[Paper Review] Lattices in finite real reflection groups

Thomas A. Brady, Colum Watt|ArXiv.org|Jan 28, 2005

Advanced Combinatorial Mathematics11 references17 citations

TL;DR

This paper presents a uniform, classification-free proof that the interval $[I, \gamma]$ in a finite real reflection group $W$, ordered by reflection length, forms a lattice. Using a newly constructed simplicial complex embedded in the sphere $S^{n-1}$, the authors establish a geometric model for this interval and show its isomorphism to the generalized associahedron, thereby providing a new, intrinsic proof of the lattice property and the spherical nature of the associahedron.

ABSTRACT

For a finite real reflection group $W$ with Coxeter element $γ$ we give a uniform proof that the closed interval, $[I, γ]$ forms a lattice in the partial order on $W$ induced by reflection length. The proof involves the construction of a simplicial complex which can be embedded in the type W simplicial generalised associahedron.

Motivation & Objective

• To provide a uniform, classification-independent proof of the lattice property for the interval $[I, \gamma]$ in finite real reflection groups.
• To construct a simplicial complex $X(\gamma)$ embedded in $S^{n-1}$ that geometrically models the poset $[I, \gamma]$.
• To establish a direct isomorphism between this complex and the generalized associahedron, thereby proving its spherical structure without relying on classification.
• To unify and generalize the theory of Garside structures in Artin groups by using the full reflection generating set.

Proposed method

• Define a simplicial complex $X(\gamma)$ using positive roots and dot product conditions derived from the action of the Coxeter element $\gamma$, ensuring it lies in $S^{n-1}$.
• Use the reflection length order and properties of moved and fixed subspaces $M(\alpha)$ and $F(\alpha)$ to characterize the poset structure on $W$.
• Introduce subcomplexes $X(\sigma)$ for $\sigma \leq \gamma$ and characterize their geometric realizations via the dot product condition $\rho_i \cdot \mu_j = 0$.
• Leverage the Petrie polygon and the action of $\gamma$ to define a cyclic ordering of roots and determine edge relations in the complex.
• Prove that the edge structure of $X(\gamma)$ matches that of the generalized associahedron by showing equivalence between the edge criterion $R(\rho_i)R(\rho_j) \leq \gamma$ and the orthogonality condition $\rho_i \cdot \mu_j = 0$.
• Use the fact that two simplicial complexes with isomorphic one-skeleta are equal to conclude that $X(\gamma)$ is isomorphic to the generalized associahedron.

Experimental results

Research questions

• RQ1 Can the lattice property of the interval $[I, \gamma]$ in finite real reflection groups be proven uniformly, without relying on the classification of such groups?
• RQ2 What geometric structure models the poset $[I, \gamma]$ under the reflection length order?
• RQ3 How is the generalized associahedron related to the simplicial complex $X(\gamma)$ constructed from the Coxeter element $\gamma$?
• RQ4 Is the generalized associahedron a spherical simplicial complex, and can this be proven without classification?
• RQ5 What is the precise correspondence between the edge sets of the generalized associahedron and the simplicial complex $X(\gamma)$?

Key findings

• The interval $[I, \gamma]$ forms a lattice under the reflection length order, proven uniformly across all finite real reflection groups without case-by-case analysis.
• A new simplicial complex $X(\gamma)$ is constructed in $S^{n-1}$ whose vertices are positive roots and whose faces are determined by orthogonality conditions under the action of $\gamma$, providing a geometric model for $[I, \gamma]$.
• The complex $X(\gamma)$ is isomorphic to the generalized associahedron $GA(W)$, establishing a direct geometric link between the poset structure and the associahedron.
• The proof shows that the generalized associahedron is a spherical simplicial complex, independent of the classification of finite reflection groups.
• The edge structure of $X(\gamma)$ matches that of $GA(W)$: two vertices $\rho_i, \rho_j$ are connected by an edge if and only if $R(\rho_i)R(\rho_j) \leq \gamma$, which is equivalent to $\rho_i \cdot \mu_j = 0$.
• The construction provides a new, intrinsic proof of the asphericity of the $K(\pi,1)$-space for the Artin group $A(W)$, relying on the lattice property and the geometric model.