Local Representations of the Flat Virtual Braid Group
Mohamad N. Nasser, Mohammad Y. Chreif, Malak M. Dally
TL;DR
This work classifies local and homogeneous local representations of the flat virtual braid group $FVB_n$ into matrix groups across three regimes: $GL_2(\mathbb{C})$, $GL_n(\mathbb{C})$, and $GL_{n+1}(\mathbb{C})$. It provides a complete enumeration for $FVB_2 \to GL_2(\mathbb{C})$ via twelve ${\lambda_i}$ types with explicit irreducibility and faithfulness criteria, and it extends to $FVB_n \to GL_n(\mathbb{C})$ (types ${\gamma_i}$) and $FVB_n \to GL_{n+1}(\mathbb{C})$ (types ${\delta_i}$), detailing reducibility and unfaithfulness conditions in each case. The results sharpen the understanding of how flat virtual braids admit linear representations, clarifying when such representations decompose or fail to be faithful, and thus contribute to the broader linearity questions for braid-like groups. Overall, the paper offers a structured classification framework that connects classical braid representations to the flat virtual setting, with precise parameter-driven criteria for irreducibility and faithfulness.
Abstract
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 $λ_i: FVB_2 \rightarrow GL_2(\mathbb{C})$, $1\leq i\leq 12$. We find necessary and sufficient conditions that guarantee the irreducibility of representations of type $λ_i$, $1\leq i\leq 5$, and we prove that representations of type $λ_i$, $6\leq i\leq 12$, are reducible. Regarding faithfulness, we find necessary and sufficient conditions for representations of type $λ_6$ or $λ_7$ to be faithful. Moreover, we give sufficient conditions for representations of type $λ_1$, $λ_2$, or $λ_4$ to be unfaithful, and we show that representations of type $λ_i$, $i=3, 5, 8, 9, 10, 11, 12$ 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 $γ_i: FVB_n \rightarrow GL_n(\mathbb{C})$, $i=1, 2$. We then prove that representations of type $γ_1: FVB_n \rightarrow GL_n(\mathbb{C})$ are reducible for $n\geq 6$, while representations of type $γ_2: FVB_n \rightarrow GL_n(\mathbb{C})$ are reducible for $n\geq 3$. Then, we show that representations of type $γ_1$ are unfaithful for $n\geq 3$ and that representations of type $γ_2$ 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 $δ_i: FVB_n \rightarrow GL_{n+1}(\mathbb{C})$, $1\leq i\leq 8$. We prove that these representations are reducible for $n\geq 10$. Then, we show that representations of types $δ_i$, $i\neq 5, 6$, are unfaithful, while representations of types $δ_5$ or $δ_6$ are unfaithful if $x=y$.