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 (The author cites an as yet unpublished paper which claims that \(P_\Gamma\) is simple.)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.]

\[ h(dP)(\alpha,t) = \left(\frac{\partial}{\partial\alpha} + \frac{\partial}{\partial t}\right)h(P)(\alpha,t) \] (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.]

Reviewer: Peter McMullen (London)

##### MSC:

52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |

13N15 | Derivations and commutative rings |

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\textit{V. M. Buchstaber}, Proc. Steklov Inst. Math. 263, 13--37 (2008; Zbl 1184.52012); translation from Tr. Mat. Inst. Steklova 263, 18--43 (2008)

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##### References:

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