[Paper Review] From Permutahedron to Associahedron

Thomas A. Brady, Colum Watt|ArXiv.org|Apr 15, 2008

Advanced Combinatorial Mathematics3 references4 citations

TL;DR

This paper constructs the $c$-Cambrian fan for a finite irreducible real reflection group $W$ by embedding the simplicial generalized associahedron $AX(c)$ into the permutahedron via the linear map $\mu = 2(I - c)^{-1}$, showing that the image $\mu(AX(c))$ yields a fan isomorphic to the $c$-Cambrian fan. The key contribution is a geometric, isometric construction of the Cambrian fan that establishes a tractable bijection between facets of the associahedron and the non-crossing partition lattice $\mathrm{NCP}_c$.

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 $W$.

Motivation & Objective

• Develop a geometric construction of the $c$-Cambrian fan for a finite irreducible real reflection group $W$ without relying on Coxeter-sorting.
• Establish a direct, isometric link between the simplicial generalized associahedron and the $c$-Cambrian fan via the linear transformation $\mu = 2(I - c)^{-1}$.
• Provide a more tractable and geometrically intuitive bijection between the facets of the generalized associahedron and the elements of the non-crossing partition lattice $\mathrm{NCP}_c$.
• Show that the fan defined by the image $\mu(AX(c))$ coincides with the $c$-Cambrian fan, thereby giving a new realization of this important polyhedral fan.
• Reconcile the cluster fan and the Cambrian fan through a geometric construction rooted in root systems and reflection group theory.

Proposed method

• The construction begins with the simplicial generalized associahedron $EX(c)$, defined via a total order on roots and a reflection-length condition, and applies the involution $c_+$ to obtain the complex $AX(c)$.
• The linear map $\mu = 2(I - c)^{-1}$ is applied to $AX(c)$, yielding the complex $\mu(AX(c))$, which is shown to be isometric and to preserve the simplicial structure.
• Each facet of $\mu(AX(c))$ is proven to lie on a union of reflecting hyperplanes from the permutahedron fan, establishing that the fan of $\mu(AX(c))$ is a coarsening of the permutahedron fan.
• A bijection is defined between elements $w \in \mathrm{NCP}_c$ and regions $F(w)$ in $\mathbb{R}^n$, where $F(w)$ is defined by inequalities involving the simple roots of the parabolic subgroups $W_w$ and $W_{cw^{-1}}$.
• The rays of each $F(w)$ are shown to be generated by images under $\mu$ of specific roots $\rho_i$, and the facet structure of $\mu(AX(c))$ is matched to these $F(w)$ cones via the map $\phi$.
• Finally, the map $L$ from the cluster fan to the Cambrian fan is shown to coincide with $\frac{1}{2} \mu \circ c_+$, proving that the fan of $\mu(AX(c))$ is linearly isomorphic to the $c$-Cambrian fan.

Experimental results

Research questions

• RQ1 How can the $c$-Cambrian fan be geometrically realized as a coarsening of the permutahedron fan for a finite irreducible real reflection group $W$?
• RQ2 What is the precise relationship between the simplicial generalized associahedron $AX(c)$ and the $c$-Cambrian fan in terms of linear isomorphism?
• RQ3 Can a more tractable and geometrically intuitive bijection be constructed between the facets of the generalized associahedron and the non-crossing partition lattice $\mathrm{NCP}_c$?
• RQ4 How does the linear transformation $\mu = 2(I - c)^{-1}$ relate to the cluster fan and the Cambrian fan, and does it yield a canonical realization of the latter?
• RQ5 Does the fan defined by the cones $F(w)$, indexed by $w \in \mathrm{NCP}_c$, coincide with the $c$-Cambrian fan for a bipartite Coxeter element $c$?

Key findings

• The fan defined by the cones $F(w)$ for $w \in \mathrm{NCP}_c$ is a complete simplicial fan that is linearly isomorphic to the cluster fan.
• The image $\mu(AX(c))$ of the simplicial generalized associahedron under the map $\mu = 2(I - c)^{-1}$ forms a polyhedral complex whose normal fan is the $c$-Cambrian fan.
• Each facet of $\mu(AX(c))$ lies on a union of reflecting hyperplanes from the permutahedron, confirming that the fan of $\mu(AX(c))$ is a coarsening of the permutahedron fan.
• The map $L$ from the cluster fan to the $c$-Cambrian fan is shown to coincide with $\frac{1}{2} \mu \circ c_+$, establishing the linear isomorphism between the two fans.
• The construction provides a new, geometrically intuitive bijection between the facets of the generalized associahedron and the elements of $\mathrm{NCP}_c$, which is more tractable than previous combinatorial constructions.
• In the $C_3$ case, the cyclohedron (type $C_3$ associahedron) is explicitly embedded in the permutahedron via $\mu(AX(c))$, with a stereographic projection illustrating the fan structure and facet labeling.