Rooted Clusters for Graph LP Algebras
LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a f...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211815 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Rooted Clusters for Graph LP Algebras. Esther Banaian, Sunita Chepuri, Elizabeth Kelley and Sylvester W. Zhang. SIGMA 18 (2022), 089, 30 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862699943205011456 |
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| author | Banaian, Esther Chepuri, Sunita Kelley, Elizabeth Zhang, Sylvester W. |
| author_facet | Banaian, Esther Chepuri, Sunita Kelley, Elizabeth Zhang, Sylvester W. |
| citation_txt | Rooted Clusters for Graph LP Algebras. Esther Banaian, Sunita Chepuri, Elizabeth Kelley and Sylvester W. Zhang. SIGMA 18 (2022), 089, 30 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called rooted clusters. We prove positivity for these clusters by giving explicit formulas for each cluster variable. We also give a combinatorial interpretation for these expansions using a generalization of -paths.
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| first_indexed | 2026-03-18T12:28:24Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211815 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T12:28:24Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Banaian, Esther Chepuri, Sunita Kelley, Elizabeth Zhang, Sylvester W. 2026-01-12T10:17:09Z 2022 Rooted Clusters for Graph LP Algebras. Esther Banaian, Sunita Chepuri, Elizabeth Kelley and Sylvester W. Zhang. SIGMA 18 (2022), 089, 30 pages 1815-0659 2020 Mathematics Subject Classification: 05E15; 05C70 arXiv:2107.14785 https://nasplib.isofts.kiev.ua/handle/123456789/211815 https://doi.org/10.3842/SIGMA.2022.089 LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a graph. For the graph LP algebra defined by a tree, we define a family of clusters called rooted clusters. We prove positivity for these clusters by giving explicit formulas for each cluster variable. We also give a combinatorial interpretation for these expansions using a generalization of -paths. We would like to thank Pavlo Pylyavskyy for suggesting this problem, Trevor Karn for organizing the 2021 Minnesota Combinatorics Working Group, and Kayla Wright and Libby Farrell for generating initial computational data. We thank the anonymous referees for their thoughtful comments, which significantly improved the paper. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the second author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Combinatorial Algebraic Geometry program. The third author was partially supported by NSF Grant No. DMS-1937241. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Rooted Clusters for Graph LP Algebras Article published earlier |
| spellingShingle | Rooted Clusters for Graph LP Algebras Banaian, Esther Chepuri, Sunita Kelley, Elizabeth Zhang, Sylvester W. |
| title | Rooted Clusters for Graph LP Algebras |
| title_full | Rooted Clusters for Graph LP Algebras |
| title_fullStr | Rooted Clusters for Graph LP Algebras |
| title_full_unstemmed | Rooted Clusters for Graph LP Algebras |
| title_short | Rooted Clusters for Graph LP Algebras |
| title_sort | rooted clusters for graph lp algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211815 |
| work_keys_str_mv | AT banaianesther rootedclustersforgraphlpalgebras AT chepurisunita rootedclustersforgraphlpalgebras AT kelleyelizabeth rootedclustersforgraphlpalgebras AT zhangsylvesterw rootedclustersforgraphlpalgebras |