Polytopal Realizations of Generalized Associahedra

In [5], a complete simplicial fan was associated to an arbitrary finite root system. It was conjectured that this fan is the normal fan of a simple convex polytope (a generalized associahedron of the corresponding type). Here we prove this conjecture by explicitly exhibiting a family of such polytopal realizations (see Theorems 1.4–1.5 and Corollary 1.9 below). The name “generalized associahedron” was chosen because for the type An the construction in [5] produces the n-dimensional associahedron (also known as the Stasheff polytope). Its face complex was introduced by J. Stasheff [13] as a basic tool for the study of homotopy associative H-spaces. The fact that this complex can be realized by a convex polytope was established much later in [8, 6]. Note that the realizations given in Corollary 1.9 are new even in this classical case. The face complex of a generalized associahedron of type Bn (or Cn) is another familiar polytope: the n-dimensional “cyclohedron.” It was first introduced by R. Bott and C. Taubes [1] (and given its name by J. Stasheff [14]) in connection with the study of link invariants; an alternative combinatorial construction was independently given by R. Simion [11, 12]. Polytopal realizations of cyclohedra were constructed explicitly by M. Markl [9] (cf. also [14, Appendix B]) and R. Simion [12]; again, our construction in Corollary 1.9 gives a new family of such realizations. Associahedra of types A and B have a number of remarkable connections with algebraic geometry [6], topology [13], moduli spaces, knots and operads [1, 3], combinatorics [10], etc. It would be interesting to extend these connections to the type D and the exceptional types. As explained in [5], the construction of generalized associahedra given there was motivated by the theory of cluster algebras, introduced in [4] as a device for studying dual canonical bases and total positivity in semisimple Lie groups. This