Tetrahedron equation and Schur functions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916249301155840 |
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| author | Iwao, Shinsuke Motegi, Kohei Ohkawa, Ryo |
| author_facet | Iwao, Shinsuke Motegi, Kohei Ohkawa, Ryo |
| contents | The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang-Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as $q$-oscillator valued vertex models with matrix elements of the $L$-operators given by generators of the $q$-oscillator algebra acting on the Fock space. Using one of the $q=0$-oscillator valued vertex models introduced by Bazhanov-Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov-Faddeev algebra provided by Kuniba-Okado-Maruyama. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10011 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tetrahedron equation and Schur functions Iwao, Shinsuke Motegi, Kohei Ohkawa, Ryo Mathematical Physics Quantum Algebra 05E05, 16T25, 82B83 The tetrahedron equation introduced by Zamolodchikov is a three-dimensional generalization of the Yang-Baxter equation. Several types of solutions to the tetrahedron equation that have connections to quantum groups can be viewed as $q$-oscillator valued vertex models with matrix elements of the $L$-operators given by generators of the $q$-oscillator algebra acting on the Fock space. Using one of the $q=0$-oscillator valued vertex models introduced by Bazhanov-Sergeev, we introduce a family of partition functions that admits an explicit algebraic presentation using Schur functions. Our construction is based on the three-dimensional realization of the Zamolodchikov-Faddeev algebra provided by Kuniba-Okado-Maruyama. Furthermore, we investigate an inhomogeneous generalization of the three-dimensional lattice model. We show that the inhomogeneous analog of (a certain subclass of) partition functions can be expressed as loop elementary symmetric functions. |
| title | Tetrahedron equation and Schur functions |
| topic | Mathematical Physics Quantum Algebra 05E05, 16T25, 82B83 |
| url | https://arxiv.org/abs/2405.10011 |