Permutahedra and generalized associahedra

By Christophe Hohlweg, Carsten E. M. C. Lange, Hugh Thomas. 2007

125 citations recorded by the source corpus.

Given a finite Coxeter system (W,S)(W,S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W , we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan Fc, settling a conjecture of Reading that this is possible. We call this polytope the c-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type A c-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the c-singleton cones, the cones in the c-Cambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if W is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to W, then we show that our realizations have integer coordinates in this lattice.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Physiology

View Paper: https://www.semanticscholar.org/paper/5fff12ec7e218060b401e3013428dd98ea9f4a45

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Physiology

Citations in the corpus

Hopf Monoids and Generalized Permutahedra (2017, background): “Theorem7.1) the structure of Loday’s associahedron with respect to the 1 S directions (Lemma10.3) Interestingly, there are many other realizations of the associahedron as a generalized permutahedron [21,22,43,44,45,52,60,61]. These have isomorphic face posets, but they lead to dierent Hopf structure”
Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors (2019, methodology): “These realizations were then extended by Bazier-Matte, Chapelier-Laget, Douville, Mousavand, Thomas, and Yıldırım [17] in the context of finite-type cluster algebras using tools from representation theory of quivers.”
Brick polytopes of spherical subword complexes and generalized associahedra (2011, methodology): “brian fan. Although Corollary1.10follows from [IS10,IT09] for simply laced types and Corollary1.11could be deduced from the connection between the brick polytope B(cw (c)) and the c-associahedron of [HLT11], the subword complex approach BRICK POLYTOPES OF SPHERICAL SUBWORD COMPLEXES 7 provides new i”