Compatibility Fans for Graphical Nested Complexes

Wednesday, June 3, 2015 - 2:30pm - 3:30pm
Lind 305
Vincent Pilaud (École Polytechnique)
Graph associahedra are natural generalizations of the classical associahedra. They provide polytopal realizations of the nested complex of a graph G, defined as the simplicial complex whose vertices are the tubes (i.e. connected induced subgraphs) of G and whose faces are the tubings (i.e. collections of pairwise nested or non-adjacent tubes) of G. They appeared in the work of M. Carr and S. Devadoss, and were further studied by A. Postnikov, by E.-M. Feichtner and B. Sturmfels, and by A. Zelevinsky. Recently, they also appeared in the work of T. Lam and P. Pylyavskyy on linear Laurent phenomenon algebras.
This talk deals with geometric realizations of graphical nested complexes. All known constructions for graph associahedra are based on the nested fan which coarsens the normal fan of the permutahedron. In view of the combinatorial and geometric variety of simplicial fan realizations of the classical associahedra, it is tempting to search for alternative fans realizing graphical nested complexes. Motivated by the analogy between finite type cluster complexes and graphical nested complexes, we transpose S. Fomin and A. Zelevinsky’s construction of compatibility fans from the former to the latter setting. For this, we define a compatibility degree between two tubes of a graph G, and we show that the compatibility vectors of all tubes of G with respect to an arbitrary maximal tubing on G support a complete simplicial fan realizing the nested complex of G. In particular, when the graph G is reduced to a path, our compatibility degree lies in {−1, 0, 1} and we recover F. Santos’ Catalan many simplicial fan realizations of the associahedron.
MSC Code: