Matrix representations of the twisted virtual braid group and its extensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916778321379328 |
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| author | Nasser, Mohamad N. Keshari, Vaibhav Prabhakar, Madeti |
| author_facet | Nasser, Mohamad N. Keshari, Vaibhav Prabhakar, Madeti |
| contents | 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 $Φ$-type extension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03806 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix representations of the twisted virtual braid group and its extensions Nasser, Mohamad N. Keshari, Vaibhav Prabhakar, Madeti Representation Theory Primary: 20F36, 20F38, Secondary: 57K12 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 $Φ$-type extension. |
| title | Matrix representations of the twisted virtual braid group and its extensions |
| topic | Representation Theory Primary: 20F36, 20F38, Secondary: 57K12 |
| url | https://arxiv.org/abs/2506.03806 |