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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2503.14840 |
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| _version_ | 1866913787133558784 |
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| author | Negami, Haru |
| author_facet | Negami, Haru |
| contents | In this paper, we establish a correspondence between algebraic and analytic approaches to constructing representations of the braid group $B_n$, namely Katz-Long-Moody construction and multiplicative middle convolution for Knizhnik-Zamolodchikov (KZ)-type equations, respectively. The Katz-Long-Moody construction yields an infinite sequence of representations of $F_n \rtimes B_n$. On the other hand, the fundamental group of the domain of the $n$-valued KZ-type equation is isomorphic to the pure braid group $P_n$. The multiplicative middle convolution for the KZ-type equation provides an analytical framework for constructing (anti-)representations of $P_n$. Furthermore, we show that this construction preserves unitarity relative to a Hermitian matrix and establish an algorithm to determine the signature of the Hermitian matrix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-Moody construction of braid group representations and Haraoka's multiplicative middle convolution for KZ-type equations Negami, Haru Mathematical Physics Geometric Topology Representation Theory In this paper, we establish a correspondence between algebraic and analytic approaches to constructing representations of the braid group $B_n$, namely Katz-Long-Moody construction and multiplicative middle convolution for Knizhnik-Zamolodchikov (KZ)-type equations, respectively. The Katz-Long-Moody construction yields an infinite sequence of representations of $F_n \rtimes B_n$. On the other hand, the fundamental group of the domain of the $n$-valued KZ-type equation is isomorphic to the pure braid group $P_n$. The multiplicative middle convolution for the KZ-type equation provides an analytical framework for constructing (anti-)representations of $P_n$. Furthermore, we show that this construction preserves unitarity relative to a Hermitian matrix and establish an algorithm to determine the signature of the Hermitian matrix. |
| title | Long-Moody construction of braid group representations and Haraoka's multiplicative middle convolution for KZ-type equations |
| topic | Mathematical Physics Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2503.14840 |