Compatibility degree of cluster complexes
[Degré de compatibilité des complexes de clusters]
Annales de l'Institut Fourier, Online first, 56 p.

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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Révisé le :
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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
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Fu, Changjian; Gyoda, Yasuaki. Compatibility degree of cluster complexes. Annales de l'Institut Fourier, Online first, 56 p.

[1] Amiot, Claire Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier, Volume 59 (2009) no. 6, pp. 2525-2590 https://aif.centre-mersenne.org/articles/10.5802/aif.2499/ | DOI | MR | Zbl

[2] Barot, M.; Kussin, D.; Lenzing, H. The cluster category of a canonical algebra, Trans. Amer. Math. Soc., Volume 362 (2010) no. 8, pp. 4313-4330 | DOI | MR | Zbl

[3] Buan, A. B.; Iyama, O.; Reiten, I.; Scott, J. Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math., Volume 145 (2009) no. 4, pp. 1035-1079 | DOI | MR | Zbl

[4] Buan, Aslak Bakke; Marsh, Bethany R.; Reiten, Idun Cluster mutation via quiver representations, Comment. Math. Helv., Volume 83 (2008) no. 1, pp. 143-177 | DOI | MR | Zbl

[5] Buan, Aslak Bakke; Marsh, Robert; Reineke, Markus; Reiten, Idun; Todorov, Gordana Tilting theory and cluster combinatorics, Adv. Math., Volume 204 (2006) no. 2, pp. 572-618 | DOI | MR | Zbl

[6] Caldero, Philippe; Chapoton, Frédéric; Schiffler, Ralf Quivers with relations and cluster tilted algebras, Algebr. Represent. Theory, Volume 9 (2006) no. 4, pp. 359-376 | DOI | MR | Zbl

[7] Cao, Peigen 𝒢-systems, Adv. Math., Volume 377 (2021), 107500, 68 pages | DOI | MR | Zbl

[8] Cao, Peigen; Li, Fang The enough g-pairs property and denominator vectors of cluster algebras, Math. Ann., Volume 377 (2020) no. 3-4, pp. 1547-1572 | DOI | MR | Zbl

[9] Ceballos, Cesar; Pilaud, Vincent Denominator vectors and compatibility degrees in cluster algebras of finite type, Trans. Amer. Math. Soc., Volume 367 (2015) no. 2, pp. 1421-1439 | DOI | MR | Zbl

[10] Cerulli Irelli, Giovanni; Keller, Bernhard; Labardini-Fragoso, Daniel; Plamondon, Pierre-Guy Linear independence of cluster monomials for skew-symmetric cluster algebras, Compos. Math., Volume 149 (2013) no. 10, pp. 1753-1764 | DOI | MR | Zbl

[11] Chapoton, Frédéric; Fomin, Sergey; Zelevinsky, Andrei Polytopal realizations of generalized associahedra, Canad. Math. Bull., Volume 45 (2002) no. 4, pp. 537-566 | DOI | MR | Zbl

[12] Dehy, Raika; Keller, Bernhard On the combinatorics of rigid objects in 2-Calabi–Yau categories, Int. Math. Res. Not. IMRN (2008) no. 11, rnn029, 17 pages | DOI | MR | Zbl

[13] Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc., Volume 23 (2010) no. 3, pp. 749-790 | DOI | MR | Zbl

[14] Fomin, Sergey; Shapiro, Michael; Thurston, Dylan Cluster algebras and triangulated surfaces Part I: Cluster complexes, Acta Math., Volume 201 (2008) no. 1, pp. 83-146 | DOI | MR | Zbl

[15] Fomin, Sergey; Thurston, Dylan Cluster algebras and triangulated surfaces Part II: Lambda lengths, Mem. Amer. Math. Soc., 255, American Mathematical Society, 2018 no. 1223, v+97 pages | DOI | MR | Zbl

[16] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl

[17] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. II. Finite type classification, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | DOI | MR | Zbl

[18] Fomin, Sergey; Zelevinsky, Andrei Y-systems and generalized associahedra, Ann. Math. (2), Volume 158 (2003) no. 3, pp. 977-1018 | DOI | MR | Zbl

[19] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. IV. Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | DOI | MR | Zbl

[20] Fu, Changjian; Geng, Shengfei On cluster-tilting graphs for hereditary categories, Adv. Math., Volume 383 (2021), 107670, 26 pages | DOI | MR | Zbl

[21] Fu, Changjian; Keller, Bernhard On cluster algebras with coefficients and 2-Calabi–Yau categories, Trans. Amer. Math. Soc., Volume 362 (2010) no. 2, pp. 859-895 | DOI | MR | Zbl

[22] Fujiwara, Shogo; Gyoda, Yasuaki Duality between final-seed and initial-seed mutations in cluster algebras, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 15 (2019), 040, 24 pages | DOI | MR | Zbl

[23] Geigle, Werner; Lenzing, Helmut A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) (Lecture Notes in Math.), Volume 1273, Springer, Berlin, 1987, pp. 265-297 | DOI | MR | Zbl

[24] Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim Canonical bases for cluster algebras, J. Amer. Math. Soc., Volume 31 (2018) no. 2, pp. 497-608 | DOI | MR | Zbl

[25] Gyoda, Yasuaki Relation between f-vectors and d-vectors in cluster algebras of finite type or rank 2, Ann. Comb., Volume 25 (2021) no. 3, pp. 573-594 | DOI | MR | Zbl

[26] Gyoda, Yasuaki; Yurikusa, Toshiya F-matrices of cluster algebras from triangulated surfaces, Ann. Comb., Volume 24 (2020) no. 4, pp. 649-695 | DOI | MR | Zbl

[27] Happel, Dieter A characterization of hereditary categories with tilting object, Invent. Math., Volume 144 (2001) no. 2, pp. 381-398 | DOI | MR | Zbl

[28] Iyama, Osamu; Yoshino, Yuji Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math., Volume 172 (2008) no. 1, pp. 117-168 | DOI | MR | Zbl

[29] Keller, Bernhard On triangulated orbit categories, Doc. Math., Volume 10 (2005), pp. 551-581 | MR | Zbl

[30] Keller, Bernhard; Reiten, Idun Cluster-tilted algebras are Gorenstein and stably Calabi–Yau, Adv. Math., Volume 211 (2007) no. 1, pp. 123-151 | DOI | MR | Zbl

[31] Labardini-Fragoso, Daniel Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. (3), Volume 98 (2009) no. 3, pp. 797-839 | DOI | MR | Zbl

[32] Labardini-Fragoso, Daniel Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions, Selecta Math. (N.S.), Volume 22 (2016) no. 1, pp. 145-189 | DOI | MR | Zbl

[33] Ladkani, Sefi On Jacobian algebras from closed surfaces (2012) (https://arxiv.org/abs/1207.3778)

[34] Lee, Kyungyong; Li, Li; Zelevinsky, Andrei Greedy elements in rank 2 cluster algebras, Selecta Math. (N.S.), Volume 20 (2014) no. 1, pp. 57-82 | DOI | MR | Zbl

[35] Muller, Greg The existence of a maximal green sequence is not invariant under quiver mutation, Electron. J. Combin., Volume 23 (2016) no. 2, 2.47, 23 pages | MR

[36] Nakanishi, Tomoki Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams (2021) (https://arxiv.org/abs/2103.16309)

[37] Nakanishi, Tomoki; Zelevinsky, Andrei On tropical dualities in cluster algebras, Algebraic groups and quantum groups (Contemp. Math.), Volume 565, Amer. Math. Soc., Providence, RI, 2012, pp. 217-226 | DOI | MR | Zbl

[38] Palu, Yann Cluster characters for 2-Calabi–Yau triangulated categories, Ann. Inst. Fourier, Volume 58 (2008) no. 6, pp. 2221-2248 http://aif.cedram.org/item?id=AIF_2008__58_6_2221_0 | DOI | MR | Zbl

[39] Qiu, Yu; Zhou, Yu Cluster categories for marked surfaces: punctured case, Compos. Math., Volume 153 (2017) no. 9, pp. 1779-1819 | DOI | MR | Zbl

[40] Reading, Nathan Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc., Volume 359 (2007) no. 12, pp. 5931-5958 | DOI | MR | Zbl

[41] Reading, Nathan Scattering fans, Int. Math. Res. Not. IMRN (2020) no. 23, pp. 9640-9673 | DOI | MR | Zbl

[42] Reading, Nathan; Stella, Salvatore Initial-seed recursions and dualities for d-vectors, Pacific J. Math., Volume 293 (2018) no. 1, pp. 179-206 | DOI | MR | Zbl

[43] Trepode, Sonia; Valdivieso-Díaz, Yadira On finite dimensional Jacobian algebras, Bol. Soc. Mat. Mex. (3), Volume 23 (2017) no. 2, pp. 653-666 | DOI | MR | Zbl

[44] Yurikusa, Toshiya Combinatorial cluster expansion formulas from triangulated surfaces, Electron. J. Combin., Volume 26 (2019) no. 2, 2.33, 39 pages | MR

[45] Yurikusa, Toshiya Density of g-vector cones from triangulated surfaces, Int. Math. Res. Not. IMRN (2020) no. 21, pp. 8081-8119 | DOI | MR | Zbl

[46] Zhang, Jie; Zhou, Yu; Zhu, Bin Cotorsion pairs in the cluster category of a marked surface, J. Algebra, Volume 391 (2013), pp. 209-226 | DOI | MR | Zbl

[47] Zhou, Yu; Zhu, Bin Maximal rigid subcategories in 2-Calabi–Yau triangulated categories, J. Algebra, Volume 348 (2011), pp. 49-60 | DOI | MR | Zbl

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