Quantum superalgebras and the free-fermionic Yang-Baxter equation
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| Format: | Preprint |
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2025
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| _version_ | 1866915368223637504 |
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| author | Brubaker, Ben Bump, Daniel Gustafsson, Henrik P. A. |
| author_facet | Brubaker, Ben Bump, Daniel Gustafsson, Henrik P. A. |
| contents | The free-fermion point refers to a $\operatorname{GL}(2)\times\operatorname{GL}(1)$ parametrized Yang-Baxter equation within the six-vertex model. It has been known for a long time that this is connected with the quantum group $U_q(\mathfrak{gl}(1|1))$. We demonstrate that $R$-matrices from the finite quantum superalgebra $U_q(\mathfrak{gl}(1|1))$ produce a dense subset of the free-fermionic Yang-Baxter equations of the six-vertex model, matching those of the prime, simple modules in the affine quantum superalgebra $U_q(\widehat{\mathfrak{gl}}(1|1))$. Either of these quantum groups can be used to generate the full free-fermion point, and we discuss them both. Our discussion includes 6 families of six-vertex models used by Brubaker, Bump, and Friedberg in connection with Tokuyama's theorem, a deformation of the Weyl character formula. Thus our work gives quantum group interpretations for those models, known informally as Tokuyama ice. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_24189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum superalgebras and the free-fermionic Yang-Baxter equation Brubaker, Ben Bump, Daniel Gustafsson, Henrik P. A. Representation Theory Quantum Algebra 82B23 (Primary) 16T25, 05E10, 17B37, 17B10 (Secondary) The free-fermion point refers to a $\operatorname{GL}(2)\times\operatorname{GL}(1)$ parametrized Yang-Baxter equation within the six-vertex model. It has been known for a long time that this is connected with the quantum group $U_q(\mathfrak{gl}(1|1))$. We demonstrate that $R$-matrices from the finite quantum superalgebra $U_q(\mathfrak{gl}(1|1))$ produce a dense subset of the free-fermionic Yang-Baxter equations of the six-vertex model, matching those of the prime, simple modules in the affine quantum superalgebra $U_q(\widehat{\mathfrak{gl}}(1|1))$. Either of these quantum groups can be used to generate the full free-fermion point, and we discuss them both. Our discussion includes 6 families of six-vertex models used by Brubaker, Bump, and Friedberg in connection with Tokuyama's theorem, a deformation of the Weyl character formula. Thus our work gives quantum group interpretations for those models, known informally as Tokuyama ice. |
| title | Quantum superalgebras and the free-fermionic Yang-Baxter equation |
| topic | Representation Theory Quantum Algebra 82B23 (Primary) 16T25, 05E10, 17B37, 17B10 (Secondary) |
| url | https://arxiv.org/abs/2503.24189 |