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\usepackage{xspace}
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indention=0pt, singlelinecheck=true,labelfont=normalfont}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{lem}[theorem]{Lemma}
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\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ep}{\varepsilon}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\ra}{\rightarrow}
\newcommand{\se}{\subseteq}
\newcommand{\abs}[1]{\vert #1 \vert}
\newcommand{\la}{\mathbf{a}}
\newcommand{\B}{{\mathcal B}}
\newcommand{\Span}{\text{\rm Span}}
\newcommand{\Sym}{\text{\rm Sym}} 

\newcommand{\macd}[2]{j^{#1}_{#2}({\rm{sgn}})} 
\newcommand{\macrep}{\macd{\Phi}{nA_1}}

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arrows, decorations.pathmorphing, calc} 
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size=0.3em,inner sep=0.5pt}} 
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size=2em,fill=blue!20,inner sep=0pt}} 
\tikzset{small node/.style={circle,draw,minimum size=0.5em,inner
sep=2pt,font=\sffamily\bfseries}}

%% define your title in the usual way
\title{Orthogonal roots, Macdonald representations, and quasiparabolic $W$-sets}

%% 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{\texorpdfstring{R. M. Green\thanks{\href{mailto:rmg@colorado.edu}{rmg@colorado.edu}.}\addressmark{1} \and
 Tianyuan Xu\thanks{\href{mailto:tianyuan.xu@richmond.edu}{tianyuan.xu@richmond.edu}.}\addressmark{2}}{R. M. Green and
 Tianyuan Xu}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Colorado,
Boulder, Colorado, 80309, U.S.A.\\ 
\addressmark{2}Department of Mathematics and Statistics, University of Richmond,
Richmond, Virginia, 23173, U.S.A.}

%% put the date of submission here
\received{November 12, 2024}

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%\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{Let $W$ be a finite Weyl group with root system $\Phi$ and of rank $n>1$.
We study the maximal sets of orthogonal positive roots of $\Phi$ with
cardinality $n$, which exist if and only if $W$ has type $E_7, E_8$, or $D_n$ for
$n$ even. We show that in these types, the set $X$ of all such maximal
orthogonal sets forms a quasiparabolic $W$-set in the sense of Rains--Vazirani.
The quasiparabolic structure can be described in terms of certain quadruples of
orthogonal roots that we call crossings, nestings, and alignments. This leads to
noncrossing and nonnesting bases of a suitable irreducible representation of $W$
known as a Macdonald representation, as well as some highly structured partially
ordered sets, including the strong Bruhat poset of symmetric groups.  In type
$E_8$, we use the set $X$ to give a concise description of a graph that is known
to be non-isomorphic but quantum isomorphic to the orthogonality graph of the
$E_8$ root system.}

%% 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{root system, Macdonald representation, canonical basis, quasiparabolic
$W$-set, Specht module}

%% 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
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\addbibresource{nroots.bib}
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\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

\section{Introduction}
Root systems and Weyl groups are both fundamental objects in algebraic
combinatorics. In 1972, Macdonald \cite{macdonald72} described a simple yet
powerful method to construct an irreducible $\Q$-representation
$\macd{\Phi}{\Psi}$ of a finite Weyl group $W$ from the root system $\Phi$ of
$W$ and any subsystems $\Psi$ of $\Phi$. This paper studies such representations
arising from a proper subsystem $\Psi$ of $\Phi$ of type
$nA_1$ where $n$ is the rank of $\Phi$, whence $\Psi$ is of the form
$\{\pm\al_1\}\cup\{\pm\al_2\}\cup\dots\cup\{\pm\al_n\}$ for pairwise orthogonal
roots $\al_1,\al_2,\dots,\al_n$ in $\Phi$.  Such proper subsystems exist
precisely when $W$ has type $E_7, E_8$ or $D_{n}$ for $n$ even
(\Cref{fig:ade}), and in these cases the corresponding Macdonald
representation, which we henceforth denote by $\macrep$ as it does not depend on the choice
of $\Psi$, is spanned by what we call \emph{positive $n$-roots}.
As we explain in \Cref{sec:nroots}, the positive $n$-roots can be
identified in a natural and precise way  with the subsets of $\Phi$ consisting
of $n$ pairwise orthogonal positive roots.


\begin{figure}[h!]
    \centering
  %  \subfloat[{$A_n$}]
    %{
%\begin{tikzpicture}

            %\node[main node] (1) {};
            %\node[main node] (2) [right=1cm of 1] {};
      %%\node (33) [right=0.5cm of 32] {$\cdots$};
            %\node[main node] (3) [right=1.5cm of 2] {};
            %\node[main node] (4) [right=1cm of 3] {};

            %\node (11) [below=0.1cm of 1] {\small{$1$}};
            %\node (22) [below=0.1cm of 2] {\small{$2$}};
            %\node (33) [below=0.1cm of 3] {\small{$n-1$}};
            %\node (44) [below=0.15cm of 4] {\small{$n$}};

            %\path[draw]
            %(1)--(2)
            %(3)--(4);

            %\path[draw,dashed]
            %(2)--(3);
%\end{tikzpicture}
%}
%\quad\quad\quad
\subfloat[{$D_n$}]
{
\begin{tikzpicture}

            \node[main node] (1) {};
            \node[main node] (2) [right=1cm of 1] {};
      %\node (33) [right=0.5cm of 32] {$\cdots$};
            \node[main node] (3) [right=1.5cm of 2] {};
            \node[main node] (4) [right=1cm of 3] {};
            \node[main node] (5) [above right=0.7cm and 0.9cm of 4] {};
            \node[main node] (6) [below right=0.7cm and 0.9cm of 4] {};

            \node (11) [below=0.1cm of 1] {\small{$1$}};
            \node (22) [below=0.1cm of 2] {\small{$2$}};
            \node (33) [below=0.1cm of 3] {\small{$n-3$}};
            \node (44) [right=0.1cm of 4] {\small{$n-2$}};
            \node (55) [right=0.1cm of 5] {\small{$n-1$}};
            \node (66) [right=0.1cm of 6] {\small{$n$}};

            \path[draw]
            (1)--(2)
            (3)--(4)--(5)
            (4)--(6);

            \path[draw,dashed]
            (2)--(3);
\end{tikzpicture}
}\\
%\quad\quad\quad
%\subfloat[{$E_6$}]
%{
%\begin{tikzpicture}
    %\node[main node] (1) {};
    %\node[main node] (2) [right=1cm of 1] {};
            %\node[main node] (3) [right=1cm of 2] {};
            %\node[main node] (4) [right=1cm of 3] {};
            %\node[main node] (5) [right=1cm of 4] {};
            %\node[main node] (6) [above=1cm of 3] {};
            %%\node[main node] (8) [right=1cm of 6] {};

            %\path[draw]
            %(1)--(2)--(3)--(4)--(5)
            %(3)--(6);

            %\node (11) [below=0.1cm of 1] {\small{$1$}};
            %\node (22) [below=0.1cm of 2] {\small{$2$}};
            %\node (33) [below=0.1cm of 3] {\small{$3$}};
            %\node (44) [below=0.1cm of 4] {\small{$4$}};
            %\node (55) [below=0.1cm of 5] {\small{$5$}};
            %\node (66) [above=0.1cm of 6] {\small{$6$}};
%\end{tikzpicture}
%}\\
%\quad\quad\quad
\subfloat[{$E_7$}]
{
\begin{tikzpicture}
    \node[main node] (1) {};
    \node[main node] (2) [right=1cm of 1] {};
            \node[main node] (3) [right=1cm of 2] {};
            \node[main node] (4) [right=1cm of 3] {};
            \node[main node] (5) [right=1cm of 4] {};
            \node[main node] (6) [right=1cm of 5] {};
            \node[main node] (7) [above=1cm of 3] {};
            %\node[main node] (8) [right=1cm of 6] {};

            \path[draw]
            (1)--(2)--(3)--(4)--(5)--(6)
            (3)--(7);

            \node (11) [below=0.1cm of 1] {\small{$1$}};
            \node (22) [below=0.1cm of 2] {\small{$2$}};
            \node (33) [below=0.1cm of 3] {\small{$3$}};
            \node (44) [below=0.1cm of 4] {\small{$4$}};
            \node (55) [below=0.1cm of 5] {\small{$5$}};
            \node (66) [below=0.1cm of 6] {\small{$6$}};
            \node (77) [above=0.1cm of 7] {\small{$7$}};
\end{tikzpicture}
}
\quad
\subfloat[{$E_8$}]
{
\begin{tikzpicture}
    \node[main node] (1) {};
            \node[main node] (3) [right=1cm of 1] {};
            \node[main node] (4) [right=1cm of 3] {};
            \node[main node] (5) [right=1cm of 4] {};
            \node[main node] (6) [right=1cm of 5] {};
            \node[main node] (7) [right=1cm of 6] {};
            \node[main node] (8) [right=1cm of 7] {};
            \node[main node] (2) [above=1cm of 4] {};

            \path[draw]
            (1)--(3)--(4)--(5)--(6)--(7)--(8)
            (2)--(4);

            \node (11) [below=0.1cm of 1] {\small{$1$}};
            \node (22) [above=0.1cm of 2] {\small{$2$}};
            \node (33) [below=0.1cm of 3] {\small{$3$}};
            \node (44) [below=0.1cm of 4] {\small{$4$}};
            \node (55) [below=0.1cm of 5] {\small{$5$}};
            \node (66) [below=0.1cm of 6] {\small{$6$}};
            \node (77) [below=0.1cm of 7] {\small{$7$}};
            \node (88) [below=0.1cm of 8] {\small{$8$}};

\end{tikzpicture}
}
\caption{Dynkin diagrams for root systems of type $D_n, E_7$ and $E_8$.}
\label{fig:ade}
\end{figure}


The goal of this paper is to explore the rich combinatorial properties of
positive $n$-roots and use them to elucidate the structure of
the representation $\macrep$.  Our results have natural connections to many
previous works.  In type $D_{n}$ for $n=2k$ even, the positive $n$-roots can be
identified with the perfect matchings of the set $[n]=\{1,2,\dots,n\}$ and with
the fixed-point-free involutions of the symmetric group $S_n$. The Macdonald
representation $\macrep$ in this case is a lift of the Specht module $S^{(k,k)}$
of $S_n$ indexed by the two-row partition $(k,k)\vdash n$ studied recently in
\cite{rhoades17,russell19,hwang23}, and our results have implications for
$S^{(k,k)}$ (\Cref{rmk:specht}).  In type $E_7$, $\macrep$ is known as the
Coble representation \cite[Proposition 4.12]{colombo10}, which has a long
history going back to the work of Coble in 1916 on the G\"opel variety
\cite[(65)]{coble1916}. In type $E_8$, we use $8$-roots to give a convenient
construction of a certain strongly regular graph
recently studied by Schmidt \cite{schmidt24} in connection to quantum
isomorphisms of graphs in the sense of Atserias et al \cite{atserias19} (\Cref{prop:XI}.3). 

Let $W$ be a Weyl group of type $E_7$, $E_8$, or $D_{n}$ for $n$ even, and let
$\Phi$ be the root system of $W$. The positive $n$-roots of $W$ can be
identified with the subsets of $\Phi$ consisting of $n$ pairwise orthogonal
positive roots, and the collection $X$ of all such subsets admits a $W$-action
induced by the action of $W$ on $\Phi$. The set $X$ turns out to have a rich
structure as a $W$-set: \Cref{thm:qp}, our first main theorem, shows that
$X$ is a quasiparabolic $W$-set in the sense of Rains--Vazirani \cite{rains13}.
A {quasiparabolic $W$-set} for a Weyl group $W$ is a $W$-set equipped with an
integer-valued level function satisfying certain axioms that specify how the
action of a reflection changes the level (\Cref{def:qpset}). These
axioms generalize properties satisfied by the natural action of $W$ on any
quotient $W/W_I$ of $W$ by a parabolic subgroup $W_I$, and they enable the
deformation of the $W$-action on $X$ to create a module for the
Iwahori--Hecke algebra of $W$ %\cite{rains13}. 
\cite[Section 7]{rains13}. 
In our setting, the
$W$-action on $X$ in type $D_{n}$ for $n$ even induces an action of $S_n$ on
$X$, and \Cref{thm:qp} implies that the fixed-point-free involutions in $S_n$ form
a quasiparabolic $S_n$-set, which is the one of the original motivating examples
for the definition of quasiparabolic $W$-sets %\cite{rains13}.  
\cite[Section~4]{rains13}.  

To understand the quasiparabolic structure of $X$, it is useful to consider
certain special quadruples of orthogonal positive roots that we call
\emph{alignments}, \emph{crossings}, and \emph{nestings}. These terms are
motivated by the perfect matching diagrams corresponding to the positive
$4$-roots in type $D_{4}$ (\Cref{fig:features}), but they are defined in a
type-free way and prove to be useful for understanding $X$ even in types $E_7$
and $E_8$, where no diagrammatic interpretation of $n$-roots seems to be
available.  The level function associated to the quasiparabolic set $X$ can be
obtained by counting crossings and nestings, and levels of $n$-roots in type
$D_n$ have interesting connections to combinatorial game distributions of
Steiner systems, Gaussian $q$-distributions, Laguerre polynomials, and $q$-Bessel
numbers; %see \cite{gx5}.
see \cite[Section 6.1]{gx5}. 

%$\lambda(\gamma)=C(\gamma)+2N(\gamma)$ for each $\gamma\in X$, where $C(\gamma)$
%and $N(\gamma)$ are the numbers of crossings and nestings in $\gamma$,
%respectively, and  

\begin{figure}[h!]
    \centering
\subfloat[{$\gamma_A$}]
{
    \begin{tikzpicture}    
        \draw (1,0) arc[start angle=0, end angle=-180, x radius = 0.5, y
        radius=0.8];
        \draw (2.8,0) arc[start angle=0, end angle=-180, x radius = 0.5, y
        radius=0.8];
        \draw (-0.4,0)--(3.2,0);
    \end{tikzpicture}
 }
\quad\quad\quad
\subfloat[{$\gamma_C$}]
{
\begin{tikzpicture}   
        \draw (2,0) arc[start angle=0, end angle=-180, x radius = 1, y
        radius=1];
        \draw (3,0) arc[start angle=0, end angle=-180, x radius = 1, y
        radius=1];
        \draw (-0.4,0)--(3.4,0);
    \end{tikzpicture}
} 
\quad\quad\quad
\subfloat[{$\gamma_N$}]
{
    \begin{tikzpicture}    
        \draw (3,0) arc[start angle=0, end angle=-180, x radius = 1.5, y
        radius=1];
        \draw (2,0) arc[start angle=0, end angle=-180, x radius = 0.5, y
        radius=0.5];
        \draw (-0.4,0)--(3.4,0);
    \end{tikzpicture}
}

\caption{Perfect matchings corresponding to the positive $4$-roots in type $D_4$,
    which include exactly one alignment, $\gamma_A$, one crossing, $\gamma_C$,
    and one nesting, $\gamma_N$.  These $4$-roots satisfy the three-term
relation $\gamma_C=\gamma_N+\gamma_A$, which can be interpreted as a skein
relation among the matchings.}
\label{fig:features}
\end{figure}


We define a positive $n$-root to be \emph{noncrossing}, \emph{nonnesting}, or
\emph{alignment-free} if its set of components contains no crossing, nesting, or
alignment as a subset, respectively. These ``feature-avoiding'' elements have
many remarkable properties, and we summarize some of them in
\Cref{thm:summary}, our second main result. In particular, we prove that
the noncrossing elements of $X$ form a canonical basis of $\macrep$ that behaves
in $\macrep$ somewhat like a simple system in the reflection representation of
$W$.  The set of nonnesting elements not only forms another basis of $\macrep$
dual to the noncrossing basis, but also has the structure of a distributive
lattice induced by the weak Bruhat order on $W$. The alignment-free elements
form a quasiparabolic $W_I$-set, $X_I$, for a suitable parabolic subgroup $W_I$
of $W$.  Moreover, the three types of feature-avoiding $n$-roots also fit
together in satisfying ways. We define a equivalence relation on $X$, called
\emph{$\sigma$-equivalence}, whose equivalence classes induce a canonical
bijection between the noncrossing and nonnesting elements of $X$. Any set of
$\sigma$-equivalence class representatives forms a $\Q$-basis for
$\macd{\Phi}{nA_1}$, and any two such bases, which
include the nonnesting and noncrossing bases of $\macrep$, admit a 
unitriangular change of basis matrix with integer entries. 
%The set $X_I$ turns out to be canonically isomorphic to the symmetric group
%$S_k$ as a poset in type $D_{2k}$, and to be in a canonical bijection with the
%30 inequivalent labellings of the Fano plane in type $E_7$. 

The rest of this abstract is organized as follows. \Cref{sec:prelim}
reviews preliminary materials leading to the notion of positive $n$-roots.
\Cref{sec:qp} introduces crossings, nestings, and alignments, and then
explains how the positive $n$-roots form a quasiparabolic $W$-set.
\Cref{sec:fa} discusses feature-avoiding $n$-roots, including both
type-independent results and some type-specific properties of the alignment-free
positive $n$-roots.  This abstract is based on the full paper \cite{gx5}, where
all proofs omitted in the abstract can be found. 

%\begin{figure} \centering \includegraphics[width=0.5\textwidth]{plot}
%      \caption{A plot of a function.}
%\label{fig:plot}
%\end{figure}
\section{Preliminaries}
\label{sec:prelim}
We recall the construction of Macdonald representations and explain their
connections to positive $n$-roots in this section. Throughout the abstract, we
work over the rational field $\Q$ and assume basic familiarity with root
systems. For notation and terminology related to root systems, we follow 
\cite[Chapter 1]{humphreys90}.
%except in the construction of the root
%system of type $E_7$, for which we use a convention \cite{} compatible with
%labelling of Fano planes.


\subsection{Macdonald representations}
\label{sec:mac}
Let $W$ be a finite Weyl group with a root system $\Phi$ of rank $n$. Let
$\Gamma$ be the Dynkin diagram of $\Phi$ shown in \Cref{fig:ade}, and let
$\Delta=\{\al_i: i\in \Gamma\}$ be a simple system of $\Phi$. Let $V$ be the
reflection representation of $W$ defined over $\Q$. We recall that $\Delta$ is a
basis of $V$, and that $V$ admits a positive-definite bilinear form $B$ defined
by 
\[ (\alpha_i, \alpha_j) =
    \begin{cases} 2 & \text{\ if \ } i = j;\\ -1 & \text{\ if \ $i$\ and\ $j$\
      are\ adjacent\ in\ } \Gamma;\\ 0 & \text{\ otherwise.}\\
      \end{cases} 
\]
Each root $\al\in \Phi$ gives rise to a reflection, $s_\al$, which acts on
$\Phi$ by the formula \[s_\al(\beta)=\be-B(\al,\beta)\al\] 
and extends linearly to an action on $V$. For each simple root $\al_i$, we write
$s_i$ for $s_{\al_i}$.  The bilinear form $B$ is $W$-invariant in the sense that
$B(\al, \be) = B(w.\al, w.\be)$ for all $w \in W$ and all $\al, \be \in \Phi$.
We say two roots are $\al,\beta\in \Phi$ are \emph{orthogonal} if
$B(\al,\be)=0$.
     
Let $\Psi$ be a {\it subsystem} of $\Phi$, i.e., a nonempty subset of $\Psi$
which is itself a root system. Let $\Phi^+$ be the set of positive roots in
$\Phi$, and let $\Psi^+=\Psi\cap\Phi^+$. The positive-definite form $B$ allows us to identify $V$ with
its dual, $V^*$, and the symmetric algebra $\Sym(V^*)$ is a $W$-module under
the contragredient action $(w\cdot\phi)(x) = \phi(w^{-1}(x))$.  The {\it Macdonald
representation of $W$ associated to $\Psi$}, denoted $\macd{\Phi}{\Psi}$, is the
cyclic $W$-submodule of $\Sym(V^*)$ generated by the element $\pi_\Psi =
\prod_{\al \in \Psi^+} \al.$ The representation $\macd{\Phi}{\Psi}$ is irreducible, and all 
irreducible representations of $W$ can be constructed as a Macdonald representation 
for a suitable choice of $\Psi$ in types $A_n$ and $B_n$ \cite{macdonald72}. 

\subsection{Positive \texorpdfstring{$n$}{n}-roots}
\label{sec:nroots}
When $W$ has type $E_7, E_8$ or $D_n$ for $n$ even, the root system $\Phi$
contains proper subsystems $\Psi$ of type $nA_1$, and we denote the
corresponding Macdonald representation $\macd{\Phi}{\Psi}$ as $\macrep$. 
The set $\Psi^+$ consists of $n$ orthogonal positive roots, and we call each
element of the form $w.\pi_\Psi\in \macrep$ an {\it $n$-root}. Since the
bilinear form $B$ is $W$-invariant, every $n$-root has the form $\al = \prod_{i = 1}^n
\be_i$ where the elements $\be_i$ are orthogonal roots. Conversely, every
product of $n$ orthogonal roots is an $n$-root, because $W$ acts transitively on
the set of maximal sets of orthogonal roots (\cite[Lemma 3.1.2]{gx5}).

We say an $n$-root $\al$ is {\it positive} if it can be written in the form
$\al=\prod_{i=1}^n \beta_i$ where all the factors $\be_i$ are positive, and we
call the roots $\be_i$ the \emph{components} of $\al$ in this case. The
components are well defined because they are the irreducible factors of $\al$ in
the unique factorization domain $\Q[\al_1, \al_2, \ldots, \al_n]$. It follows
from the definitions that if $\al$ is an $n$-root then $-\al$ is also an
$n$-root, and that precisely one of $\al$ and $-\al$ is positive. Whenever $\al$
is either a root or an $n$-root, we define the \emph{absolute value} of $\al$ to
be the positive element in the pair $\{\al,-\al\}$ and denote it by $\abs{\al}$.  

The set $\Phi^+_n$ of all positive $n$-roots admits a natural $W$-action given
by $w(\al)=\abs{w.\al}$. Similarly, the set $X$ of sets of $n$ orthogonal
positive roots admits a $W$-action given by
$w(\{\be_1,\cdots,\be_n\})=\{\abs{w(\be_1)}, \cdots, \abs{w(\be_n)}\}$.  The map
sending each set $\{\be_1,\cdots,\be_n\}\in X$ to the product $\prod_{i=1}^n
\be_i\in \Phi^+_n$ respects these two $W$-actions, and we will henceforth use it
to identify $X$ with $\Phi^+_n$. This identification provides a convenient
passage between the set $X$ and the representation $\macrep$.  

\subsection{Root systems of types \texorpdfstring{$A_n$}{An} and \texorpdfstring{$D_n$}{Dn}}
\label{sec:constructions}
We now review some well-known facts about root systems of types $A_n$ and $D_n$,
including explicit constructions of the root systems. Details about similar
constructions for types $E_7$ and $E_8$ are also relevant to our treatment of
$n$-roots, and they are described in \cite[Section 2.3]{gx5},
but we omit them here because they will not be directly used in this
abstract.

Let $\ep_1, \ep_2, \ldots, \ep_n$ be an orthonormal basis for $\R^n$.  The
vectors $\{\ep_i - \ep_j : 1 \leq i \ne j \leq n\}$ form a root system of type
$A_{n-1}$.  The simple roots $\{\al_1, \al_2, \ldots, \al_{n-1}\}$ are given
by $\al_i = \ep_i - \ep_{i+1}$.  The bilinear form $B$ is the usual dot product.
A root $\ep_i - \ep_j$ is positive if $i < j$ and negative if $i > j$.   The
Weyl group is isomorphic to the symmetric group $S_n$ and acts on $\Phi$ by permuting the basis
$\ep_1,..,\ep_n$.  
 

The vectors $\{\pm\ep_i \pm \ep_j : 1 \leq i < j \leq n\}$ form a root system of
type $D_n$.  The simple roots $\{\al_1, \al_2, \ldots, \al_n\}$ are given by
$\al_i = \ep_i - \ep_{i+1}$ for $i < n$, and $\al_n = \ep_i + \ep_{i+1}$. If $i
< j$, then the roots $\ep_i \pm \ep_j$ are positive, and the roots $-\ep_i \pm
\ep_j$ are negative.  The bilinear form $B$ is the usual dot product.  The Weyl
group acts on $\Phi$ by signed permutations of the orthonormal basis, with the
restriction that each element effects an even number of sign changes
\cite[Section 2.10]{humphreys90}.  There is a well-known homomorphism $\phi$
from $W$ to $S_{n}$ resulting from forgetting the signs in a signed permutation,
given as follows: 

\begin{equation}
    \label{eq:phimap} \phi(s_i) = \begin{cases} (i, i+1) & \text{if\ } i < n\\
        (n-1, n) & \text{otherwise.}
    \end{cases}
\end{equation}


\subsection{A key example}
\label{sec:key}

Suppose that $\Phi$ is a root system of type $D_n$ for an even integer $n=2k\ge
4$.  It follows from \Cref{sec:constructions} that two roots $\al, \be\in
\Phi$ are orthogonal if and only if either (a) $\al$ and $\be$ have disjoint
support or (b) $\al$ and $\be$ have the same support, but $\al \ne \pm \be$.
Using this fact, it is straightforward to show that every orthogonal set of
positive roots in $\Phi$ that is maximal with respect to set containment has the
form $$ R=\{ \ep_{i_1} - \ep_{j_1}, \ \ep_{i_1} + \ep_{j_1}, \ \ep_{i_2} -
\ep_{j_2}, \ \ep_{i_2} + \ep_{j_2}, \ \ldots, \ep_{i_k} - \ep_{j_k}, \ \ep_{i_k}
+ \ep_{j_k} \}$$ for numbers $i_1, j_1, \dots, i_k, j_k$ such that $i_1<j_1,
\dots, i_k<j_k,$ and $\{i_1,j_1,\dots,i_k,j_k\}=[n]$.  Identifying the set $R$
with the perfect matching  $\{ \{i_1, j_1\}, \ \{i_2, j_2\},\ \ldots, \{i_k,
j_k\}\}$, we obtain a bijection between the positive $n$-roots and the perfect
matchings of $[n]$.  Here, by a perfect matching of $[n]$ we mean a collection
of pairwise disjoint size-2 subsets of $[n]$ whose union is the whole of $[n]$.

The case where $\Phi$ has type $D_4$ turns out to be particularly instructive
for understanding the general theory of $n$-roots in all types.
In type $D_4$, there are exactly three $n$-roots, namely:
\[
\gamma_A=(\ep_1-\ep_2)(\ep_1+\ep_2)(\ep_3-\ep_4)(\ep_3+\ep_4)=(\ep_1^2-\ep_2^2)(\ep_3^2-\ep_4^2),\]
\[\gamma_{C}=(\ep_1-\ep_3)(\ep_1+\ep_3)(\ep_2-\ep_4)(\ep_2+\ep_4)=(\ep_1^2-\ep_3^2)(\ep_2^2-\ep_4^2),\]
and
\[
\gamma_N=(\ep_1-\ep_4)(\ep_1+\ep_4)(\ep_2-\ep_3)(\ep_2+\ep_3)=(\ep_1^2-\ep_4^2)(\ep_2^{2}-\ep_3^2).\]
Viewed as maximal orthogonal sets, these 4-roots form a partition of the 12
positive roots of $\Phi$, and they correspond to the three perfect matchings
$\{\{1,2\}\,\{3,4\}\},\{\{1,3\}\,\{2,4\}\},$ and $\{\{1,4\}\,\{2,3\}\}$ depicted
in \Cref{fig:features}, respectively. Furthermore, these $4$-roots
satisfy the relation
\[
\gamma_C=\gamma_N+\gamma_A
\]
by a straightforward application of the Ptolemy relation \[
(A-C)(B-D)=(A-D)(B-C)+(A-B)(C-D),\] and they have the property that for any
component $\al$ in one of them, the reflection $s_\al$ interchanges the other
two $4$-roots in the Macdonald representation.

The above facts about the $4$-roots in type $D_4$ are important because in all of the types $E_7$, $E_8$, and $D_n$ for $n$ even, the action of each
reflection in the Weyl group on a positive $n$-root is always
controlled by a suitable type-$D_4$ subsystem of $\Phi$ (\Cref{prop:local}, 2.).  The three-term relation $\gamma_C=\gamma_N+\gamma_A$
provides an important connection between the combinatorial structure of the set
$X$ as a $W$-set and the linear structure of the Macdonald representation
$\macrep$ as a $W$-module. In particular, the three-term relation can help
establish the noncrossing and nonnesting bases of $\macrep$.


\section{Quasiparabolic Structure}
\label{sec:qp}
In this section, we recall the definition of quasiparabolic $W$-sets and explain
how the  positive $n$-roots form a quasiparabolic $W$-set in types $E_7, E_8$,
and $D_n$ for $n$ even.

\subsection{Quasiparabolic W-sets}
\label{sec:rv}
Rains and Vazirani introduced quasiparabolic $W$-sets for a general Coxeter
system $(W,S)$ and associated a partial order to them as follows:

\begin{defn} \cite[Definitions 2.1, 2.3, and 5.1]{rains13}
    \label{def:qpset} Let $W$ be a Weyl group with generating set $S$ and set of
          reflections $T$.  A {\it scaled $W$-set} is a pair $(X, \lambda)$,
          where $X$ is a $W$-set and $\lambda : X \rightarrow \Z$ is a function, called the \emph{level} function,
          such that $|\lambda(sx) - \lambda(x)| \leq 1$ for all $s \in S$.  A
          {\it quasiparabolic set} for $W$ is a scaled $W$-set $X$ satisfying
          the following two properties:
          \begin{enumerate}          
              \item {for any $r \in T$ and $x \in X$, if $\lambda(rx) =
               \lambda(x)$, then $rx = x$;}
           \item {for any $r \in T$, $x \in X$, and $s \in S$, if
               $\lambda(rx) > \lambda(x)$ and $\lambda(srx) < \lambda(sx)$, then
           $rx = sx$.}
          \end{enumerate}
For each quasiparabolic $W$-set $(X,\lambda)$,
the \emph{quasiparabolic order} on a $X$ is the weakest
partial order $\leq_Q$ such that $x \leq_Q rx$ whenever we have $x \in X$, $r \in T$,
and $\lambda(x) \leq \lambda(rx)$.
\end{defn}
We note that Rains and Vazirani call $\lambda$ the {\it height} function, and
$\leq_Q$ the {\it Bruhat order}, but we have chosen to call them the {\it level}
function and the {\it quasiparabolic order} because of the potential for confusion
in the context of this abstract. 

\subsection{Crossing, Nestings, and Alignments}
\label{sec:cna}
To explain how positive $n$-roots form a quasiparabolic $W$-set, we need to
specify the level function $\lambda$. We do so by examining certain special
quadruples of orthogonal roots:

\begin{defn}%\cite[Definition 3.1.4]{gx5}
    \label{def:coplanar}
    Let $\Phi$ be a root system of type $E_7, E_8$ or $D_n$ for $n$ even. We say
    a quadruple $Q= \{\be_1, \be_2, \be_3, \be_4\}$ of four mutually orthogonal
    positive roots in $\Phi$ is a \emph{coplanar quadruple} if the element
    $\gamma=(\be_1+\be_2+\be_3+\be_4)/2$ is a root in $\Phi$.
\end{defn}

The following proposition reveals intimate connections between coplanar
quadruples, subsystems of type $D_4$ in $\Phi$, and the $W$-action on positive
$n$-roots. Note that if $\al$ is a root in a maximal orthogonal set $R$ then
the reflection $s_\al$ fixes $R$ in the $W$-action on $X$, hence the more
interesting actions arise in the case where $\al\not\in R$, as treated in the
second part of the proposition below.

\begin{prop}
    \label{prop:local} 
Let $\Phi$ be a root system of type $E_7, E_8$ or $D_n$ for $n$ even. Let
$Q=$$\{\be_1,\be_2,\be_3,\be_3\}$ be a set of four mutually orthogonal positive
roots in $\Phi$, and let  $\gamma = (\be_1 + \be_2 + \be_3 + \be_4)/2$.  
\begin{enumerate}
    \item %\cite[Proposition 3.1.3]{gx5} 
        The following are equivalent:
\begin{enumerate}
    \item the set $Q$ is coplanar, i.e., {$\gamma$ is a root;}
 \item{the set $Q$ is contained in a subsystem $\Psi\se \Phi$ of type $D_4$;}
 \item there is a unique subsystem $\Psi$ of type $D_4$, $\Psi_Q$, such that
 $(Q\cup\{\gamma\}) \subset \Psi_Q \subseteq \Phi$.
 \end{enumerate}
 \item %\cite[Proposition 3.1.8]{gx5} 
     Let $\al\in \Phi$ and let $R$ be a set of $n$ orthogonal
     positive roots such that $\al\not\in R$.
\begin{enumerate}
\item The root $\al$ is orthogonal to all but precisely four elements $Q =
     \{\be_1, \be_2, \be_3, \be_4\}$ of $R$. The elements of $Q$ form a coplanar
     quadruple.  
 \item For the subsystem $\Psi_Q$ as defined in 1.(c), we have $\alpha\in
      \Psi_Q$, and the set $s_\al(Q)=\{s_\al(\be_i):1\le i\le 4\}$ is also a
      coplanar quadruple contained in $\Psi_Q$.
\end{enumerate}
\end{enumerate}
\end{prop}
 
To help clarify the relationship between the coplanar quadruples $Q$ and
$s_\al(Q)$ in the setting of \Cref{prop:local}.2, we introduce the
following notions:

\begin{defn}%\cite[Definition 3.2.1]{gx5}
\label{def:nestcross} Let $Q = \{\be_1, \be_2, \be_3, \be_4\}$ be a coplanar
      quadruple, let $\Psi=\Psi_Q$ be the $D_4$-subsystem associated to $Q$ as
      in \Cref{prop:local}, let $\leq$ be the partial order on $\Psi$
      relative to the induced simple roots of $\Psi$, and let $\gamma=(\be_1 +
      \be_2 + \be_3 + \be_4)/2$. We say that $Q$ is
\begin{enumerate}
\item {a {\it crossing} if $\be_i \leq \gamma$ for all $i$ and $Q$ contains the
     unique maximal element of the set $Q \cup (-s_\gamma(Q))$ with respect to $\le$;}
\item {a {\it nesting} if $\be_i \leq \gamma$ for all $i$ and $Q$ contains the
    unique minimal element of the set $Q \cup (-s_\gamma(Q))$ with respect to $\le$;}
\item {an {\it alignment} otherwise.}
\end{enumerate} 
We call each crossing, nesting, or alignment a \emph{feature}.  For each maximal
orthogonal set $R\in X$, we define $A(R), C(R)$ and $N(R)$ to be the numbers of
alignments, crossings, and nestings contained in $R$, respectively.
\end{defn}

It is straightforward to verify that in type $D_4$, the three $4$-roots
$\gamma_A, \gamma_C, \gamma_N$ from \Cref{sec:key} form an alignment, a
crossing, and a nesting, respectively, which is compatible with the diagrams of
their corresponding perfect matchings in the obvious sense.  In general types,
it can also be shown that a coplanar quadruple is always a crossing, a nesting,
or an alignment, but never multiple features at once \cite[Theorem
3.2.2.(ii)]{gx5}. The purpose of \Cref{def:nestcross} is to
distinguish the three coplanar quadruples in a type-$D_4$ subsystem using only
abstract properties of roots, and doing so allows us to develop the theory of
$n$-roots even in types $E_7$ and $E_8$, where no convenient diagrammatic
interpretation of $n$-roots seems to be available. 

\subsection{The positive \texorpdfstring{$n$}{n}-roots form a quasiparabolic \texorpdfstring{$W$}{W}-set.}

\label{sec:thmA}
We are ready to state our first main theorem precisely. Recall from \Cref{sec:nroots} that for each
root $\alpha$, we denote by $\abs{\alpha}$ the absolute value of $\alpha$.
\begin{theorem}\label{thm:qp}
%\cite[Theorem 4.1.5]{gx5}
Let $W$ be a Weyl group of type $E_7$, $E_8$, or $D_n$ for $n$ even, and let $X$
be the set of maximal orthogonal sets of positive roots of $W$, regarded as a
$W$-set under the action \[ w(\{\be_1,\cdots,\be_n\})=\{\abs{w(\be_1)}, \cdots,
\abs{w(\be_n)}\}.\] The pair $(X, \lambda)$ is a quasiparabolic $W$-set,
where $\lambda : X \rightarrow \Z$ is the level function given by $\lambda(x) = C(x) +
2N(x)$.
\end{theorem}

The proof of the theorem involves a detailed analysis of how a reflection
$s_\al$ acts on a maximal orthogonal set $R$. Using
\Cref{prop:local}, it can be shown that $s_\al$ either fixes $R$ or
changes exactly four roots in $R$ forming a coplanar quadruple $Q$ within a
type-$D_4$ subsystem $\Psi_Q$ of $\Phi$ to a different feature, but considerable
care is required to deal with other coplanar quadruples in $R$ that overlap with
$Q$, which may undergo changes not local to $\Psi_Q$ under the action of
$s_\al$.  For example, in type $D_6$, as the reflection $s_\al$ for the root
$\al=\ep_1-\ep_3$ takes the maximal orthogonal set
$R=\{\ep_1\pm\ep_4,\ep_2\pm\ep_5,\ep_3\pm\ep_6\}$ to the set $\{\ep_1\pm\ep_6,
\ep_2\pm\ep_5,\ep_3\pm\ep_4\}$, it not only changes the crossing
$Q=\{\ep_1\pm\ep_4,\ep_3\pm\ep_6\}$ consisting of the roots not orthogonal to $\al$
to a nesting, but also causes the crossings $\{\ep_1\pm\ep_4,\ep_2\pm\ep_5\}$
and $\{\ep_2\pm\ep_5,\ep_3\pm\ep_6\}$ in $R$ to become nestings.

\section{Feature-avoiding Elements}
\label{sec:fa}
We study the noncrossing, nonnesting, and alignment-free positive $n$-roots in
this section. To understand the interactions of these feature-avoiding elements,
it is helpful to define two $n$-roots to be \emph{$\sigma$-equivalent} if their
components have the same sum. This is an equivalence relation, and we call each
equivalence class of the relation a \emph{$\sigma$-class}.

\subsection{Type-independent properties}
\label{sec:ind}
The following theorem is a summary of selected results from \cite[Sections
5.1--5.5]{gx5}. We omit the proofs, but note that the three-term relation
$\gamma_C=\gamma_N+\gamma_A$ plays a crucial role in the proofs of all the
results summarized.

\begin{theorem}\label{thm:summary} Let $W$ be a Weyl group of type $E_7$, $E_8$,
      or $D_n$ for $n$ even, and let $X$ be the set of maximal orthogonal sets
      of positive roots of $W$.
\begin{enumerate}
\item %\cite[Theorems 5.2.2 and 5.3.2]{gx5} 
The noncrossing $n$-roots in $X$ form a canonical $\Q$-basis, $\B_{NC}$, for the
Macdonald representation $\macd{\Phi}{nA_1}$. Moreover, every $n$-root of $W$ is
a $\Z$-linear combination of $\B_{NC}$ with coefficients of like sign.
Equivalently, the basis $\B_{NC}$ is a sign-coherent basis for $\macrep$ in the
sense of \cite[Definition 6.12]{fomin07}.

\item  %\cite[Theorems 5.2.2 and 5.4.5]{gx5} 
The nonnesting $n$-roots in $X$ form a $\Q$-basis, $\B_{NN}$, for
$\macd{\Phi}{nA_1}$.  Moreover, the set $\B_{NN}$ has the structure of a
distributive lattice isomorphic to the lattice $L=\{v\in W: v\le_L w_N\}$ for a suitable
element $w_N$, where $\le_L$ is the left weak Bruhat order on $W$.

\item %\cite[Proposition 5.1.2, Remark 5.1.3]{gx5} 
The alignment-free $n$-roots in $X$ form a quasiparabolic $W_I$-set, $X_I$,
where $W_I$ is a suitable standard parabolic subgroup of $W$, under the
restriction of the level function $\lambda$ on $X$.  The set $X_I$ also has a
bipartite structure: the $n$-roots in $X_I$ with even levels and those with
odd levels partition $X_I$ into two equal-sized components, and these components
are interchanged by every reflection in $W_I$.


\item  %\cite[Proposition 5.5.1]{gx5} 
Each $\sigma$-class $C$ of $X$ contains a unique nonnesting $n$-root, $\be_1$,
and a unique noncrossing element, $\be_2$. Moreover, in this case $C$ coincides
with the interval \[[\be_1, \be_2] = \{ \gamma \in X : \be_1 \leq_Q \gamma \leq_Q
\beta_2 \} \se X\] with respect to the quasiparabolic order $\le_{Q}$. The
set $X_I$ coincides with the unique maximal $\sigma$-class with
respect to a certain natural partial order on the $\sigma$-classes. 

\item %\cite[Theorem 5.5.2]{gx5} 
Any set of $\sigma$-equivalence class representatives forms a $\Q$-basis for
$\macd{\Phi}{nA_1}$, and any two such bases (when suitably ordered) have a
unitriangular change of basis matrix with integer entries. In particular, this is true for the
bases $\B_{NC}$ and $\B_{NN}$. 
\end{enumerate}
\end{theorem}

In \Cref{thm:summary}, the facts that $\B_{NC}$ and $\B_{NN}$ are bases
of $\macrep$ can be established using a version of Bergman's diamond lemma
\cite{bergman78}. Simple reflections in $W$ act on $\B_{NC}$ by a simple formula
\cite[Theorem 5.3.2.(iv)]{gx5}, and the sign-coherence property of $\B_{NC}$ may
be amenable to categorification. The element $w_N$ is a fully commutative element (in the
sense of \cite{FC}) that we can identify explicitly, and the distributive lattice
$L$ is isomorphic to the lattice of ideals in the heap poset of $w_N$ (\Cref{fig:heaps}). The set $X_I$ has many interesting properties, and we
summarize some of them in \Cref{sec:specific}. The results in (4) give
rise to a canonical bijection $\be_1\leftrightarrow \be_2$ between $\B_{NC}$ and
$\B_{NN}$, and both (4) and (5) generalize known results for a Specht module, as
we explain in the following remark.


\begin{figure}[h!]
    \centering
    \subfloat[]{
\begin{tikzpicture}
\node[main node] (0) {$s_2$};
\node[main node] (1) [right=0.8cm of 0] {$s_4$};
\node[main node] (2) [right=0.8cm of 1] {$s_6$};
\node[main node] (3) [above right=0.6cm and 0.32cm of 0] {$s_3$};
\node[main node] (4) [above right=0.6cm and 0.32cm of 1] {$s_5$};
\node[main node] (5) [above right=0.6cm and 0.32cm of 3] {$s_4$};

\path[draw]
(0)--(3)--(5)--(4)--(2)
(4)--(1)--(3);
\end{tikzpicture}
    }\quad\quad\quad\quad
    \subfloat[]{
\begin{tikzpicture}
\node[main node] (0) {$s_1$};
\node[main node] (1) [right=0.8cm of 0] {$s_3$};
\node[main node] (2) [right=0.8cm of 1] {$s_5$};
\node[main node] (3) [above right=0.48cm and 0.32cm of 0] {$s_2$};
\node[main node] (4) [above right=0.48cm and 0.32cm of 1] {$s_4$};
\node[main node] (5) [above right=0.48cm and 0.32cm of 3] {$s_3$};
\node[main node] (6) [above right=0.48cm and 0.32cm of 5] {$s_7$};


\path[draw]
(0)--(3)--(5)--(4)--(2)
(4)--(1)--(3)
(5)--(6);

\end{tikzpicture}
    }\quad\quad\quad\quad
    \subfloat[]{
\begin{tikzpicture}
\node[main node] (0) {$s_1$};
\node[main node] (1) [right=0.8cm of 0] {$s_4$};
\node[main node] (2) [right=0.8cm of 1] {$s_6$};
\node[main node] (3) [right=0.8cm of 2] {$s_8$};
\node[main node] (4) [above right=0.48cm and 0.32cm of 0] {$s_3$};
\node[main node] (5) [above right=0.48cm and 0.32cm of 1] {$s_5$};
\node[main node] (6) [above right=0.48cm and 0.32cm of 2] {$s_7$};
\node[main node] (7) [above right=0.48cm and 0.32cm of 4] {$s_4$};
\node[main node] (8) [above right=0.48cm and 0.32cm of 5] {$s_6$};
\node[main node] (9) [above right=0.48cm and 0.32cm of 7] {$s_5$};
\node[main node] (10) [above left=0.48cm and 0.32cm of 7] {$s_2$};
\node[main node] (11) [above right=0.48cm and 0.32cm of 10] {$s_4$};
\node[main node] (12) [above right=0.48cm and 0.32cm of 11] {$s_3$};
\node[main node] (13) [above right=0.48cm and 0.32cm of 12] {$s_1$};


\path[draw]
(0)--(4)--(1)--(5)--(2)--(6)--(3)
(4)--(7)--(5)--(8)--(6)
(8)--(9)--(7)--(10)--(11)--(12)--(13)
(9)--(11);
\end{tikzpicture}
    }
    \caption{Hasse diagrams for the heaps of the elements $w_N$ in types $D_8, E_7$ and $E_8$.}
    \label{fig:heaps}
\end{figure}


\begin{rmk}
\label{rmk:specht} 
In type $D_{n}$ for $n=2k$ even, the action of $W$ on the representation
$\macd{D_n}{nA_1}$ factors through the map $\phi$ from
Equation~\eqref{eq:phimap} to induce an $S_n$-module structure on
$\macd{D_n}{nA_1}$. The resulting $S_n$-module is isomorphic to the Specht
module $S^{(k,k)}$, and the nonnesting and
noncrossing bases for $\macd{D_n}{nA_1}\cong S^{(k,k)}$ have been studied
extensively as the \emph{web basis} and \emph{standard basis} in the works of
Rhoades \cite{rhoades17}, Russell--Tymoczko \cite{russell19}, and
Hwang--Jang--Oh \cite{hwang23}. The canonical bijection between $\B_{NC}$ and
$\B_{NN}$ generalizes the graph isomorphism $\psi$ from \cite{russell19}, and
\Cref{thm:summary}.5 can be used to recover the results on the transition
matrix from $\B_{NN}$ to $\B_{NC}$ in \cite[Section 5]{russell19}. If we expand
the maximally crossing and maximally nesting positive $n$-roots (such $n$-roots
exist in all types by \cite[Sections 4.3 and 5.2]{gx5}) as linear
combinations of $\B_{NC}$, then the resulting coefficients give rise to
\emph{Euler numbers} \cite[A000111]{oeis} and \emph{web permutations} by the
results of \cite{hwang23}, respectively. For more details, see \cite[Remark
6.1.3]{gx5}.
\end{rmk}


\subsection{Type-specific properties}
\label{sec:specific}
The alignment-free positive $n$-roots have the following properties.
We refer the interested reader to \cite[Sections 6.1--6.3]{gx5} for more
details. 

\begin{prop}
\label{prop:XI}
Let $W$ be a Weyl group of type $E_7$, $E_8$, or $D_n$ for $n$ even, and let
$X_I$ be the set of alignment-free positive $n$-roots of $W$.
\begin{enumerate}
\item %\cite[Proposition 6.1.1]{gx5} 
If $W$ has type $D_{n}$ for $n=2k$ even, then under the restriction of the
quasiparabolic order on positive $n$-roots, the set $X_I$ is canonically
isomorphic to the symmetric group $S_k$ under the strong Bruhat order via the
map $\varphi: S_k \ra X_I$ sending each element $\tau\in S_n$ to the $n$-root
$\varphi(\tau)=\prod_{i = 1}^k (\ep_i^2 - \ep_{\tau(i) + k}^2)$.
\item %\cite[Proposition 6.2.2]{gx5} 
If $W$ has type $E_7$, then the set $X_I$ admits a canonical bijection to the 30
inequivalent labellings of the Fano plane.
\item %\cite[Remark 6.3.5]{gx5} \cite[Theorem 3.13]{schmidt24} 
If $W$ has type $E_8$, then the graphs $G_{E_8}$ and $\Gamma$ constructed as
follows are not isomorphic but quantum isomorphic in the sense of
\cite{atserias19}.  The vertices of $G_{E_8}$ are the positive roots of type
$E_8$, with two vertices being adjacent if and only if they are orthogonal. The
vertices of $\Gamma$ are the elements of $X_I$ with even levels, with two vertices
being adjacent if and only if they share no common components. 
\end{enumerate}
\end{prop}


\acknowledgements{
We thank D.C. Ernst, H. Russell, and N. Thiem for helpful conversations.
}
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