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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.21133 |
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| _version_ | 1866929651914375168 |
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| author | Sysoeva, Inna |
| author_facet | Sysoeva, Inna |
| contents | In this paper we study irreducible matrix representations of the welded braid group $WB_n$, also known as the group of conjugating automorphisms of a free group $F_n.$ We prove that $WB_n$ has no irreducible representations of dimension $r,$ where $2\leqslant r\leqslant n-2$ for $n\geqslant 5.$ We give complete classification of all extensions of irreducible representations of the braid group $B_n$ to the welded braid group $WB_n$ of dimensions $n-1$ (for $n\geqslant 7$) and $n$ (for $n\geqslant 7,$ $n\neq 8$). Classification of all extensions of the irreducible $n-1-$ dimensional reduced Burau representation is given for $n\geqslant 5,$ $n\neq 6.$ A new one-parameter family of $n-1-$ dimensional irreducible representations of $WB_n$ is discovered. Classification of all extensions of the irreducible $n-$dimensional standard representation (also known as Tong-Yang-Ma representation) is given for all $n\geqslant 3.$ New families of the 3-dimensional irreducible representations of $WB_3$ are discovered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducible representations of welded braid group Sysoeva, Inna Group Theory 20F36 In this paper we study irreducible matrix representations of the welded braid group $WB_n$, also known as the group of conjugating automorphisms of a free group $F_n.$ We prove that $WB_n$ has no irreducible representations of dimension $r,$ where $2\leqslant r\leqslant n-2$ for $n\geqslant 5.$ We give complete classification of all extensions of irreducible representations of the braid group $B_n$ to the welded braid group $WB_n$ of dimensions $n-1$ (for $n\geqslant 7$) and $n$ (for $n\geqslant 7,$ $n\neq 8$). Classification of all extensions of the irreducible $n-1-$ dimensional reduced Burau representation is given for $n\geqslant 5,$ $n\neq 6.$ A new one-parameter family of $n-1-$ dimensional irreducible representations of $WB_n$ is discovered. Classification of all extensions of the irreducible $n-$dimensional standard representation (also known as Tong-Yang-Ma representation) is given for all $n\geqslant 3.$ New families of the 3-dimensional irreducible representations of $WB_3$ are discovered. |
| title | Irreducible representations of welded braid group |
| topic | Group Theory 20F36 |
| url | https://arxiv.org/abs/2412.21133 |