Order and substitution on graph associahedra

Abstract: M. Carr and S. Devadoss introduced in [1] associated a finite partially ordered set to any simple finite graph, whose geometric realization is a convex polytope ${\mathcal K}\Gamma$, the graph-associahedron. Their construction include many well-known families of polytopes, liked permutahedra, associahedra, cyclohedra and the standard simplexes.

The goal of the present work is to give an algebraic description of graph associahedra. We introduce a substitution operation on Carr and Devadoss tubings, which allows us to describe graph associahedra as a free object on the set of all connected simple graphs, for a type of colored operad generated by pairs of a finite connected graph and a connected subgraph of it.

We show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories. We describe an order on the faces of graph-associahedra, different from the one given by Carr and Devadoss, which allows us to construct a standard triangulation of graph associahedra, following [2].

(joint work with Stefan Forcey)

[1] M. Carr, S. Devadoss, Coxeter complexes and graph associahedra, Topol. and its Applic. 153 (1-2) (2006) 2155–2168.
[2] J.-L. Loday, Parking functions and triangulation of the associahedron, Proceedings of the Street’s fest 2006, Contemp. Math. AMS 431 (2007), 327–340.

algebraic topologycategory theoryquantum algebra

Audience: learners

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