no. 2
Compatibility degree of cluster complexes
[Degré de compatibilité des complexes de clusters]
Fu, Changjian1 ; Gyoda, Yasuaki2
1 Department of Mathematics Sichuan University Chengdu, 610064 PR (China)
2 Graduate School of Mathematics Nagoya University Chikusa-ku, Nagoya 464-8602 Japan
Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 663-718.

Nous introduisons une nouvelle fonction sur l’ensemble des paires de variables de cluster via les vecteurs f, qui est appelée le degré de compatibilité (des complexes de cluster). Le degré de compatibilité est une généralisation naturelle du degré de compatibilité classique introduit par Fomin et Zelevinsky. En particulier, nous prouvons que le degré de compatibilité possède la propriété de dualité, la propriété de symétrie, la propriété d’encastrement et la propriété de compatibilité, que possède le degré classique. Nous conjecturons également que le degré de compatibilité possède la propriété d’échangeabilité. Comme éléments de preuve de cette conjecture, nous établissons la propriété d’échangeabilité pour les algèbres à grappes de rang 2, les algèbres à grappes acycliques asymétriques, les algèbres à grappes provenant de lignes projectives pondérées et les algèbres à grappes provenant de surfaces marquées.

We introduce a new function on the set of pairs of cluster variables via f-vectors, which is called the compatibility degree (of cluster complexes). The compatibility degree is a natural generalization of the classical compatibility degree introduced by Fomin and Zelevinsky. In particular, we prove that the compatibility degree has the duality property, the symmetry property, the embedding property and the compatibility property, which the classical one has. We also conjecture that the compatibility degree has the exchangeability property. As pieces of evidence of this conjecture, we establish the exchangeability property for cluster algebras of rank 2, acyclic skew-symmetric cluster algebras, cluster algebras arising from weighted projective lines, and cluster algebras arising from marked surfaces.

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DOI : 10.5802/aif.3596
Classification : 13F60, 16G70
Keywords: Cluster algebra, mutation, $f$-vector, compatibility degree, cluster complex.
Mot clés : Algèbre des clusters, mutation, vecteur $f$, degré de compatibilité, complexe de clusters.

Fu, Changjian 1 ; Gyoda, Yasuaki 2

1 Department of Mathematics Sichuan University Chengdu, 610064 PR (China)
2 Graduate School of Mathematics Nagoya University Chikusa-ku, Nagoya 464-8602 Japan
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Compatibility degree of cluster complexes},
     journal = {Annales de l'Institut Fourier},
     pages = {663--718},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Fu, Changjian; Gyoda, Yasuaki. Compatibility degree of cluster complexes. Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 663-718. doi : 10.5802/aif.3596. https://aif.centre-mersenne.org/articles/10.5802/aif.3596/

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