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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
##### Keywords:
flag polytope; nestahedron; building set
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