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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Інститут математики НАН України
2015
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| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/147118 |
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| Zitieren: | 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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irk-123456789-1471182019-02-14T01:26:33Z Populations of Solutions to Cyclotomic Bethe Equations Varchenko, A. Young, C.A.S. 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. 2015 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147118 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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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. |
| format |
Article |
| author |
Varchenko, A. Young, C.A.S. |
| spellingShingle |
Varchenko, A. Young, C.A.S. Populations of Solutions to Cyclotomic Bethe Equations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Varchenko, A. Young, C.A.S. |
| author_sort |
Varchenko, A. |
| title |
Populations of Solutions to Cyclotomic Bethe Equations |
| title_short |
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_sort |
populations of solutions to cyclotomic bethe equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147118 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2025-07-11T01:23:54Z |
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