From Permutahedron to Associahedron

TL;DR

This paper constructs a new, more tractable bijection between associahedra and permutahedra for finite reflection groups, linking combinatorial and geometric structures like non-crossing partitions and Cambrian fans.

Contribution

It identifies a simplicial associahedron inside the permutahedron for any finite reflection group, establishing a more accessible bijection and connecting it with Cambrian fans.

Findings

Bijection between associahedron facets and non-crossing partitions

Associating the associahedron with the Cambrian fan

Simplification over previous bijections

Abstract

For each finite real reflection group $W$, we identify a copy of the type-$W$ simplicial generalised associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalised associahedron and the elements of the type $W$ non-crossing partition lattice which is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for $W$.