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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.06607 |
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| _version_ | 1866909038107688960 |
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| author | Nasser, Mohamad N. Chreif, Mohammad Y. Dally, Malak M. |
| author_facet | Nasser, Mohamad N. Chreif, Mohammad Y. Dally, Malak M. |
| contents | 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 irreducible if and only if $b\neq y$, 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 $x=y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local Representations of the Flat Virtual Braid Group Nasser, Mohamad N. Chreif, Mohammad Y. Dally, Malak M. Representation Theory 20F36 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 irreducible if and only if $b\neq y$, 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 $x=y$. |
| title | Local Representations of the Flat Virtual Braid Group |
| topic | Representation Theory 20F36 |
| url | https://arxiv.org/abs/2503.06607 |