Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group

TL;DR

This work develops SBDW triangulations of the $W$-permutahedron by weaving together Bruhat order, the noncrossing partition lattice, and Cambrian congruences, with a height function from total linear stability ensuring regularity (in simply-laced types) and enabling folding to all finite types. It provides a concrete mechanism to relate Artin and dual braid presentations via the triangulation, linking combinatorial models of hyperplane complements to explicit group-theoretic relations. Key contributions include a detailed combinatorial and geometric construction of a regular triangulation, a deep analysis of concordancy and cluster structures, and a blueprint toward uniform, type-uniform presentations for the pure braid group. The approach integrates quiver stability, positive cluster complexes, and Cambrian combinatorics to illuminate the braid group’s structure and its representations, with potential implications for explicit homotopies between braid-space models and for broader polyhedral realizations of noncrossing phenomena.

Abstract

For each finite Coxeter group $W$ and each standard Coxeter element of $W$, we construct a triangulation of the $W$-permutahedron. For particular realizations of the $W$-permutahedron, we show that this is a regular triangulation induced by a height function coming from the theory of total linear stability for Dynkin quivers. We also explore several notable combinatorial properties of these triangulations that relate the Bruhat order, the noncrossing partition lattice, and Cambrian congruences. Each triangulation gives an explicit mechanism for relating two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation). This is a step toward uniformly proving conjectural simple, explicit, and type-uniform presentations for the corresponding pure braid group.

Figures (8)

• Figure 1: On the top right is the $c$-noncrossing partition lattice of $\mathfrak{S}_3$, where ${c=s_1s_2}$. On the bottom right is its order complex. On the left is the permutahedron $\mathrm{Perm}_\mathfrak{y}(\mathfrak{S}_3)$, triangulated by $\mathsf{SBDW}_\mathfrak{y}(\mathfrak{S}_3,c)$. Each simplex $\mathrm{conv}(w\vec{\pi} \mathfrak{y})$ is drawn on the left in the same color as the edges in the chain $\vec{\pi}$ on the right. There are two ways of walking from $\mathfrak{y}$ up to $w_\circ\mathfrak{y}$ along the $1$-skeleton of $\mathrm{Perm}_\mathfrak{y}$; these paths are colored differently.
• Figure 2: The Salvetti complex for the symmetric group $\mathfrak{S}_3$ has six $0$-cells (one for each vertex of $\mathrm{Perm}_\mathfrak{y}$), twelve $1$-cells (one for each orientation of each edge of $\mathrm{Perm}_\mathfrak{y}$), and six $2$-cells (one for each of the oriented copies of $\mathrm{Perm}_\mathfrak{y}$ shown here). Quotienting by the action of $\mathfrak{S}_3$ yields the quotient Salvetti complex. Each light blue path corresponds to the expression ${\color{Teal}\mathbf{s}_1\mathbf{s}_2\mathbf{s}_1}$, while each orange path corresponds to the expression ${\color{Orange}\mathbf{s}_2\mathbf{s}_1\mathbf{s}_2}$. One can push a light blue path through a gray 2-cell and onto an orange path, obtaining a topological manifestation of the relation ${\color{Teal}\mathbf{s}_1\mathbf{s}_2\mathbf{s}_1}={\color{Orange}\mathbf{s}_2\mathbf{s}_1\mathbf{s}_2}$ in the Artin presentation of $\mathbf{B}_{\mathfrak{S}_3}$.
• Figure 3: The Bessis--Brady--Watt complex ${\sf{BBW}}(\mathfrak{S}_3,s_1s_2)$ is a pure simplicial complex with $3!\times 3=18$ maximal (2-dimensional) simplices, shown here as colored triangles. Each edge of these simplices is oriented from some element $w\in\mathfrak{S}_3$ (represented by a blue square) to the element $ws_1s_2$ (represented by a blue star). Each maximal simplex is of the form $\{w\pi_0,w\pi_1,w\pi_2\}$, where $\{\pi_0\lessdot_T \pi_1\lessdot_T \pi_2\}$ is one of the maximal chains in the noncrossing partition lattice shown on the right in \ref{['fig:S3']}. Quotienting by the action of $\mathfrak{S}_3$ yields the quotient Bessis--Brady--Watt complex.
• Figure 4: The hyperplane arrangement $\mathcal{H}$ for $W=\mathfrak{S}_4$, represented via a stereographic projection of great circles. The region $\Delta_{s_1s_2s_3}^+$ is shaded in cyan. We have labeled each region $w\mathbb{B}$ by $w$. Each vertex $\delta_{(i\,j)}$ (for $(i\,j)\in T$) is represented by .
• Figure 5: Fix $W=\mathfrak{S}_4$ and $c=s_1s_2s_3$. On the left is $\Delta_c^+$, with regions indexed by inverses of elements of $W_c^+$ and with edges from $\mathrm{Clus}^+(W,c^{-1})$ (note that half of the vertical red edge corresponding to a piece of $H_{24}$ is dotted, indicating that it is not an edge of the positive $c^{-1}$-cluster complex). On the right is a simplex representing $(u,\vec{\pi}) \in \Omega(W,c)$, with $u=s_3s_2$ and $\vec{\pi}$ chosen so that $\mathrm{rw}(\vec{\pi})=(1\,3)(1\,2)(3\,4)$. The triangle representing $\Delta(u)$ is marked with a white star. The blue triangle is the image $\Delta(A)$ of the positive cluster $A \in \mathrm{MClus}^+(W,c^{-1})$ whose corresponding $c^{-1}$-Cambrian congruence class $W^+_c(A)$ contains $u^{-1}=s_2s_3$ (note that $s_2$ is also contained in this congruence class). The triangle representing the cone $\Delta(\vec{\pi})=\Delta(\mathcal{C}_{\vec{\pi}})$ is the union of the blue and purple triangles.

Theorems & Definitions (118)

• Remark 1.1
• Conjecture 1.2: N. Williams
• Definition 1.3
• Example 1.4
• Theorem 1.5
• Remark 1.6
• Definition 1.7
• Theorem 1.8
• Theorem 1.9
• Corollary 1.10