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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.

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