The virtual singular twin monoid and group: presentations and representations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
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2026
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| _version_ | 1866918271995871232 |
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| author | Caprau, Carmen Nasser, Mohamad N. |
| author_facet | Caprau, Carmen Nasser, Mohamad N. |
| contents | In this article, we introduce the algebraic definitions and presentations of the virtual singular twin monoid and virtual singular twin group, denoted by $VSTM_n$ and $VST_n$, respectively, for a positive integer $n$. These structures extend the twin group $T_n$ in close analogy to how the virtual singular braid monoid and virtual singular braid group extend the classical braid group. We then construct and study representations of the group $VST_n$, for $n \geq 3$, focusing in particular on extending the representations $η_1$ and $η_2$ of $T_n$, introduced by M. Nasser, to $VST_n$ via the $2$-local extension method. To analyze the resulting representations, $η_1'$ and $η_2'$, and their properties, we establish necessary and sufficient conditions for irreducibility and show that both $η_1'$ and $η_2'$ are unfaithful. Additionally, we classify all complex homogeneous $2$-local representations of $VST_n$ for every integer $n\geq 3$, providing a foundation for further investigation into representations of $VST_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01707 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The virtual singular twin monoid and group: presentations and representations Caprau, Carmen Nasser, Mohamad N. Representation Theory 20F36 In this article, we introduce the algebraic definitions and presentations of the virtual singular twin monoid and virtual singular twin group, denoted by $VSTM_n$ and $VST_n$, respectively, for a positive integer $n$. These structures extend the twin group $T_n$ in close analogy to how the virtual singular braid monoid and virtual singular braid group extend the classical braid group. We then construct and study representations of the group $VST_n$, for $n \geq 3$, focusing in particular on extending the representations $η_1$ and $η_2$ of $T_n$, introduced by M. Nasser, to $VST_n$ via the $2$-local extension method. To analyze the resulting representations, $η_1'$ and $η_2'$, and their properties, we establish necessary and sufficient conditions for irreducibility and show that both $η_1'$ and $η_2'$ are unfaithful. Additionally, we classify all complex homogeneous $2$-local representations of $VST_n$ for every integer $n\geq 3$, providing a foundation for further investigation into representations of $VST_n$. |
| title | The virtual singular twin monoid and group: presentations and representations |
| topic | Representation Theory 20F36 |
| url | https://arxiv.org/abs/2601.01707 |