The Kashaev Equation and Related Recurrences
The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising mod...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210063 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Kashaev Equation and Related Recurrences / A. Leaf // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 12 назв. — англ. |
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Leaf, A. 2025-12-02T09:34:09Z 2019 The Kashaev Equation and Related Recurrences / A. Leaf // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E99; 13F60; 15A15; 30G25; 52C22 arXiv: 1805.04197 https://nasplib.isofts.kiev.ua/handle/123456789/210063 https://doi.org/10.3842/SIGMA.2019.012 The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence are restricted to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads to new results about principal minors of symmetric matrices. We describe and study other recurrences whose behavior is similar to that of the Kashaev equation and hexahedron recurrence. These include equations that appear in the study of s-holomorphicity, as well as other recurrences which, like the hexahedron recurrence, can be related to cluster algebras. I would like to thank my Ph.D. advisor, Sergey Fomin, for his invaluable mathematical insights and the countless hours he dedicated to our meetings while I was writing this paper. I am also grateful to Dmitry Chelkak for pointing out the connection with s-holomorphicity, and to Thomas Lam and John Stembridge for helpful discussions and editorial suggestions. Finally, I would like to thank the anonymous referees for their thorough reading of this manuscript and for their suggestions that improved the quality of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Kashaev Equation and Related Recurrences Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The Kashaev Equation and Related Recurrences |
| spellingShingle |
The Kashaev Equation and Related Recurrences Leaf, A. |
| title_short |
The Kashaev Equation and Related Recurrences |
| title_full |
The Kashaev Equation and Related Recurrences |
| title_fullStr |
The Kashaev Equation and Related Recurrences |
| title_full_unstemmed |
The Kashaev Equation and Related Recurrences |
| title_sort |
kashaev equation and related recurrences |
| author |
Leaf, A. |
| author_facet |
Leaf, A. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
| format |
Article |
| description |
The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence are restricted to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads to new results about principal minors of symmetric matrices. We describe and study other recurrences whose behavior is similar to that of the Kashaev equation and hexahedron recurrence. These include equations that appear in the study of s-holomorphicity, as well as other recurrences which, like the hexahedron recurrence, can be related to cluster algebras.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210063 |
| citation_txt |
The Kashaev Equation and Related Recurrences / A. Leaf // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 12 назв. — англ. |
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AT leafa thekashaevequationandrelatedrecurrences AT leafa kashaevequationandrelatedrecurrences |
| first_indexed |
2025-12-07T21:24:15Z |
| last_indexed |
2025-12-07T21:24:15Z |
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1850886225136713728 |