Exponents Associated with Y-Systems and their Relationship with q-Series
Let ᵣ be a finite type Dynkin diagram, and ℓ be a positive integer greater than or equal to two. The -system of type ᵣ with level ℓ is a system of algebraic relations whose solutions have been proved to have periodicity. For any pair (ᵣ,ℓ), we define an integer sequence called exponents using the fo...
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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/123456789/210582 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Exponents Associated with Y-Systems and their Relationship with q-Series. Yuma Mizuno. SIGMA 16 (2020), 028, 42 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862700483962994688 |
|---|---|
| author | Mizuno, Yuma |
| author_facet | Mizuno, Yuma |
| citation_txt | Exponents Associated with Y-Systems and their Relationship with q-Series. Yuma Mizuno. SIGMA 16 (2020), 028, 42 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let ᵣ be a finite type Dynkin diagram, and ℓ be a positive integer greater than or equal to two. The -system of type ᵣ with level ℓ is a system of algebraic relations whose solutions have been proved to have periodicity. For any pair (ᵣ,ℓ), we define an integer sequence called exponents using the formulation of the -system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type ᵣ, and prove this conjecture for (A₁,ℓ) and (Aᵣ,2) cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from q-series identities for this relationship.
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| first_indexed | 2025-12-17T12:04:16Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210582 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:16Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mizuno, Yuma 2025-12-12T10:29:11Z 2020 Exponents Associated with Y-Systems and their Relationship with q-Series. Yuma Mizuno. SIGMA 16 (2020), 028, 42 pages 1815-0659 2020 Mathematics Subject Classification: 13F60;17B22;81R10 arXiv:1812.05863 https://nasplib.isofts.kiev.ua/handle/123456789/210582 https://doi.org/10.3842/SIGMA.2020.028 Let ᵣ be a finite type Dynkin diagram, and ℓ be a positive integer greater than or equal to two. The -system of type ᵣ with level ℓ is a system of algebraic relations whose solutions have been proved to have periodicity. For any pair (ᵣ,ℓ), we define an integer sequence called exponents using the formulation of the -system by cluster algebras. We give a conjectural formula expressing the exponents by the root system of type ᵣ, and prove this conjecture for (A₁,ℓ) and (Aᵣ,2) cases. We point out that a specialization of this conjecture gives a relationship between the exponents and the asymptotic dimension of an integrable highest weight module of an affine Lie algebra. We also give a point of view from q-series identities for this relationship. The author gratefully acknowledges the help provided by Yuji Terashima. This work is supported by JSPS KAKENHI Grant Number JP18J22576. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Exponents Associated with Y-Systems and their Relationship with q-Series Article published earlier |
| spellingShingle | Exponents Associated with Y-Systems and their Relationship with q-Series Mizuno, Yuma |
| title | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_full | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_fullStr | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_full_unstemmed | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_short | Exponents Associated with Y-Systems and their Relationship with q-Series |
| title_sort | exponents associated with y-systems and their relationship with q-series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210582 |
| work_keys_str_mv | AT mizunoyuma exponentsassociatedwithysystemsandtheirrelationshipwithqseries |