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\usepackage{multicol} \usepackage{paracol} \usepackage{etoolbox} \usepackage{caption} % \usepackage{hyperref} % \hypersetup{colorlinks=true, citecolor=LightBlue, linkcolor=NewTurquoise,urlcolor=NewTurquoise} \newcommand{\dfn}[1]{\textcolor{NewBlue}{\emph{#1}}} \newcommand{\defn}[1]{\textcolor{NewBlue}{\emph{#1}}} %% define your title in the usual way \title[Triangulations and Braid Groups]{Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group} %% define your authors in the usual way %% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details %% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author \author[]{Colin Defant\thanks{\href{colindefant@gmail.com}{colindefant@gmail.com}. Colin Defant was supported by the National Science Foundation under Award No.\ 2201907.}\addressmark{1}, Melissa Sherman-Bennett\thanks{\href{mshermanbennett@ucdavis.edu}{mshermanbennett@ucdavis.edu}. Melissa Sherman-Bennett was partially supported by the National Science Foundation under Award No.~2444020.}\addressmark{2}, \and Nathan Williams\thanks{\href{nathan.f.williams@gmail.com}{nathan.f.williams@gmail.com}. Nathan Williams was partially supported by the National Science Foundation under Award No.~2246877}\addressmark{3}} %% then use \addressmark to match authors to institutions here \address{\addressmark{1}Department of Mathematics, Harvard University, Cambridge, MA 02138, USA \\ \addressmark{2}Department of Mathematics, University of California Davis, Davis, CA 95616, USA \\ \addressmark{3}Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080} %% put the date of submission here \received{\today} %% leave this blank until submitting a revised version %\revised{} %% put your English abstract here, or comment this out if you don't have one yet %% please don't use custom commands in your abstract / resume, as these will be displayed online %% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year) \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 the combinatorial properties of these triangulations, relating 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. %% put your French abstract here, or comment this out if you don't have one %\resume{\lipsum[2]} %% put your keywords here, or comment this out if you don't have them yet \keywords{permutahedron, triangulation, braid group, Bruhat order, noncrossing partition lattice, total linear stability} %% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier %% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text %% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex %% but you can use bibtex if you want \usepackage[backend=bibtex]{biblatex} \addbibresource{sample.bib} %% note the \printbibliography command at the end of the file which goes with these biblatex commands \articlenumber{13} \footnotelayout{m} \begin{document} \maketitle %% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above %This extended abstract concerns new\footnotemark{} triangulations of Coxeter permutahedra into simplices coming from the interplay between Bruhat order and the lattice of noncrossing partitions. %\footnotetext{See~\Cref{rem:new} and~\Cref{sec:past_and_present}.} \section{Introduction}\label{sec:intro} The complexified complements of finite real simplicial hyperplane arrangements and the complements of hyperplane arrangements for finite well-generated complex reflection groups share the remarkable property of being $K(\pi,1)$ spaces: the fundamental group---called the \defn{pure braid group}---is the only nontrivial homotopy group of the space. \columnratio{0.43} \setlength{\columnsep}{1em} \setlength{\columnseprule}{0.4pt} \begin{paracol}{2} \noindent Building on Garside's single mathematical paper~\cite{garside1969braid} (``que je suis souvent de tr\`es pr\`es''~\cite{deligne1972immeubles}), Deligne proved that the complexified complement of a real finite simplicial hyperplane arrangement is a $K(\pi,1)$ space~\cite{deligne1972immeubles}. \switchcolumn \noindent Building on the structure of Deligne's proof~\cite{deligne1972immeubles} (``The general architecture of our proof is borrowed from Deligne’s original approach, but the details are quite different''~\cite{bessis2015finite}), Bessis proved that the complement of the hyperplane arrangement of a finite complex reflection group is $K(\pi,1)$~\cite{bessis2015finite}. \end{paracol} The proofs that these spaces are $K(\pi,1)$ rely on two different combinatorial models for the hyperplane complements. \medskip \columnratio{0.5} \begin{paracol}{2} The complexified hyperplane complement of a real simplicial arrangement has a combinatorial model called the \emph{Salvetti complex}. In~\cite{salvetti1987topology}, by choosing a real basepoint for the fundamental group, Salvetti used the real chamber geometry of any real finite hyperplane arrangement---not just simplicial arrangements---to construct a CW complex that is a deformation retract of the complexified hyperplane complement. \switchcolumn The hyperplane complement for a finite well-generated complex reflection group has a combinatorial model we call the \emph{Bessis--Brady--Watt complex}. In~\cite{bessis2015finite}, by choosing an eigenvector for a Coxeter element as the basepoint for the fundamental group, Bessis used the noncrossing partition lattice as a replacement for the chamber geometry for real arrangements (see also Brady and Brady--Watt~\cite{brady2000artin,brady2001partial,brady2002k}). \end{paracol} \medskip An arrangement that lies at the intersection of simplicial arrangements and well-generated complex reflection arrangements---that is, a finite Coxeter arrangement---has \emph{both} combinatorial models: that is, the hyperplane complement is homotopic to both the Salvetti complex and the Bessis--Brady--Watt complex. Associated to quotients of the Salvetti and Bessis--Brady--Watt complexes are the Artin and dual presentations for the braid group of the arrangement, as we now explain. Let $W$ be a finite Coxeter group acting in its reflection representation on a real vector space $V$ of dimension $r$. Write $\mathcal{H}$ for the set of its $N$ reflecting hyperplanes and $V^\mathrm{reg}:=V\setminus \bigcup\mathcal{H}$ for the hyperplane complement. Choose a connected component of $V^\mathrm{reg}$ to call the \dfn{base region} $\BB$; this induces a bijection from $W$ to the set of connected components of $V^\mathrm{reg}$ given by $w \mapsto w\BB$. Let $\Vreg_\CC$ be the complexification of $\Vreg$. The \dfn{pure braid group} of type $W$ is the fundamental group ${\bf P}_W:=\pi_1\!\left(\Vreg\right)$. The \dfn{braid group} of type $W$ is $\B_W:=\pi_1\!\left(V^\mathrm{reg}_\CC/W\right)$, the fundamental group of the quotient of $\Vreg_\CC$ by the natural action of $W$. These fit into a natural short exact sequence \[0\to {\bf P}_W \to {\bf B}_W \to W \to 0.\] \medskip \begin{paracol}{2} The group $W$ is generated by its set $S$ of $r$ \dfn{simple reflections}, which act on $V$ via reflections through the walls of $\BB$. The \defn{length} $\ell_S(w)$ of an element $w \in W$ is the minimum number of simple reflections required to write $w$. Equivalently, $\ell_S(w)$ is the number of hyperplanes in $\HH$ separating $\BB$ from $w\BB$. \switchcolumn The group $W$ is generated by its set $T$ of $N$ \dfn{reflections}, which act on $V$ via reflections through the hyperplanes in $\HH$. The \defn{absolute length} $\ell_T(w)$ of an element $w \in W$ is the minimum number of reflections required to write $w$. Equivalently (as $W$ is real), $\ell_T(w)$ is the dimension of the image of $w-1$. \switchcolumn* The \dfn{long element} $w_\circ$ is the unique element of $W$ with $\ell_S(w_\circ)=N$. For $\yy\in \BB$, the \dfn{$W$-permutahedron} $\Perm_\yy=\Perm_\yy(W)\subseteq V$ is the polytope defined as the convex hull of the $W$-orbit $W\yy$. The 1-skeleton of $\Perm_\yy$ is the \defn{weak order}, the interval $\Weak(W):=[e,w_\circ]_S$ between the identity $e$ and the long element in the right Cayley graph of $W$ generated by $S$ (with edges labeled by simple reflections). \switchcolumn A \dfn{standard Coxeter element} is an element $c$ of $W$ obtained as a product of the simple reflections in some order. We have $\ell_T(c)=r= \ell_S(c)$. The order of $c$ is denoted $h$. The \dfn{$c$-noncrossing partition lattice} is the interval $\NC(W,c):=[e,c]_T$ between the identity $e$ and the Coxeter element $c$ in the right Cayley graph of $W$ generated by $T$ (with edges labeled by reflections). Elements of $\NC(W,c)$ are called \defn{$c$-noncrossing partitions}. \switchcolumn* Write $\mathbf{S}:=\{ \mathbf{s} : s \in S\}$ for a formal copy of the set $S$ of simple reflections. The \dfn{Artin presentation} of the braid group is \[\BA = \langle \mathbf{S} : [\mathrm{Red}_S(w_\circ)] \rangle,\] where $[\mathrm{Red}_S(w_\circ)]$ stands for the relations setting equal all reduced $S$-words for $w_\circ$ (replacing each $s \in S$ by the $\mathbf{s} \in \mathbf{S})$. \switchcolumn Write $\dT:=\{ \dt : t \in T\}$ for a formal copy of the set $T$ of reflections. Bessis's \dfn{dual presentation} of the braid group is \[\BD = \langle \dT : [\mathrm{Red}_T(c)] \rangle,\] where $[\mathrm{Red}_T(c)]$ stands for the relations setting equal all reduced $T$-words for $c$ (replacing each $t \in T$ by the $\dt \in \dT)$. \switchcolumn* \end{paracol} \medskip These two different presentations for the braid group come from the two different combinatorial models for the complexified hyperplane complement. In more detail: \medskip \begin{paracol}{2} For a finite Coxeter group $W$, the Salvetti complex $\Sal(W)$ can be taken to be invariant with respect to the action of the group, giving a homotopy equivalence between the quotient complex and the orbit space~\cite{salvetti1994homotopy}. The Artin presentation for $\B_{W}$ is associated to the \dfn{quotient Salvetti complex}: informally, this complex is formed by gluing together oriented copies of $\Perm_\yy(W)$---one for each element of $W$---and then quotienting by the action of $W$. The edges of this complex can be viewed as oriented edges of the $W$-permutahedron, so they are labeled by simple reflections, while relations come from higher-dimensional faces of the permutahedron. \switchcolumn For a Coxeter element $c$ of $W$, the Bessis--Brady--Watt complex $\Bessis(W,c)$ is a simplicial complex built from the noncrossing partition lattice~\cite[Examples 2.2 \& 2.3]{brady2018noncrossing}; it carries an action of $W$. Bessis's dual presentation for $\B_{W}$ is associated to the \dfn{quotient Bessis--Brady--Watt complex}: informally, this complex is formed by gluing together oriented copies of the order complex of the noncrossing partition lattice $\NC(W,c)$---one for each element of $W$---and then quotienting by the action of $W$. The edges of this complex can be viewed as oriented edges in the noncrossing partition lattice, while relations come from longer chains. \end{paracol} \medskip Our triangulation of the permutahedron (depending on a Coxeter element $c$) subdivides $\Sal(W)$ into simplices used in the Bessis--Brady--Watt complex. \begin{remark} Since one can identify $\mathbf{S}$ as a subset of $\dT$, it is reasonable to restrict the Cayley graph of the \emph{dual braid monoid} to the image of the \emph{positive Artin monoid}. After drawing some highly suggestive low-rank pictures, the third author was led to search the internet for the term ``triangulation of the permutahedron'' in the summer of 2021. This search yielded some excellent slides by Brady, Delucchi, and Watt~\cite{slides} that appeared to have the same construction as these low-rank drawings and a similar construction to the one we present here for bipartite $c$. (In this bipartite case, the height function is particularly nice to describe.) After some number of years, it was suggested to us by several parties that we write up our version---this is the second step after~\cite{defant2022pop} to proving certain conjectures of the third author giving explicit, elegant, type-uniform presentations for the pure braid group of $W$, relying on the interaction of Salvetti's presentation with noncrossing Coxeter--Catalan combinatorics. \label{rem:new} \end{remark} \begin{comment} \subsubsection{Presentations of the pure braid group} The Artin and dual presentations of the braid group are explicit, elegant, and type-uniform. By contrast, it is notoriously difficult to find ``nice'' presentations for the pure braid group. Artin gave a normal form of pure braids, which yields a presentation of the pure braid group in type $A$~\cite{artin1947theory}. This presentation is at least somewhat unsatisfactory---in Artin's own words~\cite{artin1947theory}: \emph{``Although it has been proved that every braid can be deformed into a similar normal form the writer is convinced that any attempt to carry this out on a living person would only lead to violent protests and discrimination against mathematics. He would therefore discourage such an experiment.''} Extending the fiber-type technique from type $A$, Cohen gave a presentation of the pure braid group in type $B$~\cite[Theorem 1.4.3]{cohen2001monodromy} (see also Digne--Gomi~\cite{digne2001presentation,digne2015pr}). Markushevich claimed to have found a presentation in type $D$ in~\cite{markushevich1991d}, but Falk and Randall later determined it was incorrect~\cite{falk2000homotopy}. These approaches share similar problems with Artin's presentation: they are type-specific and give complicated, unmotivated presentations. For example, Artin's presentation uses five families of relations, while the presentations of Cohen and Digne--Gomi each use nine families. Salvetti's presentation in~\cite{salvetti1987topology} for the complexified complement of any real arrangement is unwieldy without very precise combinatorial control over all shards of the arrangement (see~\cite{reading2011noncrossing} and especially~\cite[Theorem 1.1]{defant2022pop}). Nearly a decade ago, in unpublished notes, the third author conjectured explicit, elegant, type-uniform presentations for the pure braid group of $W$ relying on the interaction of Salvetti's presentation with noncrossing Coxeter--Catalan combinatorics. He proved these presentations in types $A$, $B$, $H_3$, and $I_2(m)$ using the precise combinatorial control over noncrossing shards given by Coxeter--Catalan combinatorics. In more detail, using the inclusion $\mathbf{P}_W \hookrightarrow \mathbf{B}_W$, write $\mathbb{t}_c:=\mathbf{t}^2_c$ for the square of the generator of the dual braid group corresponding to the reflection $t$, and let $\mathbb{T}_c:=\{\mathbb{t}_c : t \in T\}$. It is not difficult~\cite{defant2022pop} to use Salvetti's construction together with Reading's theory of Coxeter-sorting words to see that $\mathbb{T}_c$ generates ${\bf P}_W$. The \defn{full twist} $\mathbb{c}:={\bf w}_\circ^2=\mathbf{c}^h \in \mathbf{P}_W$ generates the center of ${\bf P}_W$. \begin{conjecture}[{N.~Williams}]\label{conj:pure} For any finite Coxeter group $W$, the pure braid group has presentation \[{\bf P}_W = \langle \mathbb{T}_c : [\Red_{\mathbb{T}_c}(\mathbb{c})] \rangle,\] where $[\Red_{{\mathbb{T}_c}}(\mathbb{c})]$ stands for the relations setting equal all reduced words in $\mathbb{T}_c$ for the full twist $\mathbb{c}$.\nathan{the full twist is not displaying} \end{conjecture} We give more context for this conjecture in~\cref{subsec:pure}, including a conjectural characterization of $[\Red_{{\mathbb{T}_c}}(\mathbb{c})]$ as the EL-shelling orders of the noncrossing partition lattice (\Cref{conj:total_orders}). We view our results in this article, together with recent work of the first and third authors~\cite{defant2022pop}, as steps toward resolving these conjectures. \end{comment} \begin{figure}[htbp] \begin{center}{\includegraphics[height=5.745cm]{SBWDPIC1NNN}}\qquad\qquad\raisebox{-1.43cm}{\includegraphics[height=8.003cm]{SBWDPIC2N}} \end{center} \caption{On the top right is the $c$-noncrossing partition lattice of $\SSS_3$, where ${c=s_1s_2}$. On the bottom right is its order complex. On the left is the permutahedron $\Perm_\yy(\SSS_3)$, triangulated by $\SBDW_\yy(\SSS_3,c)$. Each simplex $\conv(w\C \yy)$ is drawn on the left in the same color as the edges in the chain $\C$ on the right. There are two ways of walking from $\yy$ up to $\wo\yy$ along the $1$-skeleton of $\Perm_\yy$; these paths are colored differently.}\label{fig:S3} \end{figure} \begin{comment} \begin{figure}[htbp] \begin{center}\includegraphics[height=9.937cm]{SBWDPIC4} \end{center} \caption{The Salvetti complex for the symmetric group $\SSS_3$ has six $0$-cells (one for each vertex of $\Perm_\yy$), twelve $1$-cells (one for each orientation of each edge of $\Perm_\yy$), and six $2$-cells (one for each of the oriented copies of $\Perm_\yy$ shown here). Quotienting by the action of $\SSS_3$ yields the quotient Salvetti complex. Each {\color{Teal}light blue} path corresponds to the expression ${\color{Teal}\mathbf{s}_1\mathbf{s}_2\mathbf{s}_1}$, while each {\color{Orange}orange} path corresponds to the expression ${\color{Orange}\mathbf{s}_2\mathbf{s}_1\mathbf{s}_2}$. One can push a {\color{Teal}light blue} path through a {\color{Gray}gray} 2-cell and onto an {\color{Orange}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 $\B_{\SSS_3}$.}\label{fig:6Salvetti} \end{figure} \begin{figure}[htbp] \begin{center}\includegraphics[height=9.937cm]{SBWDPIC3} \end{center} \caption{The Bessis--Brady--Watt complex $\Bessis(\SSS_3,s_1s_2)$ 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 $w\in\SSS_3$ (represented by a {\color{blue}blue} square) to the element $ws_1s_2$ (represented by a {\color{blue}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 \cref{fig:S3}. Quotienting by the action of $\SSS_3$ yields the quotient Bessis--Brady--Watt complex. }\label{fig:6BBW} \end{figure} \end{comment} \section{Triangulations of the permutahedron and linear stability} Recall that a \dfn{subdivision} of a polytope $P$ is a collection $\Theta$ of polytopes of the same dimension as $P$ such that $\bigcup\Theta=P$ and such that for all $P_1,P_2\in\Theta$, the intersection $P_1\cap P_2$ is a (possibly empty) common face of $P_1$ and $P_2$. A subdivision $\Theta$ is a \dfn{triangulation} if all polytopes in $\Theta$ are simplices. Fix a standard Coxeter element $c$ of $W$ and a point $\yy\in \BB$. The \dfn{Bruhat order} is the partial order $\leqB$ on $W$ with a cover relation $u \lessdot_\hB v$ if and only if $\ell_S(v)= \ell_S(u) + 1$ and $v=ut$ for some $t \in T$. We will construct a triangulation of $\Perm_\yy$ by leveraging an interplay between the noncrossing partition lattice $\NC(W,c)$ and the Bruhat order. Let \begin{equation}W_c^+:=\{u\in W:\ell_S(uc)=\ell_S(u)+\ell_S(c)\}.\label{eq:wplus}\end{equation} (One can show that $W_c^+$ is the set of elements less than or equal to $w_\circ c^{-1}$ in left weak order.) For each $u\in W_c^+$, the Bruhat interval $[u,uc]_\hB$---consisting of those elements between $u$ and $uc$ in the Bruhat order---is necessarily isomorphic to a subposet of $\NC(W,c)$ under the map $x \mapsto u^{-1}x$. A maximal chain $\{u=u_0\lessdot_\hB u_1\lessdot_\hB \cdots \lessdot_\hB u_r=uc\}$ in the interval $[u,uc]_\hB$ now defines a polytope $\conv\big(\{u_0\yy,u_1\yy,\ldots,u_r\yy\}\big)$, where $\conv(\cdot)$ denotes convex hull. This polytope is in fact an $r$-dimensional simplex. \begin{definition}\label{def:SBDW} Fix a standard Coxeter element $c$ of $W$ and a point $\yy\in \BB$. We define the \dfn{Salvetti--Brady--Delucchi--Watt triangulation} (or \dfn{SBDW triangulation}) $\SBDW_\yy(W,c)$ to be the set of simplices in $\Perm_\yy(W)$ of the form $\conv(\{u_0\yy,u_1\yy,\ldots,u_r\yy\})$, where $u\in W_c^+$ and ${\{u=u_0\lessdot_\hB u_1\lessdot_\hB \cdots \lessdot_\hB u_r=uc\}}$ is a maximal chain of the Bruhat interval $[u,uc]_\hB$. \end{definition} \begin{example} \cref{fig:S3} illustrates \cref{def:SBDW} for the symmetric group $\SSS_3$, with Coxeter element $c=s_1s_2$ (where $s_i=(i\,\,i+1)$). \end{example} It is not obvious that $\SBDW_\yy(W,c)$ is a triangulation of the $W$-permutahedron $\Perm_\yy$. Indeed, it is not clear that the union of the simplices is the entire $W$-permutahedron, nor that two simplices do not intersect in their interior, nor that intersections occur along common faces. Remarkably, these simplices coming from the interaction of $\NC(W,c)$ and the Bruhat order do form a triangulation of $\Perm_\yy$---to steal a phrase from Bessis, the noncrossing partition lattice is ``somehow real''~\cite{bessisslides}. \begin{theorem}\label{thm:triangulation} Let $W$ be a finite Coxeter group. For each standard Coxeter element $c$ of $W$ and each point $\yy\in\BB$, $\SBDW_\yy(W,c)$ is a triangulation of the $W$-permutahedron $\Perm_\yy(W)$. \end{theorem} To prove \cref{thm:triangulation}, we first assume that $W$ is simply-laced (i.e., of type $A$, $D$, or $E$). Under this assumption, we provide a type-uniform argument that relies on recent developments in quiver representation theory. Specifically, we invoke \emph{totally stable linear stability functions}, the existence of which was established in \cite{apruzzese2020stability,HuangHu,kinser2022total} in type~$A$ and in \cite{chang2024geometric} in general. Geometrically, specifying a totally stable linear stability function amounts to finding a point $\yy^\star \in \BB$ and a direction $\gamma$ so that shooting a laser from $\yy^\star$ in the direction of $\gamma$ crosses a certain piece of each hyperplane in the arrangement $\HH$ in a certain order (more precisely, the laser crosses the \emph{$c$-noncrossing shards} in the order specified by the inversion sequence of the $c$-sorting word for $w_\circ$). The time at which the hyperplane piece is crossed gives a height function, and the existence of such a function allows us to show that $\SBDW_{\yys}(W,c)$ is a \emph{regular} triangulation of $\Perm_{\yys}$ (for the particular $\yys\in\BB$ used in the stability function)---we refer the reader to \href{https://personal.utdallas.edu/~nxw170830/docs/Bessalvetti/laser.html}{an interactive visualization}~\cite{nathansite2}. We also establish a result that allows us to deform $\Perm_{\yys}$ into $\Perm_{\yy}$ for an arbitrary choice of $\yy$, thereby proving \cref{thm:triangulation} in general for simply-laced Coxeter groups. We use \emph{folding} to deduce the theorem for the remaining finite Coxeter groups. \section{Combinatorics of the SBDW triangulation}\label{subsec:properties-of-triang} While our height function gives us a coarse global understanding of the SBDW triangulation, we actually have very fine combinatorial control of the dual graph (see~\cite{nathansite} for an interactive animation, and \Cref{ex:visualization,fig:TrianglesLinear} for a static visualization for $\SSS_4$). To analyze the combinatorics of the SBDW triangulation, it is helpful to use the following alternative description. Let $\MCh(W,c)$ denote the set of maximal chains in the $c$-noncrossing partition lattice $\NC(W,c)$. There is a straightforward bijection from $\MCh(W,c)$ to the set $\Red_T(c)$ of reduced $T$-words for $c$ that sends the maximal chain $\C=\{\pi_0\lessdot_T \pi_1 \lessdot_T \cdots \lessdot_T \pi_r\}$ to the word $\rw(\C)=t_1\cdots t_r$, where $t_i=\pi_{i-1}^{-1}\pi_i$. \begin{definition} We say that a chain $\C=\{\pi_0\lessdot_T \pi_1 \lessdot_T \cdots \lessdot_T \pi_r\}\in\MCh(W,c)$ and an element $u \in W_c^+$ are \dfn{concordant} if $u\pi_0\leqB u\pi_1\leqB\cdots\leqB u\pi_r$. Let $\Omega(W,c)$ denote the set of pairs $(u,\C)\in W_c^+\times \MCh(W,c)$ such that $\C$ and $u$ are concordant. \end{definition} In this language, we have $\SBDW_\yy(W,c)=\{\conv(u\C \yy):(u,\C)\in\Omega(W,c)\},$ where $\conv(u\C\yy)=\conv(\{u\pi\yy:\pi\in\C\})\subseteq\Perm_\yy.$ We say two chains $\bpi,\bpi'\in\MCh(W,c)$ are \dfn{commutation equivalent} if their corresponding reduced $T$-words $\rw(\bpi)$ and $\rw(\bpi')$ are related by commutation moves (equivalently, these words use the same set of reflections). Write $\Class_c$ for the set of commutation classes of maximal chains in $\MCh(W,c)$, and let $\CL_{\bpi} \in \Class_c$ denote the commutation equivalence class of $\bpi$. For $u\in W_c^+$, let $\Class_c(u)$ denote the set of commutation equivalence classes of chains in $\MCh(W,c)$ that are concordant with $u$. We say $u$ and $\CL$ are concordant if $\CL \in \Class_c(u)$. There is a partial order $\preceq_c$ on $T$ coming from the heap of the $c$-sorting word for the long element $w_\circ$. We say a chain $\bpi\in\MCh(W,c)$ with $\rw(\bpi)=t_1\cdots t_r$ is \dfn{$\preceq_{c}$-increasing} (respectively, \dfn{$\preceq_c$-decreasing}) if there do not exist $1\leq i