Free quantum analogue of Coxeter group $D_4$
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866929234783502336 |
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| author | Gromada, Daniel |
| author_facet | Gromada, Daniel |
| contents | We define the quantum group $D_4^+$ -- a free quantum version of the demihyperoctahedral group $D_4$ (the smallest representative of the Coxeter series $D$). In order to do so, we construct a free analogue of the property that a $4\times4$ matrix has determinant one. Such analogues of determinants are usually very hard to define for free quantum groups in general and our result only holds for the matrix size $N=4$. The free $D_4^+$ is then defined by imposing this generalized determinant condition on the free hyperoctahedral group $H_4^+$. Moreover, we give a detailed combinatorial description of the representation category of $D_4^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_13242 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Free quantum analogue of Coxeter group $D_4$ Gromada, Daniel Quantum Algebra Combinatorics 20G42 (Primary), 05C25, 18D10 (Secondary) We define the quantum group $D_4^+$ -- a free quantum version of the demihyperoctahedral group $D_4$ (the smallest representative of the Coxeter series $D$). In order to do so, we construct a free analogue of the property that a $4\times4$ matrix has determinant one. Such analogues of determinants are usually very hard to define for free quantum groups in general and our result only holds for the matrix size $N=4$. The free $D_4^+$ is then defined by imposing this generalized determinant condition on the free hyperoctahedral group $H_4^+$. Moreover, we give a detailed combinatorial description of the representation category of $D_4^+$. |
| title | Free quantum analogue of Coxeter group $D_4$ |
| topic | Quantum Algebra Combinatorics 20G42 (Primary), 05C25, 18D10 (Secondary) |
| url | https://arxiv.org/abs/2011.13242 |