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Cubical realizations of flag nestohedra and proof of Gal’s conjecture for them. (English. Russian original) Zbl 1204.52014
Russ. Math. Surv. 65, No. 1, 188-190 (2010); translation from Usp. Mat. Nauk. 65, No. 1, 183-184 (2010).
For a \(d\)-dimensional polytope \(P\) the \(f\)-polynomial is given by \(f_P(t) = \sum_{i=0}^df_it^i\) where \(f_i\) is the number of \(i\)-dimensional faces of \(P\). The \(h\)-polynomial of \(P\) can then be defined by \(h_P(t) := f_P(t-1)\). The Dehn-Sommerville equations for simple polytopes are equivalent to the reciprocity of their \(h\)-polynomials; \(h_P(t) = \sum_{i=0}^{\lfloor d/2\rfloor} \gamma_it^i(1+t)^{n-2i}\). The polynomial \(\gamma_P(t) = \sum_{i=0}^{\lfloor d/2\rfloor} \gamma_it^i\) is called the \(\gamma\)-polynomial of \(P\). A simple polytope is a flag polytope if any set of pairwise intersecting facets has a non-empty intersection. A set \({\mathcal B}\) of subsets of \([n+1] = \{1,\ldots,n+1\}\) is a building set on \([n+1]\) if (i) \(\{i\} \in {\mathcal B}\) for each \(i\in [n+1]\), and (ii) \(S_1,S_2\in {\mathcal B}\) and \(S_1\cap S_2\neq\emptyset\) implies \(S_1\cup S_2\in {\mathcal B}\). If \(\Delta_S\) is the convex hull of \(\{e_i : i\in S\}\subseteq {\mathbb R}^{n+1}\), then the Minkowski sum \(P_{\mathcal B} = \sum_{S\in {\mathcal B}} \Delta_S\) is the nestohedron corresponding to the building set \({\mathcal B}\). Such a building set \({\mathcal B}\) is connected if \([n+1]\in {\mathcal B}\).
The main results of this short note are: (i) If \(P_{\mathcal B}\) is a flag nestohedron of dimension \(n\), then \(0\leq \gamma_i(P_{\mathcal B})\leq \gamma_i(Pe^n)\) for each \(i\), where \(Pe^n\) is the permutahedron in \({\mathbb R}^{n+1}\) of dimension \(n\). (ii) If \({\mathcal B}_1\subseteq {\mathcal B}_2\) are connected building sets on \([n+1]\) where \(P_{{\mathcal B}_1}\) and \(P_{{\mathcal B}_2}\) are both flag polytopes, then \(\gamma_i( P_{{\mathcal B}_1})\leq \gamma_i( P_{{\mathcal B}_2})\) for each \(i\).

MSC:
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B70 Polyhedral manifolds
52B11 \(n\)-dimensional polytopes
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