On representations of the triplet group and some of its extensions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866915784562835456 |
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| author | Nasser, Mohamad N. Chbili, Nafaa Qazaqzeh, Khaled |
| author_facet | Nasser, Mohamad N. Chbili, Nafaa Qazaqzeh, Khaled |
| contents | In this paper, we study the representations of the triplet group $L_n$, where $n$ is a positive integer, and its extensions to the virtual and welded triplet groups $VL_n$ and $WL_n$, respectively. We first introduce $L_n$, its extensions, and its pure subgroup. We then investigate several representations, proving the irreducibility of the classical Tits representation $Θ: L_n \to \mathrm{GL}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $μ: L_n \to \mathrm{Aut}(\mathbb{F}_n)$, where $\mathbb{F}_n$ is the free group of rank $n$. For the representation $μ$, we determine its matrix form, faithfulness, and irreducibility. We also classify all complex homogeneous $2$-local representations of $L_n$ for $n \ge 3$ and all non-homogeneous $2$-local representations of $L_3$, establishing connections with the complex specialization of the representation $μ$. Finally, we examine extensions of $L_n$ representations to $VL_n$ and $WL_n$, proving their existence, classifying non-trivial complex homogeneous $2$-local representations, and analyzing their faithfulness and irreducibility. The paper concludes with an open question regarding further extension of representation of $L_n$ to $VL_n$ and $WL_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07863 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On representations of the triplet group and some of its extensions Nasser, Mohamad N. Chbili, Nafaa Qazaqzeh, Khaled Representation Theory Group Theory In this paper, we study the representations of the triplet group $L_n$, where $n$ is a positive integer, and its extensions to the virtual and welded triplet groups $VL_n$ and $WL_n$, respectively. We first introduce $L_n$, its extensions, and its pure subgroup. We then investigate several representations, proving the irreducibility of the classical Tits representation $Θ: L_n \to \mathrm{GL}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $μ: L_n \to \mathrm{Aut}(\mathbb{F}_n)$, where $\mathbb{F}_n$ is the free group of rank $n$. For the representation $μ$, we determine its matrix form, faithfulness, and irreducibility. We also classify all complex homogeneous $2$-local representations of $L_n$ for $n \ge 3$ and all non-homogeneous $2$-local representations of $L_3$, establishing connections with the complex specialization of the representation $μ$. Finally, we examine extensions of $L_n$ representations to $VL_n$ and $WL_n$, proving their existence, classifying non-trivial complex homogeneous $2$-local representations, and analyzing their faithfulness and irreducibility. The paper concludes with an open question regarding further extension of representation of $L_n$ to $VL_n$ and $WL_n$. |
| title | On representations of the triplet group and some of its extensions |
| topic | Representation Theory Group Theory |
| url | https://arxiv.org/abs/2602.07863 |