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\title{Actions of the 0-Hecke monoids of affine symmetric groups}

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\author{Eric Marberg\thanks{\href{mailto:eric.marberg@gmail.com}{eric.marberg@gmail.com}}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong}

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\abstract{
There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde S_n$
on involutions whose cycles are labeled periodically by nonnegative integers. 
Using these actions we construct two bijections, which are length-preserving in an appropriate sense,
 from the set of involutions in $\tilde S_n$ to the set of $\mathbb{N}$-weighted
matchings in the $n$-element cycle graph. As an application, we show that the bivariate generating 
function counting the involutions in $\tilde S_n$ by length and absolute length
is a rescaled Lucas polynomial. The 0-Hecke monoid of $\tilde S_n$ also acts on involutions (without any cycle labelling) 
by Demazure conjugation. The atoms of an involution $z \in \tilde S_n$ are the minimal length permutations $w$ 
which transform the identity to $z$ under this action.  We prove that the set of atoms for an involution in $\tilde S_n$ is naturally a bounded,  graded poset, and give a formula for the set's minimum and maximum elements. 
}

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%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Affine symmetric groups, 0-Hecke monoids, involutions, Bruhat order}

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%\usepackage[backend=bibtex]{biblatex}
%\addbibresource{65-Marberg.bib}
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\begin{document}

\maketitle
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\section{Introduction}

For each integer $n\geq 1$, let  $\tS_n$ be the \emph{affine symmetric group} of rank $n$, consisting of the bijections $w : \ZZ \to \ZZ$
with
$w(i+n) = w(i) + n$ for all $i \in \ZZ$ and $w(1) + w(2) + \dots + w(n) = \binom{n+1}{2}$.
If $n=1$ then $\tS_1 = \{1\}$.
Assume $n \geq 2$, and
 define $s_i \in \tS_n$ for $i \in \ZZ$
as the permutation which exchanges $i +mn$ and $i+1 +mn$ for each $m \in \ZZ$, while fixing every integer
not congruent to $i$ or $i+1$ modulo $n$. 
The elements $s_1,s_2,\dots,s_n$ generate $\tS_n$, and with respect to these generators $\tS_n$ is the Coxeter group of type $\tilde A_{n-1}$. 


If $W$ is any Coxeter group with simple generating set $S$ and length function $\ell :W \to \NN$,
then there is a unique associative product $\circ : W \times  W \to W$ 
such that $w\circ s = w$ if $\ell(ws) < \ell(w)$ and $w\circ s = ws$ if $\ell(ws) > \ell(w)$
for
$w \in W$ and $s \in S$ \cite[Theorem 7.1]{Humphreys}.
The product $\circ$ is often called the \emph{Demazure product},
and the pair $(W,\circ)$ is usually referred to as the \emph{0-Hecke
monoid} or \emph{Richardson--Springer monoid} of $(W,S)$.
This extended abstract discusses three actions of the 0-Hecke monoid of $\tS_n$.
Each action will be on objects related to the group's involutions, 
that is, the elements $z \in \tS_n$ with $z^2=1$.

Let $I_n $ be the set of involutions in the finite symmetric group $S_n$,
which we identify with the parabolic subgroup of $\tS_n$ generated by $s_1,s_2,\dots,s_{n-1}$.
%Recall that a \emph{matching} in a graph is a subset of edges with no shared vertices;
%with slight abuse abuse of notation, a \emph{matching} on a set is a matching in the complete graph on that set.
Elements of $I_n$  may be
viewed as matchings on $\{1,2,\dots,n\}$. For example, 
\be\label{match-eq}\barr{c} \\[-8pt]  (1,4)(2,7)(3,6) \in I_8\qquad\text{corresponds to}\qquad
\arcstart
{
*{\bullet} \arc{0.6}{rrr}     & *{\bullet}    \arc{1.5}{rrrrr} & *{\bullet}   \arc{0.6}{rrr}  & *{\bullet}     & *{\bullet}  & *{\bullet}    & *{\bullet}  & *{\bullet} \\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8
} 
\arcstop.
\earr
\ee
One adapts this model
 to the elements of   $\tI_n = \{ z\in \tS_n : z^2=1\}$
by representing $z \in \tI_n$
 as the matching on $\ZZ$ in which $i$ and $j$ are connected by an edge whenever $z(i) = j \neq i = z(j)$.
 This gives a bijection between $\tI_n$ and matchings on $\ZZ$ which are ``$n$-periodic''
 in the sense of having $\{ i,j\}$ as an edge if and only if $\{i+n,j+n\}$ is also an edge.
 We can make this model more compact by converting $n$-periodic matchings on $\ZZ$
  to $\ZZ$-weighted matchings on $\{1,2,\dots,n\}$: to represent $z \in \tI_n$, include the edge $\{i,j\}$ labeled by $m \in \ZZ$
whenever $i<j$ and  $z(i) = j + mn$ and $z(j) = i - mn$. For example,
\be\label{z-picture}
\arcstart
{
*{\bullet} \arc{0.6}{rrr}^1     & *{\bullet}    \arc{1.5}{rrrrr}^{-1} & *{\bullet}   \arc{0.6}{rrr}^0  & *{\bullet}     & *{\bullet}  & *{\bullet}    & *{\bullet}  & *{\bullet} \\
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8
} 
\arcstop
\ee
  corresponds to 
 $z= \prod_{m \in \ZZ} (1+mn,12+mn)(7+mn,10+mn)(3+mn,6+mn) \in \tI_8$.
Diagrams of this type are most useful when $\tS_n$ is viewed as a semidirect product $S_n \ltimes \ZZ^{n-1}$. 
When the structure of $\tS_n$ as a Coxeter group is significant,
a better approach is to view $n$-periodic matchings as \emph{winding diagrams}.
To construct the \emph{winding diagram} of $z \in \tI_n$, arrange $1,2,\dots,n$
clockwise on a circle, and
whenever $i < z(i) \equiv j \modu n)$, connect $i$ to $j$ by an arc winding $\frac{z(i)-i}{n}$ times in the clockwise direction around the circle's exterior.  
For the involution in \eqref{z-picture}, this produces the picture
  \be\label{winding-picture}
 % code: z = AffinePermutation([12,-1,6,-7,5,3,10,8]); WindingDiagram(z)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,transform shape,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_1$};
\node at (2.82842712475, 2.82842712475) {$_\bullet$};
\node at (1.97989898732, 1.97989898732) {$_2$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_3$};
\node at (2.82842712475, -2.82842712475) {$_\bullet$};
\node at (1.97989898732, -1.97989898732) {$_4$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_5$};
\node at (-2.82842712475, -2.82842712475) {$_\bullet$};
\node at (-1.97989898732, -1.97989898732) {$_6$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_7$};
\node at (-2.82842712475, 2.82842712475) {$_\bullet$};
\node at (-1.97989898732, 1.97989898732) {$_8$};
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 4.95))}, {(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 4.95))});
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (90.0 + \x * 1.35))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (90.0 + \x * 1.35))});
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (270.0 + \x * 1.35))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (270.0 + \x * 1.35))});
}
\end{tikzpicture}.
\ee
 Formally, a winding diagram is a collection of continuous paths between disjoint pairs of marked points on the boundary of the plane minus an open disc, up to homotopy.
Each winding diagram corresponds to a unique involution in some affine symmetric group.

Write $\ell(w)$ for the usual Coxeter length of $w \in \tS_n$,
and define the \emph{absolute length} $\ell'(z)$ of $z \in \tI_n$ to be the number of arcs in its winding diagram.
 Our first main result, Theorem~\ref{main-thm},   identifies
two  bijections $\omega_R$  and $\omega_L$ from $\tI_n$ to the set $\cWmin_n$ of $\NN$-weighted matchings 
 in $\cC_n$, the cycle graph on $n$ vertices.
 These bijections 
 preserve length and absolute length, where
 the absolute length of an $\NN$-weighted matching
 is its number of edges and the length is its number of edges plus twice the sum of their weights.
The images of the element $z \in \tI_8$ in our running example \eqref{winding-picture} are
\[
\omega_R(z) =\
% code: z = AffinePermutation([0,2,4,3,6,5,7,9]); WindingDiagram(z, labels=['8', '1','2'])
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_{1}$};
\node at (2.82842712475, 2.82842712475) {$_\bullet$};
\node at (1.97989898732, 1.97989898732) {$_{2}$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_{3}$};
\node at (2.82842712475, -2.82842712475) {$_\bullet$};
\node at (1.97989898732, -1.97989898732) {$_{4}$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_{5}$};
\node at (-2.82842712475, -2.82842712475) {$_\bullet$};
\node at (-1.97989898732, -1.97989898732) {$_{6}$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_{7}$};
\node at (-2.82842712475, 2.82842712475) {$_\bullet$};
\node at (-1.97989898732, 1.97989898732) {$_{8}$};
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{$\ \ 8$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (90.0 + \x * 0.45))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (90.0 + \x * 0.45))});
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{$1\ \ $}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (180.0 + \x * 0.45))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (180.0 + \x * 0.45))});
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{$2\ $}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (315.0 + \x * 0.45))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (315.0 + \x * 0.45))});
}
\end{tikzpicture} 
\qquand
\omega_L(z) =\ 
% code: z = AffinePermutation([2,1,3,5,4,6,8,7]); WindingDiagram(z, labels=['8', '2','1'])
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_{1}$};
\node at (2.82842712475, 2.82842712475) {$_\bullet$};
\node at (1.97989898732, 1.97989898732) {$_{2}$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_{3}$};
\node at (2.82842712475, -2.82842712475) {$_\bullet$};
\node at (1.97989898732, -1.97989898732) {$_{4}$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_{5}$};
\node at (-2.82842712475, -2.82842712475) {$_\bullet$};
\node at (-1.97989898732, -1.97989898732) {$_{6}$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_{7}$};
\node at (-2.82842712475, 2.82842712475) {$_\bullet$};
\node at (-1.97989898732, 1.97989898732) {$_{8}$};
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{$8$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 0.45))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 0.45))});
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{$2$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (135.0 + \x * 0.45))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (135.0 + \x * 0.45))});
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{$1\ \ $}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (270.0 + \x * 0.45))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (270.0 + \x * 0.45))});
}
\end{tikzpicture}
\]
and indeed it holds that $\ell'(z) = 3$ and $\ell(z ) = 25$.
The statement of Theorem~\ref{main-thm} relies on the construction
of a left and right action of the 0-Hecke monoid of $\tS_n$ on the set of \emph{weighted involutions}.
As an application, we show that 
$ \sum_{z \in \tI_n} q^{\ell(z)} x^{\ell'(z)} = \tfrac{1}{1+q^n} \Luc_n(1+q,qx)$ where $\Luc_n(x,s)$ is the $n$th bivariate \emph{Lucas polynomial};
see Corollary~\ref{A-cor}.
This is an analogue of a more complicated identity proved in \cite{MW}.


The 0-Hecke monoid  of $\tS_n$ also acts directly on $\tI_n$ by \emph{Demazure conjugation}: the right action mapping
$(z,w) \mapsto w^{-1} \circ z \circ w$ for $z \in \tI_n$ and $w \in \tS_n$.
This monoid action is a degeneration of the Iwahori--Hecke algebra representation studied by Lusztig and Vogan in \cite{Lu,LV}.
The orbit of the identity under Demazure conjugation is all of $\tI_n$, and we define $\cA(z)$ for $z \in \tI_n$ as the set of  elements $w \in \tS_n$ of minimal length such that $z = w^{-1} \circ w$. Following \cite{HMP1,HMP2}, we call these permutations the \emph{atoms} of $z$. There are a few reasons why these elements merit further study, beyond their interesting combinatorial properties. The sets $\cA(z)$ may be defined for involutions in any Coxeter group
and, in the case of finite Weyl groups,  are closely related to the sets
$W(Y)$ which Brion  \cite{B} attaches to $B$-orbit closures $Y$ in a spherical homogeneous space $G/H$ (where $G$ is a connected complex reductive group, $B$ a Borel subgroup, and $H$ a spherical subgroup).
Results of Hultman \cite{H1,H2}, extending work of Richardson and Springer \cite{RS}, show the atoms to be intimately connected to the Bruhat order of a Coxeter group restricted to its involutions. 
Finally, the atoms of involutions in $S_n$ play a central role in recent work
of Can, Joyce, Wyser, and Yong on the geometry of the orbits of the orthogonal group on the type $A$ flag variety; see \cite{CJ,CJW,WY}.


Our second objective is to generalise a number of results about the atoms of involutions
in finite symmetric groups to the affine case.
In Section~\ref{dem-sect}, extending results in \cite{HMP2,HZ}, we show that there is a natural partial order which makes $\cA(z)$ for $z \in \tI_n$
into a bounded, graded poset. We conjecture that this poset is a lattice.
Generalising results of Can, Joyce, and Wyser \cite{CJ,CJW}, we describe in Section~\ref{local-sect}  
  a ``local'' criterion for membership in $\cA(z)$ involving a notion of \emph{(affine) standardisation}; see Theorem~\ref{std-cor}.
Using this result, one can prove that involutions in $\tS_n$ have what we call the \emph{Bruhat covering property}.
For $i < j \not\equiv i \modu n)$, let $t_{ij}= t_{ji} \in \tS_n$ be the permutation
which interchanges $i + mn$ and $j + mn$ for each $m \in \ZZ$ and which fixes all
integers not in $\{i,j\}+n\ZZ$.
The elements $t_{ij}$ are precisely the
 \emph{reflections} in $\tS_n$. 

\begin{theorem}[Bruhat covering property] \label{bruh-cov-thm}
If $y \in \tI_n$ and $t \in \tS_n$ is a reflection, then there exists at most
one $z \in \tI_n$ such that $\left\{wt : w \in \cA(y)\text{ and }\ell(wt)=\ell(w)+1\right\} \cap \cA(z) \neq\varnothing$.
\end{theorem}

The analogue of this result for involutions in $S_n$ was shown in \cite{HMP3}, 
and served as a key lemma  in proofs of ``transition formulas'' for certain \emph{involution Schubert polynomials}. We conjecture that the same property holds for arbitrary Coxeter systems, in the following sense.
%
Let $(W,S)$ be a  Coxeter system with length function $\ell : W \to \NN$
and Demazure product $\circ : W\times W \to W$.
Suppose $w\mapsto w^*$ is an automorphism of $W$ with $S^* = S$.
The corresponding set of \emph{twisted involutions} is  $I_* = \{w \in W : w^{-1}=  w^*\}$.
For $y \in I_*$ let $\cA_*(y)$ be the set of elements of minimal length with $(w^*)^{-1} \circ w = y$.


\begin{conjecture}
If $y \in I_*$ is a twisted involution in an arbitrary Coxeter group and $t \in \{ wsw^{-1} : w \in W,\ s \in S\}$,
then there exists at most one twisted involution $z \in I_*$ such that $
\left\{wt : w \in \cA_*(y) \text{ and } \ell(wt) = \ell(w)+1\right\} \cap \cA_*(z) \neq \varnothing$.
\end{conjecture}

We anticipate that these results will be useful in developing 
a theory of \emph{affine involution Stanley symmetric functions}, simultaneously generalising \cite{Lam} and \cite{HMP1,HMP4,HMP5}.
For the sake of brevity, we have omitted in this extended abstract most proofs, which will appear elsewhere.




 

\section{Weighted involutions}\label{weighted-sect}

Throughout,  let $\ZZ$ be the set of all integers and let $\NN = \{0,1,2,\dots\}$,
and define  $[n]=\{1,2,\dots,n\}$ for $n \in \NN$.
For any map $w: \ZZ\to\ZZ$, let 
\[\inv(w) =\left\{(i,j) \in \ZZ\times \ZZ :\text{$i<j$ and $w(i) > w(j)$}\right\}.\]
For $w \in \tS_n$,
it then holds that  $\ell(w)$ is the number of equivalence classes in $\inv(w)$ 
under the relation on $\ZZ\times \ZZ$ generated by $(i,j) \sim (i+n,j+n)$
\cite[Section 8.3]{BB}.
For $z \in \tI_n$, let 
\[\cC(z) = \{  (i,j) \in \ZZ\times \ZZ : i < j = z(i)\}\] 
The absolute length $\ell'(z)$ is then similarly the number of equivalence classes in $\cC(z)$ under the relation on $\ZZ\times \ZZ$
generated by $(i,j) \sim (i+n,j+n)$.

\begin{definition}\label{weighted-def}
A \emph{weighted involution} in $\tS_n$ is a pair $(w, \phi)$
where $w \in \tI_n$ and $\phi$ is a map $\cC(w) \to \NN$ with $\phi(i,j) = \phi(i+n,j+n)$ for all $(i,j) \in \cC(w)$.
We refer to $\phi$ as the \emph{weight map} of $(w,\phi)$.
Define 
 the \emph{weight} of $(w,\phi)$ as the number $\wt(w,\phi) = \sum_\gamma \phi(\gamma)$ where the sum is over a set of cycles $\gamma$ representing the distinct equivalence classes in $\cC(w)$ under the relation  $(i,j) \sim (i+n,j+n)$.
Let $\cW_n$ be the set of all weighted involutions in $\tS_n$.
\end{definition}


\begin{example}\label{W5-ex}
We draw a weighted involution $(w,\phi) \in \cW_n$ as the winding diagram 
of $w$ with its arcs labeled by the values  of $\phi$. For example, if $\theta_1, \theta_2, \theta_3 \in \cW_5$
are 
\[
% code: z = AffinePermutation([2,1,10,4,-2]); WindingDiagram(z,labels=[2,3])
\begin{tikzpicture}[baseline=0,scale=0.3,label/.style={postaction={ decorate,transform shape,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.83697019872e-16, 3.0) {$_1$};
\node at (3.80422606518, 1.2360679775) {$_\bullet$};
\node at (2.85316954889, 0.927050983125) {$_2$};
\node at (2.35114100917, -3.2360679775) {$_\bullet$};
\node at (1.76335575688, -2.42705098312) {$_3$};
\node at (-2.35114100917, -3.2360679775) {$_\bullet$};
\node at (-1.76335575688, -2.42705098312) {$_4$};
\node at (-3.80422606518, 1.2360679775) {$_\bullet$};
\node at (-2.85316954889, 0.927050983125) {$_5$};
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{2}}] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 0.72))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 0.72))});
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{3}}] plot ({(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (144.0 + \x * 5.04))}, {(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (144.0 + \x * 5.04))});
}
\end{tikzpicture}
\qquad
% code: z = AffinePermutation([-7,0,11,4,7]); WindingDiagram(z,labels=[2,2])
\begin{tikzpicture}[baseline=0,scale=0.3,label/.style={postaction={ decorate,transform shape,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.83697019872e-16, 3.0) {$_1$};
\node at (3.80422606518, 1.2360679775) {$_\bullet$};
\node at (2.85316954889, 0.927050983125) {$_2$};
\node at (2.35114100917, -3.2360679775) {$_\bullet$};
\node at (1.76335575688, -2.42705098312) {$_3$};
\node at (-2.35114100917, -3.2360679775) {$_\bullet$};
\node at (-1.76335575688, -2.42705098312) {$_4$};
\node at (-3.80422606518, 1.2360679775) {$_\bullet$};
\node at (-2.85316954889, 0.927050983125) {$_5$};
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{2}}] plot ({(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (144.0 + \x * 5.76))}, {(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (144.0 + \x * 5.76))});
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{2}}] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (288.0 + \x * 1.44))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (288.0 + \x * 1.44))});
}
\end{tikzpicture}
\qquad
% code: z = AffinePermutation([-8,11,0,4,8]); WindingDiagram(z,labels=[2,1])
\begin{tikzpicture}[baseline=0,scale=0.3,label/.style={postaction={ decorate,transform shape,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.83697019872e-16, 3.0) {$_1$};
\node at (3.80422606518, 1.2360679775) {$_\bullet$};
\node at (2.85316954889, 0.927050983125) {$_2$};
\node at (2.35114100917, -3.2360679775) {$_\bullet$};
\node at (1.76335575688, -2.42705098312) {$_3$};
\node at (-2.35114100917, -3.2360679775) {$_\bullet$};
\node at (-1.76335575688, -2.42705098312) {$_4$};
\node at (-3.80422606518, 1.2360679775) {$_\bullet$};
\node at (-2.85316954889, 0.927050983125) {$_5$};
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{2}}] plot ({(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (72.0 + \x * 6.48))}, {(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (72.0 + \x * 6.48))});
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{1}}] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (288.0 + \x * 2.16))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (288.0 + \x * 2.16))});
}
\end{tikzpicture}
\]
and   $\theta_i = (w_i, \phi_i)$, then  $w_1 = t_{1,2} t_{3,10}$, $w_2 = t_{0,2}t_{3,11} $, and
$w_3 = t_{0,3}t_{2,11}$, while $\phi_1(1,2) =2$ and $\phi_1(3,10)=3$,
$\phi_2(3,11)=\phi_2(5,7)=2$, and $\phi_3(2,11)= 2$ and $\phi_3(5,8)=1$.
\end{example}

We identify $\tI_n$ with the subset of weighted involutions of the form 
 $(w,0) \in \cW_n$ with $0$ denoting the unique weight map $\cC(w) \to \{0\}$.
We extend $\ell $ and $\ell' $ to $\cW_n$ by setting 
\[ \ell(\theta) = \ell(w) + 2\wt(\theta) \qquand \ell'(\theta) = \ell'(w)\qquad\text{for }\theta = (w,\phi) \in \cW_n.\]
Given $(w, \phi) \in \cW_n$, define the \emph{right form} of $\phi$ to be the map $\phi_R : \ZZ \to \NN$ with
$\phi_R(i) =\phi(w(i),i)$ if $w(i) <i$ and with $\phi_R(i)=0$ otherwise.
Likewise, define the \emph{left form} of $\phi$ to be the map
$\phi_L : \ZZ \to \NN$ with $\phi_L(i) = \phi(i, w(i))$ if $i<w(i)$ and with $\phi_L(i) = 0$ otherwise.
Clearly $\phi_L$ and $\phi_R$ each determine $\phi$, given $w$.
%We now define operators $\pi_1,\pi_2,\dots, \pi_n$ which act on $\cW_{n}$ on the right and left.

\begin{definition}
Let $\theta =(w, \phi) \in \cW_{n}$ and $i \in \ZZ$.  We define $\theta\pi_i, \pi_i\theta \in \cW_n$ as follows:
\ben
\item[(a)] If $\phi_R(i) >\phi_R(i+1)$ then let $\theta\pi_i = (s_iws_i, \psi) \in \cW_n$ where
$\psi $ is the unique weight map with 
$\psi_R (j) = \phi_R(i)-1$ if $j\equiv i+1\modu n)$, $\psi_R(j) = \phi_R(i+1)$ if $j \equiv i \modu n)$, and $\psi_R(j)= \phi_R(j)  $ otherwise.
If $\phi_R(i) \leq\phi_R(i+1)$ then  let $\theta \pi_i= \theta$. 

\item[(b)] If  $\phi_L(i+1)>\phi_L(i)$ then  let $\pi_i\theta = (s_iws_i, \chi) \in \cW_n$ where
$\chi $ is the unique weight map with 
$\chi_L(j) =  \phi_L(i+1)-1$if $j\equiv i \modu n)$, $\chi_L(j) = \phi_L(i)$ if $j\equiv i+1 \modu n)$, and $\chi_L(j) =  \phi_L(j)$ otherwise.
If $\phi_L(i+1) \leq \phi_L(i)$ then let $\pi_i\theta = \theta$. 
  \een
\end{definition}

\begin{example}
Define $\theta_1,\theta_2,\theta_3 \in \cW_5$ as in Example~\ref{W5-ex}.
Then $\theta_1 \pi_5 = \theta_2 \pi_1 = \theta_2$ and $\theta_2\pi_2 = \theta_3$.
Form $\theta_2' \in \cW_5$ from $\theta_2$ by replacing the label of the short arc in the picture in Example~\ref{W5-ex} by 1 and the label of the long arc by 3.
Then $\pi_5 \theta_1 =\theta_2'$ and $\pi_2 \theta_2' = \theta_3$.
\end{example}

It may hold that $(\pi_i \theta) \pi_j \neq \pi_i (\theta\pi_j)$;
for example,
if $\theta =(w,\phi)$ where $w = s_1 \in \tS_2$ and $\phi(1,2) = 1$,
then $\pi_0\theta  = (\pi_0\theta)\pi_2 \neq \pi_0 (\theta \pi_2) = \theta \pi_2$.
 We always have $\pi_i = \pi_{i+n}$.
 
\begin{proposition}
The left (respectively, right) operators $\pi_i$ satisfy 
(a) $\pi_i^2  = \pi_i$, (b) $\pi_i\pi_j  = \pi_j\pi_i $ if $i \not\equiv j\pm 1 \modu n)$, and 
(c) $\pi_i\pi_{i+1}\pi_i  = \pi_{i+1} \pi_i\pi_{i+1}$
for all $i,j \in \ZZ$.
\end{proposition}

For each $g \in \tS_n$, 
we may therefore define a right (respectively, left) operator $\pi_g$ on $\cW_n$ by 
setting $\pi_g = \pi_{i_1} \pi_{i_2} \cdots \pi_{i_k}$
where $g= s_{i_1} s_{i_2}\cdots s_{i_k}$ is any reduced expression.
%Recall the definition of the Demazure product $\circ : \tS_n \times \tS_n \to \tS_n$ from the introduction.

\begin{corollary}
The map $g \mapsto \pi_g$ defines a right and left action of $(\tS_n, \circ)$ on $\cW_{n}$.
\end{corollary}









\begin{thmdef}\label{omega-thm}
Let $\theta =(w,\phi) \in \cW_n$. 
 There are unique permutations $g,h \in \tS_n$ with $\ell(g) = \ell(h)=\wt(\theta)$
and $\wt(\pi_g \theta) = \wt(\theta\pi_h)=0$.
Define $g_L(\theta)=g^{-1}$ and $g_R(\theta)=h$,
and set
 $\omega_L(\theta) =\pi_g\theta $ and $\omega_R(\theta)=\theta\pi_h $. 
\end{thmdef}


\begin{example}
If $\theta_1,\theta_2,\theta_3 \in \cW_5$ are as in Example~\ref{W5-ex}, then we have 
\[\omega_R(\theta_1) = \omega_R(\theta_2) = \omega_R(\theta_3) = t_{1,13} t_{5,9} =\
%code: z = AffinePermutation([13,2,-9,0,9]); WindingDiagram(z)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,transform shape,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_{1}$};
\node at (3.80422606518, 1.2360679775) {$_\bullet$};
\node at (2.66295824563, 0.86524758425) {$_{2}$};
\node at (2.35114100917, -3.2360679775) {$_\bullet$};
\node at (1.64579870642, -2.26524758425) {$_{3}$};
\node at (-2.35114100917, -3.2360679775) {$_\bullet$};
\node at (-1.64579870642, -2.26524758425) {$_{4}$};
\node at (-3.80422606518, 1.2360679775) {$_\bullet$};
\node at (-2.66295824563, 0.86524758425) {$_{5}$};
\draw [blue,-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 8.64))}, {(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 8.64))});
\draw [red,-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (288.0 + \x * 2.88))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (288.0 + \x * 2.88))});
}
\end{tikzpicture}
.\]
%The arcs in the winding diagram are coloured red and blue to make them easier to distinguish.
\end{example}

There is a simple relationship between the left and right versions of these constructions.
Define 
 $ \tau : \ZZ \to \ZZ$
by $\tau(i) = n+1-i$ and let $w^* = \tau w \tau$ for $w \in \tS_n$.
For  $\phi : \cC(w) \to \NN$ let $\phi^*$ be the map $\cC(w^*) \to \NN$
given by $(\tau(j),\tau(i))\mapsto \phi(i,j)$.
Extend $*$ to $\cW_n$ by setting $\theta^* = (w^*, \phi^*)$ for 
$\theta=(w,\phi) \in \cW_n$. Clearly $(\theta^*)^* = \theta$.


\begin{lemma}\label{*omega-lem}
If $\theta \in \cW_n$ then $g_L(\theta^*) = g_R(\theta)^*$ and $\omega_L(\theta^*) = \omega_R(\theta)^*$.
\end{lemma}




An involution $w \in \tI_n$ has  $\ell(w) = \ell'(w)$ if and only if $w$ is a product of commuting simple generators,
i.e., $w=s_{i_1}s_{i_2}\cdots s_{i_l}$  where $i_j \not \equiv i_k \pm 1 \modu n)$ for all $j,k \in [l]$.

\begin{definition}
Define $\cWmin_{n}$ as the set of  $\theta=(w,\phi) \in \cW_n$
with $\ell'(w) = \ell(w)$.
\end{definition}

The elements of $\cWmin_{n}$ 
are in bijection with 
$\NN$-weighted matchings in $\cC_n$, the cycle graph on $n$ vertices,
 which explains our notation.
The set $\cM_4$ consists of
\[
%%code: z = AffinePermutation([2,1,3,4]); WindingDiagram(z,labels=['a'],scale=0.22)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_1$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_2$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_3$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_4$};
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{$a$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 0.9))});
}
\end{tikzpicture}
\quad
%code: z = AffinePermutation([1,3,2,4]); WindingDiagram(z,labels=['a'],scale=0.22)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_1$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_2$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_3$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_4$};
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{$a$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (90.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (90.0 + \x * 0.9))});
}
\end{tikzpicture}
\quad
%code: z = AffinePermutation([1,2,4,3]); WindingDiagram(z,labels=['a'],scale=0.22)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_1$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_2$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_3$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_4$};
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{$a$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (180.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (180.0 + \x * 0.9))});
}
\end{tikzpicture}
\quad
%code: z = AffinePermutation([0,2,3,5]); WindingDiagram(z,labels=['a'],scale=0.22)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_1$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_2$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_3$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_4$};
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{$a$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (270.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (270.0 + \x * 0.9))});
}
\end{tikzpicture}
\quad
% code: z = AffinePermutation([2,1,4,3]); WindingDiagram(z,labels=['b','a'],scale=0.22)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_1$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_2$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_3$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_4$};
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{$b$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 0.9))});
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{$a$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (180.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (180.0 + \x * 0.9))});
}
\end{tikzpicture}
\quad
% z = AffinePermutation([0,3,2,5]); WindingDiagram(z,labels=['b','a'],scale=0.22)
\begin{tikzpicture}[baseline=0,scale=0.22,label/.style={postaction={ decorate,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.71450551881e-16, 2.8) {$_1$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (2.8, 0.0) {$_2$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.71450551881e-16, -2.8) {$_3$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-2.8, -3.42901103761e-16) {$_4$};
\draw [-,>=latex,domain=0:100,samples=100,label={[below]{$b$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (90.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (90.0 + \x * 0.9))});
\draw [-,>=latex,domain=0:100,samples=100,label={[above]{$a$}}] plot ({(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (270.0 + \x * 0.9))}, {(4.0 + 0.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (270.0 + \x * 0.9))});
}
\end{tikzpicture}
\]
where $a,b \in \NN$ are arbitrary.
The following is well-known; see \cite[A034807]{OEIS}.
\begin{proposition}\label{match-prop}
There are $\frac{n}{n-k}\binom{n-k}{k}$ distinct $k$-element matchings in $\cC_n$.
\end{proposition}





Recall the definition of $t_{ij} \in \tS_n$ for $i<j \not \equiv i \modu n)$.
%Write $u \lessdot v$ if $v$ covers $u$ in the Bruhat order on $\tS_n$, i.e., if $v = ut_{ij}$ for some $i,j$ and $\ell(v) =\ell(u)+1$.
Let $\prec$ be the partial order on $\cWmin_{n}$ with $(w,\phi) \preceq (w',\phi')$
if and only if $w=w'$ and $\phi(a,b) \leq \phi'(a,b)$ for $(a,b) \in \cC(w)$.
Next, define $\prec_R$ as the transitive closure of the relation
on $\tI_n$  
with $z \prec_R  t_{ij} z t_{ij}$
whenever $z(i) < i$ and $j = \min \{e \in \ZZ : i < e \text{ and }z(i) < z(e)\}$.
Finally, define 
$\prec_L$ similarly as the transitive closure of the relation on $\tI_n$
with 
$z \prec_L  t_{ij} z t_{ij}$
whenever $j < z(j)$ and  $i = \max\{ e \in \ZZ : e <j \text{ and }z(e)<z(j)\}$.
The posets $(\tI_{n}, \prec_R)$ and $(\tI_{n}, \prec_L)$ are 
isomorphic via the map $z \mapsto z^*$.
One can show that these 
posets are graded with rank function $z \mapsto \frac{1}{2}\ell(z)$,
and that both are 
 subposets  of the Bruhat order restricted to $\tI_n$.




%\begin{lemma}\label{min-lem}
%An element $ z\in \tI_n$ is minimal in $ \prec_R$
%(respectively, $\prec_L$) if and only if $\ell(z) = \ell'(z)$.
%\end{lemma}



\begin{theorem}\label{main-thm}
The maps 
$
 \omega_R : (\cWmin_{n}, {\prec}) \to (\tI_{n},{\prec_R})
 $
and
$ \omega_L : (\cWmin_{n}, {\prec}) \to (\tI_{n},{\prec_L})
$
are isomorphisms of partially ordered sets which preserve $\ell$ and $\ell'$.
 \end{theorem}
 
Consider the following variations of $g_L(\theta)$ and $g_R(\theta)$ from Theorem-Definition~\ref{omega-thm}.

\begin{definition}\label{alpha-def}
For  $z \in \tI_n$ let $\theta_R$ and $\theta_L$ be the unique elements of $\cM_n$ such that $\omega_R(\theta_R) =\omega_L(\theta_L)=z$,
and define
 $\alpha_R(z) = g_R(\theta_R) \in \tS_n$ and  $\alpha_L(z) = g_L(\theta_L) \in \tS_n$.
 \end{definition}

One can derive a more explicit formula for these elements.

\begin{propdef}\label{window-def}
If $a_1,a_2,\dots,a_n \in\ZZ$ represent the distinct congruence classes modulo $n$
then there is a unique  $m \in \ZZ$ and a unique 
$w \in \tS_n$ such that $w(m+i) = a_i$ for $i \in [n]$.
Moreover, it holds that $m = \frac{1}{n}\sum_{i=1}^n (a_i-i)$.
Define $\lW a_1,a_2,\dots,a_n\rW  = w\in \tS_n$.
\end{propdef}



If $a_1,a_2,\dots,a_N \in \ZZ$ represent all congruence classes modulo $n$,
and $i_1<i_2<\dots<i_n$ are the indices of the first representative of each class,
then we define $\lW a_1,a_2,\dots,a_N\rW = \lW a_{i_1},a_{i_2},\dots,a_{i_n}\rW  \in \tS_n$.
For example, if $n=3$ then $[1,0,1,3,8,4,2] = [ 1,0,8]$.
 
\begin{theorem}\label{alphaL-thm}
Let $z \in \tI_n$ and $m \in \ZZ$. Suppose 
$a_1 < a_2 <\dots <a_l
$
and
$ 
d_1 < d_2 < \dots < d_l
$ are the elements of $m+[n]$
with $a_i \leq z(a_i)  $ and $z(d_i) \leq d_i$. Define  $b_i = z(a_i)$ and $c_i = z(d_i)$.
Then 
$\alpha_R(z) = \lW a_1,b_1,a_2,b_2,\dots,a_l,b_l\rW ^{-1}$
and 
$
\alpha_L(z) = \lW c_1,d_1,c_2,d_2,\dots,c_l,d_l\rW ^{-1}.
$
\end{theorem}
 
\begin{example}
One has
$\alpha_R(1) = \alpha_L(1) = \lW 1,1,2,2,\dots,n,n\rW ^{-1} = 1$.
If $z =t_{1,8}t_{2,7} \in \tI_4$ then 
$\alpha_R(z) = \lW 1,8,2,7\rW ^{-1} =\lW 3,5,2,0\rW  $ and $ \alpha_L(z) = \lW -2,3,-3,4\rW ^{-1}=\lW 5,3,0,2\rW .$
\end{example}


Slightly abusing notation, we write $\tI_n(q,x)= \sum_{w \in \tI_n} q^{\ell(w)} x^{\ell'(w)}\in \NN[[q,x]]$.

 \begin{corollary}\label{A-cor} If $n\geq 1$ then
 $\tI_n(q,x)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \frac{n}{n-k} \binom{n-k}{k} \( \frac{qx}{1-q^2}\)^k.$
\end{corollary}

\begin{proof}
 Theorem~\ref{main-thm} implies
that $\tI_n(q,x)= \sum_{\theta \in \cWmin_n} q^{2\wt(\theta)} (qx)^{\ell'(\theta)}$. 
By Proposition~\ref{match-prop},
the coefficient of $x^k$ in
the latter power series is $\frac{n}{n-k}\binom{n-k}{k} q^k (1 + q^2 + q^4 + q^6+\dots)^k$. 
\end{proof}

\begin{corollary}
If $n\geq 3$ then
$\tI_n(q,x)= \tI_{n-1}(q,x) + \tfrac{qx}{1-q^2} \tI_{n-2}(q,x)$.
\end{corollary}


Define $\ellhat(w) = \frac{1}{2}(\ell(w) + \ell'(w))$ for $w \in \tI_n$. 
Corollary~\ref{A-cor} shows that  $\ellhat(w) \in \NN$.
Let  $\hat N_{n}(m)$   be
the number of involutions $w \in \tI_n$
with $\ellhat(w) = m$.
\begin{corollary}\label{counts-cor}
 For each $n\geq 2$ and $m\geq 1$, it holds that 
 $\hat N_{n}(m)= \sum_{j=1}^{\lfloor n/2\rfloor} \frac{n}{n-j} \binom{n-j}{j} \binom{m-1}{j-1}
.$
\end{corollary}

 

\begin{remark}
The numbers $\{ \hat N_n(n) \}_{n=1,2,3,\dots} = (0, 2, 3, 10, 25, 71, 196, 554, 1569, \dots)$ are
sequence \cite[A246437]{OEIS}, which gives the ``type $B$ analog for Motzkin sums.''
The numbers $\{ \hat N_n(2n) \}_{n=1,2,3,\dots} = (0, 2, 3, 18, 50, 215, 735, 2898, \dots)$ are sequence \cite[A211867]{OEIS}.\end{remark}






\section{Demazure conjugation}\label{dem-sect}

Recall the definition of the {Demazure product} $\circ : \tS_n \times \tS_n \to \tS_n$ from the introduction.
The operation $(z,w) \mapsto w^{-1}\circ z \circ w$ for $z \in \tI_n$ and $w \in \tS_n$
defines another right action of the monoid $(\tS_n, \circ)$,
which we call \emph{Demazure conjugation}.
If $z \in \tI_n$ and $i \in \ZZ$ then 
\be\label{dem-eq} s_i \circ z \circ s_i = \begin{cases} s_izs_i &\text{if }z(i)<z(i+1)\text{ and }(i,i+1) \notin \cC(z) \\
zs_i &\text{if }z(i) < z(i+1)\text{ and }(i,i+1) \in \cC(z) \\
z &\text{otherwise}.
\end{cases}
\ee
Every $z \in \tI_n$ can be expressed as
$z = w^{-1}\circ w $ for some $w \in \tS_n$,
and we define
 $\cA(z)$ as the set of elements $w \in \tS_n$ of shortest possible length such that $z=w^{-1}\circ w$.
For example,
 if $z = t_{0,5} = \lW -4,2,3,9\rW  \in \tI_4$ then $\cA(z) = \{ s_1s_2s_3s_4, s_2s_1s_3s_4, s_3s_2s_1s_4\}$.
 

 The set $\cA(z)$ is nonempty for all $z \in \tI_n$, and we refer to its elements as the \emph{atoms} of $z$.
Recall from Corollary~\ref{counts-cor} that  $\ellhat(z) = \frac{1}{2}(\ell(z)  + \ell'(z))$.
By  \eqref{dem-eq}, we have:

 \begin{proposition}
If $z \in \tI_n$ then $\ellhat(z)$ is the common value of $\ell(w)$ for $w \in \cA(z)$.
 \end{proposition}
 
% \begin{proof} 
% If $z \in \tI_n$ and $s \in \{s_1,s_2,\dots,s_n\}$ then
%  $\ell'(s z s)=\ell'(z)$,
%   and if $z < zs =sz$ then
% $\ell'(zs) = \ell'(z)+1$.
% From this property and \eqref{dem-eq}, the result follows by induction.
 %\end{proof}
 

Results in \cite{HMP2}, building on work of Can, Joyce, and Wyser \cite{CJ,CJW}, show that the sets $\cA(z)$ for involutions $z \in I_n \subset \tI_n$ in the finite symmetric group are naturally bounded, graded posets.
This phenomenon extends to all involutions in $\tS_n$.
Recall  
the elements $\alpha_R(z)$ and $ \alpha_L(z)$
from Definitions~\ref{alpha-def}.

\begin{definition}\label{alpha-min-def}
Given $z \in \tI_n$, 
let
$
\alpha_{\min}(z) =  \alpha_R(z)  z
$
and
$
\alpha_{\max}(z) =\alpha_L(z) z.
$
\end{definition}





\begin{corollary}\label{min-cor}
Let $z \in \tI_n$ and $m \in \ZZ$ 
and define $a_i,b_i,c_i,d_i $
%relative to $z$ and $m$
as in Theorem~\ref{alphaL-thm}. Then
$ \alpha_{\min}(z) = \lW  b_1,a_1,b_2,a_2,\dots,b_l,a_l\rW ^{-1}
$
and
$
\alpha_{\max}(z) = \lW d_1,c_1,d_2,c_2,\dots,d_l,c_l\rW ^{-1}.$
%As in Theorem~\ref{alphaL-thm}, these formulas are independent of the choice of $m$.
\end{corollary}



\begin{example}\label{alpha-min-ex}
If $z = t_{1,8}t_{2,7} \in \tI_4$ then 
$\alpha_{\min}(z) = \lW 8,1,7,2\rW ^{-1} $%=\lW 4,6,1,-1\rW  $ 
and $ \alpha_{\max}(z) = \lW 3,-2,4,-3\rW ^{-1}$. %=\lW 6,4,-1,1\rW .$
\end{example}

Let $\lessdot_\cA$ be the relation on $\tS_n$ with 
 $u \lessdot_\cA v$ if and only if $u < s_{i+1}u = s_{i}v > v$ for some $i \in \ZZ$.
Let $<_\cA$ be the transitive closure of $\lessdot_\cA$.
 

\begin{theorem}\label{poset-thm}
Let $z \in \tI_n$. Restricted to $\cA(z)$,
the relation $<_\cA$ is a bounded, graded partial order,
and  it holds that  
$\cA(z) = \left\{ w \in \tS_n : \alpha_{\min}(z) \leq_\cA w\right\} = \left\{ w \in \tS_n : w\leq_\cA \alpha_{\max}(z) \right\}.
$
\end{theorem}


The situation described by the preceding theorem has some formal
similarities to Stembridge's results in \cite[Section 4]{StemRed} about
the top and bottom classes of a permutation.

\begin{figure}[h]
{\tiny\begin{center}
\begin{tikzpicture}
  \node (min) at (0,-10) {$\lW 4, 6, 8, 7, -1, -3\rW $};
     \node (i1) at  (-2,-9) {$\lW 4, 7, 8, 5, 0, -3\rW $};  \node (i2) at (2, -9) {$\lW 5, 6, 8, 7, -3, -2\rW $};
      \node (h1) at  (-4,-8) {$\lW 4, 8, 6, 5, 1, -3\rW $};  \node (h2) at (0, -8) {$\lW 5, 7, 8, 3, 0, -2\rW $}; \node (h3) at (4,- 8) {$\lW 6, 4, 8, 7, -3, -1\rW $};
  \node (g1) at  (-4,-7) {$\lW 5, 8, 6, 3, 1, -2\rW $};  \node (g2) at (0, -7) {$\lW 6, 7, 8, 3, -2, -1\rW $}; \node (g3) at (4, -7) {$\lW 7, 4, 8, 5, -3, 0\rW $};
  %
     \node (f1) at  (-6,-6) {$\lW 5, 9, 6, 1, 2, -2\rW $};  \node (f2) at (-2,-6) {$\lW 6, 8, 4, 3, 1, -1\rW $}; \node (f3) at (2, -6) {$\lW 7, 5, 8, 3, -2, 0\rW $}; \node (f4) at (6, -6) {$\lW 8, 4, 6, 5, -3, 1\rW $};
  %
   \node (e1) at  (-4,-5) {$\lW 6, 9, 4, 1, 2, -1\rW $};  \node (e2) at (0, -5) {$\lW 7, 8, 4, 3, -1, 0\rW $}; \node (e3) at (4, -5) {$\lW 8, 5, 6, 3, -2, 1\rW $};
  %
   \node (d1) at  (-6,-4) {$\lW 6, 10, 2, 1, 3, -1\rW $};  \node (d2) at (-2,-4) {$\lW 7, 9, 4, -1, 2, 0\rW $}; \node (d3) at (2, -4) {$\lW 8, 6, 4, 3, -1, 1\rW $}; \node (d4) at (6, -4) {$\lW 9, 5, 6, 1, -2, 2\rW $};
%
    \node (c1) at  (-4,-3) {$\lW 7, 10, 2, -1, 3, 0\rW $};  \node (c2) at (0,- 3) {$\lW 8, 9, 4, -1, 0, 1\rW $}; \node (c3) at (4, -3) {$\lW 9, 6, 4, 1, -1, 2\rW $};
  \node (b1) at  (-4,-2) {$\lW 8, 10, 0, -1, 3, 1\rW $};  \node (b2) at (0, -2) {$\lW 9, 7, 4, -1, 0, 2\rW $}; \node (b3) at (4, -2) {$\lW 10, 6, 2, 1, -1, 3\rW $};
   \node (a1) at  (-2,-1) {$\lW 9, 10, 0, -1, 1, 2\rW $};  \node (a2) at (2, -1) {$\lW 10, 7, 2, -1, 0, 3\rW $};
  \node (max) at (0,0) {$\lW 10, 8, 0, -1, 1, 3\rW $};
    \draw  [->]
  (min) edge (i1)
  (min) edge (i2)
  (i1) edge (h1)
(i1) edge (h2)
(i2) edge (h3)
(h1) edge (g1)
(h2) edge (g1)
(h2) edge (g2)
(h3) edge (g3)
(g1) edge (f1)
(g1) edge (f2)
(g2) edge (f3)
(g3) edge (f3)
(g3) edge (f4)
(f1) edge (e1)
(f2) edge (e1)
(f2) edge (e2)
(f3) edge (e3)
(f4) edge (e3)
(e1) edge (d1)
(e1) edge (d2)
(e2) edge (d3)
(e3) edge (d3)
(e3) edge (d4)
(d1) edge (c1)
(d2) edge (c1)
(d2) edge (c2)
(d3) edge (c3)
(d4) edge (c3)
(c1) edge (b1)
(c2) edge (b2)
(c3) edge (b2)
(c3) edge (b3)
(b1) edge (a1)
(b2) edge (a2)
(b3) edge (a2)
  (a1) edge (max)
  (a2) edge (max)
;
\end{tikzpicture}\end{center}}
\caption{Hasse diagram of   $(\cA(z),<_\cA)$ for $z= t_{1,12}t_{2,11} t_{3,4} \in \tI_6$}
\label{poset-fig}
\end{figure}


Figure~\ref{poset-fig} shows an example of $(\cA(z),<_\cA)$. 
The lattice structure evident in this picture appears to be typical; we have used a computer to check the following conjecture
for $z \in \tI_n$ in the 333,307 cases when $0 < \ellhat(z)n \leq 100$. 
%As a graded lattice, $(\cA(z),<_\cA)$ is
%not necessarily Eulerian, distributive, or semi-modular, but
% its M\"obius function seems to always take values in $\{-1,0,1\}$.

\begin{conjecture}
The graded poset $(\cA(z), <_\cA)$ is a lattice for all $n$ and $z \in \tI_n$.
\end{conjecture}




An element $w \in \tS_n$ is \emph{321-avoiding} if no integers $a<b<c$
have $w(a) > w(b) > w(c)$,
and \emph{fully commutative}
if we cannot write $w =u  s_i s_{i+1} s_i  v$
for $u,v \in \tS_n$ and $i \in \ZZ$ with $\ell(w) = \ell(u) + \ell(v) + 3$.
The following extends  \cite[Corollary 6.11]{HMP2} to affine type $A$.

\begin{corollary}
Let $z \in \tI_n$. The following are equivalent:
(a) $|\cA(z)|=1$,
(b) $\alpha_{\min}(z) =\alpha_{\max}(z)$,
(c) $\alpha_R(z) = \alpha_L(z)$,
(d) $z$ is 321-avoiding,
and
(e) $z$ is fully commutative.
\end{corollary}

The equivalence of (d) and (e) is well-known; see the results of Green \cite[Theorem 2.7]{Green},
Lam \cite[Proposition 35]{Lam}, or Fan and Stembridge \cite{FS}.
Biagioli, Jouhet, and Nadeau \cite[Proposition 3.3]{BJN} have derived a length generating function for 
the involutions in $\tS_n$ with these equivalent properties.


\section{Local characterisations of atoms}\label{local-sect}

Fix a subset $E\subset [n]$ of size $m$. 
Let $\phi_E : [m] \to E$ and $\psi_E : E \to [m]$ be order-preserving bijections.
The \emph{standardisation} of $w \in S_n$   is the permutation $[w]_E = \psi_{w(E)} \circ w \circ \phi_E \in S_m$.
If $w^2=1$ and $w(E)=E$, then $([w]_E)^2=1$.

 The Demazure product $\circ$ on $\tS_n$ restricts to an associative product $S_n\times S_n \to S_n$
 and each involution $y \in I_n = \tI_n \cap S_n$ has $\cA(y) \subset S_n$.
Can, Joyce, and Wyser's description of $\cA(y)$ for $y \in I_n$ in \cite{CJW} 
implies that $w \in S_n$ belongs to $\cA(y)$ if and only if $[w]_E \in \cA([y]_E)$
for all subsets $E \subset [n]$ which are invariant under $y$ and contain
at most two $y$-orbits;  see Corollary $3.19$ in \cite{HMP3}.
This ``local''  criterion for membership in $\cA(y)$ was an important tool in the proofs of the main results in \cite{HMP3}.

%\begin{example}
%The atoms of the reverse permutations in $S_2$, $S_3$, and $S_4$ are given by
%$\cA(21) = \{ 21\}$,  $\cA(321) = \{ 312, 231 \}$, and $\cA(4321) = \{ 4213, 3412, 2431\}$.
%If $y = n\cdots 321 \in I_n$ and $E\subset [n]$ is $y$-invariant with at most two orbits,
%then $[y]_E$ is either $21$, $321$, or $4321$,
%and one can check that requiring $[w]_E \in \cA([y]_E)$ 
%imposes precisely the sort of conditions we saw in Theorem~\ref{excluded-thm}.
%\end{example}

This result can be extended to the affine case, provided we 
give the right definition of the standardisation of an affine permutation.
Fix $E \subset \ZZ$ with $|(E+n\ZZ) \cap [n]| = m$, 
and 
define $\tphi_{E,n}$ as the unique order-preserving
$\ZZ \to E +n\ZZ $ with $\tphi_{E,n}([m]) \subset[n]$.

\begin{lemma} \label{affine-std-lem}
Let $ w\in \tS_n$. There is a unique order-preserving bijection $\tpsi_{E,w} : w(E) + n\ZZ \to \ZZ$
with $\tpsi_{E,w} \circ w \circ \tphi_{E,n} \in \tS_m$. 
If $w \in \tI_n$ and $w(E)=E$
then $\tphi_{E,n}$ and $\tpsi_{E,w}$ are inverses.
\end{lemma}


Given $w \in \tS_n$ and $E \subset \ZZ$ with  $|(E+n\ZZ) \cap [n]| = m$, define 
$[w]_{E,n} = \tpsi_{E,w}  \circ w \circ \tphi_{E,n} \in \tS_m.$
We refer to $[w]_{E,n}$ as the \emph{(affine) standardisation} of $w$.
One has $[w]_{E, n} = [w]_{E+mn,n} $ for all $m \in \ZZ$.
When $n$ is clear from context, we write $[w]_E$ instead of $[w]_{E,n}$.  If $E \subset [n]$ and $w \in S_n\subset \tS_n$,
 then $\tphi_{E,n}|_{[m]} = \phi_E$ and $\tpsi_{E,w} |_E = \psi_{w(E)}$.
 






\begin{corollary} If $E \subset \ZZ$, 
 $y \in \tI_n$, and $y(E)=E$, then $[y]_E \in \tI_n$.
\end{corollary}



\begin{example}
Standardisation has a simple interpretation in terms of winding diagrams.
If  $E=y(E)\subset \ZZ$ then the winding diagram of $[y]_E$ is formed from that of $y \in \tI_n$
by erasing the vertices in $[n] \setminus (E+n\ZZ)$ and their incident edges, and then relabelling the remaining numbers 
as consecutive integers.
If $n=8$ and  $E = \{2,4,6,7,8\}$, then  
 \[
 % code: z = AffinePermutation([3,12,1,-6,5,8,7,6]); WindingDiagram(z)
\begin{tikzpicture}[baseline=0,scale=0.3,label/.style={postaction={ decorate,transform shape,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.83697019872e-16, 3.0) {$_1$};
\node at (2.82842712475, 2.82842712475) {$_\bullet$};
\node at (2.12132034356, 2.12132034356) {$_2$};
\node at (4.0, 0.0) {$_\bullet$};
\node at (3.0, 0.0) {$_3$};
\node at (2.82842712475, -2.82842712475) {$_\bullet$};
\node at (2.12132034356, -2.12132034356) {$_4$};
\node at (2.44929359829e-16, -4.0) {$_\bullet$};
\node at (1.83697019872e-16, -3.0) {$_5$};
\node at (-2.82842712475, -2.82842712475) {$_\bullet$};
\node at (-2.12132034356, -2.12132034356) {$_6$};
\node at (-4.0, -4.89858719659e-16) {$_\bullet$};
\node at (-3.0, -3.67394039744e-16) {$_7$};
\node at (-2.82842712475, 2.82842712475) {$_\bullet$};
\node at (-2.12132034356, 2.12132034356) {$_8$};
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 0.9))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 0.9))});
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (45.0 + \x * 4.5))}, {(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (45.0 + \x * 4.5))});
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (225.0 + \x * 0.9))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (225.0 + \x * 0.9))});
}
\end{tikzpicture}
\qquand
% code: z = AffinePermutation([7,-4,5,4,3]); WindingDiagram(z)
\begin{tikzpicture}[baseline=0,scale=0.3,label/.style={postaction={ decorate,transform shape,decoration={ markings, mark=at position .5 with \node #1;}}}]
{
\draw[fill,lightgray] (0,0) circle (4.0);
\node at (2.44929359829e-16, 4.0) {$_\bullet$};
\node at (1.83697019872e-16, 3.0) {$_1$};
\node at (3.80422606518, 1.2360679775) {$_\bullet$};
\node at (2.85316954889, 0.927050983125) {$_2$};
\node at (2.35114100917, -3.2360679775) {$_\bullet$};
\node at (1.76335575688, -2.42705098312) {$_3$};
\node at (-2.35114100917, -3.2360679775) {$_\bullet$};
\node at (-1.76335575688, -2.42705098312) {$_4$};
\node at (-3.80422606518, 1.2360679775) {$_\bullet$};
\node at (-2.85316954889, 0.927050983125) {$_5$};
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (0.0 + \x * 4.32))}, {(4.0 + 4.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (0.0 + \x * 4.32))});
\draw [-,>=latex,domain=0:100,samples=100] plot ({(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * cos(90 - (144.0 + \x * 1.44))}, {(4.0 + 2.0 * sin(180 * (0.5 + asin(-0.9 + 1.8 * (\x / 100)) / asin(0.9) / 2))) * sin(90 - (144.0 + \x * 1.44))});
}
\end{tikzpicture}
\]
represent $y =t_{1,3} t_{2,12} t_{6,8} \in \tI_8$ and $[y]_E  = t_{1,7} t_{3,5} \in \tI_5$, respectively.
\end{example}



\begin{theorem}\label{std-cor}
Let $y \in \tI_n$, $w \in \tS_n$, and $X = [n]\cup y([n])$. 
Then $w \in \cA(y)$
if and only if 
$[w]_E \in \cA([y]_E)$
for each subset $E=y(E)\subset X$ containing at most two $y$-orbits.
\end{theorem}


This result is the starting point for the proof of Theorem~\ref{bruh-cov-thm} in the introduction.

\acknowledgements{I thank Brendan Pawlowski and Graham White for helpful conversations.}

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\end{document}
