Q-system Cluster Algebras, Paths and Total Positivity
In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connecti...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2010 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2010
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146152 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Q-system Cluster Algebras, Paths and Total Positivity / P. di Francesco, R. Kedem // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862716957873143808 |
|---|---|
| author | di Francesko, P. Kedem, R. |
| author_facet | di Francesko, P. Kedem, R. |
| citation_txt | Q-system Cluster Algebras, Paths and Total Positivity / P. di Francesco, R. Kedem // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.
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| first_indexed | 2025-12-07T18:09:00Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146152 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:09:00Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | di Francesko, P. Kedem, R. 2019-02-07T19:11:41Z 2019-02-07T19:11:41Z 2010 Q-system Cluster Algebras, Paths and Total Positivity / P. di Francesco, R. Kedem // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 05E10; 13F16; 82B20 https://nasplib.isofts.kiev.ua/handle/123456789/146152 In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky. This paper is a contribution to the Proceedings of the Workshop “Geometric Aspects of Discrete and UltraDiscrete Integrable Systems” (March 30 – April 3, 2009, University of Glasgow, UK). The full collection is available at http://www.emis.de/journals/SIGMA/GADUDIS2009.html.
 We thank M. Gekhtman, S. Fomin, A. Postnikov, N. Reshetikhin and A. Vainshtein for useful discussions. RK’s research is funded in part by NSF grant DMS-0802511. RK thanks CEA/Saclay IPhT for their hospitality. PDF’s research is partly supported by the European network grant ENIGMA and the ANR grants GIMP and GranMa. PDF thanks the department of Mathematics of the University of Illinois at Urbana-Champaign for hospitality and support, and the department of Mathematics of the University of California Berkeley for hospitality en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Q-system Cluster Algebras, Paths and Total Positivity Article published earlier |
| spellingShingle | Q-system Cluster Algebras, Paths and Total Positivity di Francesko, P. Kedem, R. |
| title | Q-system Cluster Algebras, Paths and Total Positivity |
| title_full | Q-system Cluster Algebras, Paths and Total Positivity |
| title_fullStr | Q-system Cluster Algebras, Paths and Total Positivity |
| title_full_unstemmed | Q-system Cluster Algebras, Paths and Total Positivity |
| title_short | Q-system Cluster Algebras, Paths and Total Positivity |
| title_sort | q-system cluster algebras, paths and total positivity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146152 |
| work_keys_str_mv | AT difranceskop qsystemclusteralgebraspathsandtotalpositivity AT kedemr qsystemclusteralgebraspathsandtotalpositivity |