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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2504.00260 |
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| _version_ | 1866918142779850752 |
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| author | Frenkel, Edward Hernandez, David |
| author_facet | Frenkel, Edward Hernandez, David |
| contents | The character of every irreducible finite-dimensional representation of a simple Lie algebra has the highest weight property. The invariance of the character under the action of the Weyl group W implies that there is a similar "extremal weight property" for every weight obtained by applying an element of W to the highest weight. In this paper we conjecture an analogous "extremal monomial property" of the q-characters of simple finite-dimensional modules over the quantum affine algebras, using the braid group action on q-characters defined by Chari. In the case of the identity element of W, this is the highest monomial property of q-characters proved in arXiv:math/9911112. Here we prove it for simple reflections. Somewhat surprisingly, the extremal monomial property for each w in W turns out to be equivalent to polynomiality of the "X-series" corresponding to w, which we introduce in this paper. We show that these X-series are equal to certain limits of the generalized Baxter operators for all w in W. Thus, we find a new bridge between q-characters and the spectra of XXZ-type quantum integrable models associated to quantum affine algebras. This leads us to conjecture polynomiality of all generalized Baxter operators, extending the results of arXiv:1308.3444. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal monomial property of q-characters and polynomiality of the X-series Frenkel, Edward Hernandez, David Quantum Algebra Statistical Mechanics High Energy Physics - Theory Representation Theory Exactly Solvable and Integrable Systems The character of every irreducible finite-dimensional representation of a simple Lie algebra has the highest weight property. The invariance of the character under the action of the Weyl group W implies that there is a similar "extremal weight property" for every weight obtained by applying an element of W to the highest weight. In this paper we conjecture an analogous "extremal monomial property" of the q-characters of simple finite-dimensional modules over the quantum affine algebras, using the braid group action on q-characters defined by Chari. In the case of the identity element of W, this is the highest monomial property of q-characters proved in arXiv:math/9911112. Here we prove it for simple reflections. Somewhat surprisingly, the extremal monomial property for each w in W turns out to be equivalent to polynomiality of the "X-series" corresponding to w, which we introduce in this paper. We show that these X-series are equal to certain limits of the generalized Baxter operators for all w in W. Thus, we find a new bridge between q-characters and the spectra of XXZ-type quantum integrable models associated to quantum affine algebras. This leads us to conjecture polynomiality of all generalized Baxter operators, extending the results of arXiv:1308.3444. |
| title | Extremal monomial property of q-characters and polynomiality of the X-series |
| topic | Quantum Algebra Statistical Mechanics High Energy Physics - Theory Representation Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2504.00260 |