representations of type $\gamma_1$ are unfaithful for $n\geq 3$ and that representations of type $\gamma_2$ are unfaithful if $y=b$. Furthermore, we prove that any complex homogeneous local representation of the flat virtual braid group, $FVB_n$, into $GL_{n+1}(\mathbb{C})$, for all $n\geq 4$, has one of the types $\delta_i: FVB_n \rightarrow GL_{n+1}(\mathbb{C})$, $1\leq i\leq 8$. We prove that these representations are reducible for $n\geq 10$. Then, we show that representations of types $\delta_i$, $i\neq 5, 6$, are unfaithful, while representations of types $\delta_5$ or $\delta_6$ are unfaithful if $x=y$.