Populations of Solutions to Cyclotomic Bethe Equations
We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain ''...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2015 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2015
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147118 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Populations of Solutions to Cyclotomic Bethe Equations / A. Varchenko, C.A.S Young // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862726999981686784 |
|---|---|
| author | Varchenko, A. Young, C.A.S. |
| author_facet | Varchenko, A. Young, C.A.S. |
| citation_txt | Populations of Solutions to Cyclotomic Bethe Equations / A. Varchenko, C.A.S Young // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain ''extended'' master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an ''extended'' non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a Z₂-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space.
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| first_indexed | 2025-12-07T19:00:06Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147118 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:00:06Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Varchenko, A. Young, C.A.S. 2019-02-13T16:57:57Z 2019-02-13T16:57:57Z 2015 Populations of Solutions to Cyclotomic Bethe Equations / A. Varchenko, C.A.S Young // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B23; 32S22; 17B81; 81R12 DOI:10.3842/SIGMA.2015.091 https://nasplib.isofts.kiev.ua/handle/123456789/147118 We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as critical points of a cyclotomic master function, and as critical points with cyclotomic symmetry of a certain ''extended'' master function. In finite types, this yields a correspondence between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model and those of an ''extended'' non-cyclotomic Gaudin model. We proceed to define populations of solutions to the cyclotomic Bethe equations, in the sense of [Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111-163, math.QA/0209017], for diagram automorphisms of Kac-Moody Lie algebras. In the case of type A with the diagram automorphism, we associate to each population a vector space of quasi-polynomials with specified ramification conditions. This vector space is equipped with a Z₂-gradation and a non-degenerate bilinear form which is (skew-)symmetric on the even (resp. odd) graded subspace. We show that the population of cyclotomic critical points is isomorphic to the variety of isotropic full flags in this space. The research of AV is supported in part by NSF grant DMS-1362924. CY is grateful to the
 Department of Mathematics at UNC Chapel Hill for hospitality during a visit in October 2014
 when this work was initiated. CY thanks Benoit Vicedo for valuable discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Populations of Solutions to Cyclotomic Bethe Equations Article published earlier |
| spellingShingle | Populations of Solutions to Cyclotomic Bethe Equations Varchenko, A. Young, C.A.S. |
| title | Populations of Solutions to Cyclotomic Bethe Equations |
| title_full | Populations of Solutions to Cyclotomic Bethe Equations |
| title_fullStr | Populations of Solutions to Cyclotomic Bethe Equations |
| title_full_unstemmed | Populations of Solutions to Cyclotomic Bethe Equations |
| title_short | Populations of Solutions to Cyclotomic Bethe Equations |
| title_sort | populations of solutions to cyclotomic bethe equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147118 |
| work_keys_str_mv | AT varchenkoa populationsofsolutionstocyclotomicbetheequations AT youngcas populationsofsolutionstocyclotomicbetheequations |