A note on gamma triangles and local gamma vectors (with an appendix by Alin Bostan)

Chapoton, Frédéric

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 4, pp. 907-925.

Cet article introduit les gamma-triangles, qui sont liés aux F-triangles et H-triangles utilisés dans l’étude combinatoire des complexes d’amas, et dont ils sont en quelque sorte une version plus fondamentale. On démontre que les gamma-triangles s’expriment comme des sommes de gamma-vecteurs locaux, introduits par Athanasiadis comme un raffinement des h-vecteurs locaux de subdivisions simpliciales, dûs à Stanley. On calcule ensuite explicitement les gamma-triangles des complexes d’amas de type fini.

This article introduces Gamma-triangles, which are closely related to F-triangles and H-triangles that were used in the combinatorial study of cluster complexes, and in some sense are more fundamental. We prove that Gamma-triangles can be expressed as sums of local gamma-vectors, that were introduced by Athanasiadis as a refinement of the Stanley’s local h-vector of simplicial subdivisions. We compute explicitly the Gamma-triangles for cluster complexes of finite type.

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