Associahedra for finite-type cluster algebras and minimal relations between g-vectors
In: ISSN: 0024-6115, 2023
Online
academicJournal
Zugriff:
International audience ; We show that the mesh mutations are the minimal relations among the (Formula presented.) -vectors with respect to any initial seed in any finite-type cluster algebra. We then use this algebraic result to derive geometric properties of the (Formula presented.) -vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high-dimensional positive orthant with well-chosen affine spaces. This sheds a new light on and extends earlier results of Arkani-Hamed, Bai, He, and Yan in type (Formula presented.) and of Bazier-Matte, Chapelier-Laget, Douville, Mousavand, Thomas, and Yıldırım for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the (Formula presented.) -vector fans of another generalization of the associahedron: nonkissing complexes (also known as support (Formula presented.) -tilting complexes) of gentle algebras. We show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2-acyclic, and we describe in this case its facet-defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on 2-Calabi–Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of Auslander on minimal relations for Grothendieck groups of module categories.
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Associahedra for finite-type cluster algebras and minimal relations between g-vectors
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Autor/in / Beteiligte Person: | Padrol, Arnau ; Palu, Yann ; Pilaud, Vincent ; Plamondon, Pierre Guy ; Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA) ; Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS) ; Laboratoire d'informatique de l'École polytechnique Palaiseau (LIX) ; École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS) ; Laboratoire de Mathématiques de Versailles (LMV) ; Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS) ; Institut universitaire de France (IUF) ; Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.) ; Agence Nationale de la Recherche, ANR, (15 CE40 0004 01, 17 CE40 0018) ; ANR-15-CE40-0004,SC3A,Surfaces, Catégorification et Combinatoire des Algèbres Amassées(2015) ; ANR-17-CE40-0018,CAPPS,Analyse Combinatoire de Polytopes et de Subdivisions Polyédrales(2017) |
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Zeitschrift: | ISSN: 0024-6115, 2023 |
Veröffentlichung: | HAL CCSD ; London Mathematical Society, 2023 |
Medientyp: | academicJournal |
DOI: | 10.1112/plms.12543 |
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