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Proof of Gal’s conjecture for the $$D$$ series of generalized associahedra. (English. Russian original) Zbl 1227.52007
A simple $$n$$-polytope $$P$$ (that is, each vertex of $$P$$ lies in exactly $$n$$ facets) is a flag polytope if any set of pairwise intersecting facets has a common intersection. If the number of $$j$$-faces of $$P$$ is $$f_j$$ for $$j = 0,\dots,n$$, then $$f(P)(t) = \sum_{j=0}^n \, f_jt^j$$ and $$h(P)(t) = f(P)(t-1)$$ are the $$f$$- and $$h$$-polynomials of $$P$$. The Dehn-Sommerville equations $$h(P)(t) = t^nh(P)(t^{-1})$$ enable one to write the $$h$$-polynomial in the form $h(P)(t) = \sum_{i=0}^{[n/2]} \, \gamma_it^i(1 +t)^{n-2i};$ then $$\gamma(P)(t) = \sum_{i=0}^{[n/2]} \, \gamma_it^i$$ is called the $$\gamma$$-polynomial. S. R. Gal [Discrete Comput. Geom. 34, No. 2, 269–284 (2005; Zbl 1085.52005)] has conjectured that $$\gamma_i(P) \geq 0$$ for a flag polytope $$P$$, and has proved it for $$P \in \mathcal{P}^{\mathrm{cube}}$$. Polytopes in this family are obtained by successively shaving off faces of codimension $$2$$, beginning with the cube. Corresponding to the Coxeter-Dynkin diagram $$D_n$$ of the $$n$$-gon is an associahedron $$D^n$$ (whose rather complicated definition is omitted); the author here shows that $$D^n \in \mathcal{P}^{\mathrm{cube}}$$, and hence satisfies Gal’s conjecture.

##### MSC:
 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 13F60 Cluster algebras
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