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Ring of simple polytopes and differential equations. (English. Russian original) Zbl 1184.52012
Proc. Steklov Inst. Math. 263, 13-37 (2008); translation from Tr. Mat. Inst. Steklova 263, 18-43 (2008).
The author begins by forming a ring from the combinatorial equivalence classes of simple polytopes: addition is disjoint union, and multiplication is direct product. This ring has a derivation $$d$$, with $$dP$$ the sum of the facets of $$P$$, so that $$d(PQ) = (dP)Q + P(dQ)$$. If $$\dim P = n$$, then its $$h$$-polynomial is $$h(P)(\alpha,t) = \sum_{j=0}^n \, f_j\alpha^{n-j}(t - \alpha)^j$$ (what the author writes here actually deals with the dual, and so – strictly speaking – is incorrect). Corresponding to the derivation is then the formula
$h(dP)(\alpha,t) = \left(\frac{\partial}{\partial\alpha} + \frac{\partial}{\partial t}\right)h(P)(\alpha,t)$ (this already occurs – for the dual – in the reviewer’s paper [Beitr. Algebra Geom. 45, No. 1, 37–46 (2004; Zbl 1081.52011)]). More novel is the association of a simple polytope with a simple graph (no loops or multiple edges). The $$n+1$$ (say) vertices of a graph $$\Gamma$$ can be identified with the vertices of an $$n$$-simplex; then $$P_\Gamma$$ is the Minkowski sum of the faces (subsimplices) of the simplex (including the $$n$$-simplex itself if appropriate) corresponding to connected subgraphs of $$\Gamma$$. (The author cites an as yet unpublished paper which claims that $$P_\Gamma$$ is simple.) If $$I$$ is a subset of vertices of a connected graph $$\Gamma$$ whose induced subgraph $$\Gamma|_I$$ is connected, then $$\Gamma/I$$ has vertices in the complement $$I^\perp$$ of $$I$$ and edges $$\{u,v\}$$ (with $$u,v \in I^\perp$$) which are either already in $$\Gamma$$, or are such that $$\{u,v,I\}$$ induces a connected subgraph. Then
$d\Gamma = \sum_I \, \Gamma|_I \cdot \Gamma/I$ (summed over these ‘connected’ $$I$$) gives a derivation (with $$\cdot$$ denoting disjoint union), which induces a derivation on simple polytopes $$P_\Gamma$$. The author then looks at specific examples, such as the associahedron with $$\Gamma$$ a path, the permutohedron with $$\Gamma$$ a complete graph, and so on, expressing the results in terms of generating functions. These then lead to corresponding formal group laws.
[Reviewer’s comment: perhaps it would have been more helpful in some places for the original papers to be cited, rather than a Russian survey.]

##### MSC:
 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 13N15 Derivations and commutative rings
##### Keywords:
simple polytope; derivation; $$f$$-vector; $$h$$-vector; graph; group law
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##### References:
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