A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain
We study the connection between the three-color model and the polynomials ₙ() of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an exp...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2020
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211019 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. Linnea Hietala. SIGMA 16 (2020), 101, 26 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862723307405574144 |
|---|---|
| author | Hietala, Linnea |
| author_facet | Hietala, Linnea |
| citation_txt | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. Linnea Hietala. SIGMA 16 (2020), 101, 26 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the connection between the three-color model and the polynomials ₙ() of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for ₙ() in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that ₙ() has positive integer coefficients. We prove the weaker statement that ₙ(+1) and (+1)ⁿ⁽ⁿ⁺¹⁾ₙ(1/(+1)) have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.
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| first_indexed | 2026-03-21T05:39:29Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211019 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T05:39:29Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hietala, Linnea 2025-12-22T09:30:44Z 2020 A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. Linnea Hietala. SIGMA 16 (2020), 101, 26 pages 1815-0659 2020 Mathematics Subject Classification: 82B23; 05A15; 33E17 arXiv:2004.09924 https://nasplib.isofts.kiev.ua/handle/123456789/211019 https://doi.org/10.3842/SIGMA.2020.101 We study the connection between the three-color model and the polynomials ₙ() of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for ₙ() in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that ₙ() has positive integer coefficients. We prove the weaker statement that ₙ(+1) and (+1)ⁿ⁽ⁿ⁺¹⁾ₙ(1/(+1)) have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color. I would like to thank my supervisor, Hjalmar Rosengren, and my co-supervisor, Jules Lamers, for the numerous hours of support you have given me throughout the whole research process and while writing this article. I would like to thank the anonymous referees for many useful comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain Article published earlier |
| spellingShingle | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain Hietala, Linnea |
| title | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain |
| title_full | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain |
| title_fullStr | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain |
| title_full_unstemmed | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain |
| title_short | A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain |
| title_sort | combinatorial description of certain polynomials related to the xyz spin chain |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211019 |
| work_keys_str_mv | AT hietalalinnea acombinatorialdescriptionofcertainpolynomialsrelatedtothexyzspinchain AT hietalalinnea combinatorialdescriptionofcertainpolynomialsrelatedtothexyzspinchain |