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Section: New Results

A note on gamma triangles and local gamma vectors

Alin Bostan contributed to F. Chapoton's article [5] by writing an appendix, which allowed the author to complete its article. The theme of [5] is the study of simplicial complexes in algebraic combinatorics. A basic invariant is the $f$-vector that counts faces according to their dimensions. A less understood invariant is the $\gamma$-vector, introduced by Gal in 2005. Also in 2005, Chapoton, motivated by the study of the combinatorics of simplicial complexes attached to cluster algebras, considered a refined version of the $f$-vector. The main aim of [5] is to introduce the analogue in this context of the $\gamma$-vector, and a further refinement called the $\Gamma$-triangle. The author computed explicitly the $\Gamma$-triangle for all the cluster simplicial complexes of irreducible Coxeter groups. Alin Bostan contributed to the proof of an unexpected relation between the $\Gamma$-triangles of cluster fans of type $𝔹$ and $𝔻$.