The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces
The generalized cluster complex was introduced by Fomin and Reading as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex, we associate a parabolic conjugacy class of the underlying finite Coxeter group. We...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2023
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212015 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces. Theo Douvropoulos and Matthieu Josuat-Vergès. SIGMA 19 (2023), 069, 40 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862591523462316032 |
|---|---|
| author | Douvropoulos, Theo Josuat-Vergès, Matthieu |
| author_facet | Douvropoulos, Theo Josuat-Vergès, Matthieu |
| citation_txt | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces. Theo Douvropoulos and Matthieu Josuat-Vergès. SIGMA 19 (2023), 069, 40 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The generalized cluster complex was introduced by Fomin and Reading as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex, we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.
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| first_indexed | 2026-03-13T21:01:35Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212015 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T21:01:35Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Douvropoulos, Theo Josuat-Vergès, Matthieu 2026-01-22T09:18:17Z 2023 The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces. Theo Douvropoulos and Matthieu Josuat-Vergès. SIGMA 19 (2023), 069, 40 pages 1815-0659 2020 Mathematics Subject Classification: 05A15; 05E10; 20F55 arXiv:2209.12540 https://nasplib.isofts.kiev.ua/handle/123456789/212015 https://doi.org/10.3842/SIGMA.2023.069 The generalized cluster complex was introduced by Fomin and Reading as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex, we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers. This project started in Paris when both authors first moved to IRIF and discovered they shared a love for Coxeter–Catalan combinatorics. We thank Frederic Chapoton for suggesting that we investigate the generating function (defined in Section 11), which was a motivation for the whole project. We also thank Philippe Biane for our fruitful discussion throughout. Eventually, we thank the reviewers for their numerous suggestions that helped improve this article. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces Article published earlier |
| spellingShingle | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces Douvropoulos, Theo Josuat-Vergès, Matthieu |
| title | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces |
| title_full | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces |
| title_fullStr | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces |
| title_full_unstemmed | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces |
| title_short | The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces |
| title_sort | generalized cluster complex: refined enumeration of faces and related parking spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212015 |
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