# Coxeter Pop-Tsack Torsing

Colin Defant  Princeton University<EMAIL_ADDRESS>and  Nathan Williams
University of Texas at Dallas<EMAIL_ADDRESS>
###### Abstract.

Given a finite irreducible Coxeter group $W$ with a fixed Coxeter element $c$,
we define the _Coxeter pop-tsack torsing operator_ $\mathsf{Pop}_{T}:W\to W$
by $\mathsf{Pop}_{T}(w)=w\cdot\pi_{T}(w)^{-1}$, where $\pi_{T}(w)$ is the join
in the noncrossing partition lattice $\mathrm{NC}(w,c)$ of the set of
reflections lying weakly below $w$ in the absolute order. This definition
serves as a “Bessis dual” version of the first author’s notion of a Coxeter
pop-stack sorting operator, which, in turn, generalizes the pop-stack-sorting
map on symmetric groups. We show that if $W$ is coincidental or of type $D$,
then the identity element of $W$ is the unique periodic point of
$\mathsf{Pop}_{T}$ and the maximum size of a forward orbit of
$\mathsf{Pop}_{T}$ is the Coxeter number $h$ of $W$. In each of these types,
we obtain a natural lift from $W$ to the dual braid monoid of $W$. We also
prove that $W$ is coincidental if and only if it has a unique forward orbit of
size $h$. For arbitrary $W$, we show that the forward orbit of $c^{-1}$ under
$\mathsf{Pop}_{T}$ has size $h$ and is isolated in the sense that none of the
non-identity elements of the orbit have preimages lying outside of the orbit.

###### 2010 Mathematics Subject Classification:

05E16; 05A05; 37E15

##  1\. Introduction

###  1.1. Pop-stack sorting

Suppose $X$ is a set and $f:X\to X$ is a function. Define the _forward orbit_
of an element $x\in X$ under the map $f$ to be the set

$O_{f}(x)=\\{x,f(x),f^{2}(x),\ldots\\},$

where $f^{i}$ denotes the $i^{\text{th}}$ iterate of $f$. One of the primary
aims of _dynamical combinatorics_ is to understand the sizes of forward orbits
of combinatorially-defined maps.

A rich source of combinatorial dynamical systems comes from the symmetric
group $\mathfrak{S}_{n}$, which is the group of permutations of the set
$[n]:=\\{1,\ldots,n\\}$. For example, the _pop-stack sorting map_ is the
function $\mathsf{Pop}:\mathfrak{S}_{n}\to\mathfrak{S}_{n}$ that acts by
reversing all of the descending runs of a permutation while keeping different
descending runs in the same order relative to each other. This map, which
originally appeared in an article by Ungar concerning directions determined by
points in the plane [Ung82], has gained interest in the past few years among
enumerative combinatorialists [AV21, ABH21, ABB+19, EG21, CG19, CGP19, PS19].
See [Def21] and the references therein for several other dynamical systems
arising from sorting operators on permutations.

It is not difficult to show that the identity permutation $e=12\cdots n$ is
the unique fixed point of $\mathsf{Pop}:\mathfrak{S}_{n}\to\mathfrak{S}_{n}$
and that every forward orbit of this map contains $e$. However, it is
surprisingly difficult to determine the maximum size of a forward orbit; this
was done by Ungar, who proved that $\max\limits_{w\in\mathfrak{S}_{n}}\lvert
O_{\mathsf{Pop}}(w)\rvert=n$. This result was recently reproven by Albert and
Vatter [AV21].

###  1.2. Coxeter pop-stack sorting

Let $W$ be a finite Coxeter group. Let $S$ be a set of simple reflections of
$W$, and let $w_{\circ}$ be the longest element of $W$. Let $\leq_{S}$ denote
the left weak order on $W$. For $J\subseteq S$, let $w_{\circ}(J)$ be the
longest element of the parabolic subgroup of $W$ generated by $J$. Define the
_longest-element projection_ from $W$ to the set of parabolic long elements by

$\displaystyle\pi_{S}:W$ $\displaystyle\to\big{\\{}w_{\circ}(J):J\subseteq
S\big{\\}}$ (1) $\displaystyle\pi_{S}(w)$
$\displaystyle=\bigvee_{s\leq_{S}w}^{\mathrm{Weak}}s.$

That is, $\pi_{S}(w)$ is the join in left weak order of the set of simple
reflections lying weakly below $w$. Alternatively,
$\pi_{S}(w)=w_{\circ}(\mathrm{Des}_{R}(w))$ is the longest element of the
parabolic subgroup generated by the right descent set $\mathrm{Des}_{R}(w)$ of
$w$.

A natural generalization of the pop-stack sorting map to Coxeter groups was
introduced in [Def21] as

$\displaystyle\mathsf{Pop}_{S}$ $\displaystyle:W\to W$ (2)
$\displaystyle\mathsf{Pop}_{S}(w)$ $\displaystyle=w\cdot\pi_{S}(w)^{-1}.$

Indeed, as discussed in [Def21], this operator agrees with the pop-stack
sorting map when $W$ is the symmetric group $\mathfrak{S}_{n}$. Observe that
$\mathsf{Pop}_{S}$ fixes the identity element $e\in W$. On the other hand, if
$w\in W$ is not the identity element, then $\mathsf{Pop}_{S}(w)$ is strictly
less than $w$ in the right weak order. When $W$ is replaced by its braid
group, the elements of the forward orbit of $w$ under $\mathsf{Pop}_{S}$ are
the prefixes of the Brieskorn normal form of $w$ [Bri71, BS72].

One of the main results from [Def21] is the following generalization of
Ungar’s theorem to arbitrary finite irreducible Coxeter groups.

######  Theorem 1.1 ([Def21]).

Let $W$ be a finite irreducible Coxeter group with a set $S$ of simple
reflections and with Coxeter number $h$. Then $\max\limits_{w\in
W}\left|O_{\mathsf{Pop}_{S}}(w)\right|=h$.

Figure 1 illustrates the action of $\mathsf{Pop}_{S}$ on the hyperoctahedral
group $B_{3}$. As predicted by Theorem 1.1, the maximum size of a forward
orbit in this diagram is $6$, which is the Coxeter number of $B_{3}$.

$\overline{1}\overline{2}\overline{3}$$\overline{1}\overline{2}3$$\overline{2}\overline{1}\overline{3}$$2\overline{3}\overline{1}$$\overline{3}2\overline{1}$$2\overline{1}\overline{3}$$\overline{3}\overline{1}2$$3\overline{1}2$$\overline{1}32$$321$$\overline{1}\overline{3}2$$\overline{2}1\overline{3}$$2\overline{3}1$$\overline{3}21$$21\overline{3}$$\overline{1}2\overline{3}$$12\overline{3}$$1\overline{3}2$$\overline{3}12$$312$$132$$\overline{1}3\overline{2}$$\overline{1}\overline{3}\overline{2}$$13\overline{2}$$1\overline{2}3$$31\overline{2}$$\overline{2}13$$1\overline{3}\overline{2}$$\overline{3}1\overline{2}$$1\overline{2}\overline{3}$$\overline{3}\overline{2}1$$3\overline{2}1$$\overline{2}31$$\overline{2}\overline{3}1$$231$$213$$32\overline{1}$$\overline{3}\overline{1}\overline{2}$$\overline{3}\overline{2}\overline{1}$$3\overline{2}\overline{1}$$\overline{2}3\overline{1}$$3\overline{1}\overline{2}$$\overline{2}\overline{1}3$$\overline{2}\overline{3}\overline{1}$$23\overline{1}$$2\overline{1}3$$\overline{1}23$$123$

Figure 1. A diagram of the Coxeter pop-stack sorting operator
$\mathsf{Pop}_{S}:B_{3}\to B_{3}$. Each vertex in the tree is an element of
$B_{3}$, represented as a signed permutation. The root vertex is the identity
element $e$; the parent of each non-root vertex $w$ is $\mathsf{Pop}_{S}(w)$.

Let us also remark that, by construction, the elements of $W$ that get sorted
into the identity $e$ in at most one iteration of $\mathsf{Pop}_{S}$ are
exactly the longest elements of standard parabolic subgroups—hence, there are
$2^{|S|}$ such elements.

###  1.3. Coxeter pop-tsack torsing

Let $T$ be the set of all reflections of $W$, and fix a Coxeter element $c$ of
$W$ (a regular element of order $h$). Let $\leq_{T}$ denote the absolute order
on $W$. There is a general “dual” philosophy for finite Coxeter groups [Bes03]
in which the set $S$ of simple reflections is replaced by the set $T$ of all
reflections, the longest element $w_{\circ}$ is replaced by the Coxeter
element $c$, and the (left) weak order is replaced by the noncrossing
partition lattice $\mathrm{NC}(W,c)$ (the interval $[e,c]$ in absolute order).

Define the _noncrossing projection_

$\displaystyle\pi_{T}(\cdot,c)$ $\displaystyle:W\to\mathrm{NC}(W,c)$ (3)
$\displaystyle\pi_{T}(w,c)$
$\displaystyle=\bigvee_{t\leq_{T}w}^{\mathrm{NC}(W,c)}t.$

That is, $\pi_{T}(w,c)$ is the join in the noncrossing partition lattice
$\mathrm{NC}(W,c)$ of the set of reflections lying weakly below $w$ in
absolute order. It is natural to make the following definition as the “dual”
of $\mathsf{Pop}_{S}$ from (2):

######  Definition 1.2.

The _Coxeter pop-tsack torsing operator_ 111We have interchanged the roles of
$S$ and $T$. is the map

$\displaystyle\mathsf{Pop}_{T}(\cdot,c)$ $\displaystyle:W\to W$
$\displaystyle\mathsf{Pop}_{T}(w,c)$ $\displaystyle=w\cdot\pi_{T}(w,c)^{-1}.$

When $c$ is understood, we may abbreviate $\pi_{T}(w,c)$ and
$\mathsf{Pop}_{T}(w,c)$ as $\pi_{T}(w)$ and $\mathsf{Pop}_{T}(w)$,
respectively; we establish in Section 4 many structural properties of
$\mathsf{Pop}_{T}$, including that its orbit structure does not depend on $c$.
By construction, the elements $w\in W$ such that $\mathsf{Pop}_{T}(w)=e$ are
exactly the noncrossing partitions—hence, the number of such elements is the
_$W$ -Catalan number_

$\left|\mathrm{NC}(W,c)\right|=\prod_{i=1}^{n}\frac{h+d_{i}}{d_{i}},$

where $d_{1},\ldots,d_{n}$ are the degrees of $W$.

Figure 2 illustrates the action of $\mathsf{Pop}_{T}$ on $B_{3}$; additional
data for other types is given in Figure 3. Notice that every forward orbit of
$\mathsf{Pop}_{T}:B_{3}\to B_{3}$ contains the identity element $e$, which is
a fixed point. Upon inspection of Figure 2, we see that there is a unique
forward orbit of size $6$ and that all other forward orbits have size at most
$5$. This special forward orbit of size $6$ is
$O_{\mathsf{Pop}_{T}}(c^{-1})=\\{c^{-1},c^{-2},\ldots,c^{-5},e\\}$.
Furthermore, this forward orbit is “isolated” in the sense that for each
$2\leq i\leq 5$, the only preimage of $c^{-i}$ under $\mathsf{Pop}_{T}$ is
$c^{-(i-1)}$. In Section 4.4, we use uniform methods to show that this
phenomenon holds for all finite irreducible Coxeter groups.

$\overline{3}\overline{1}\overline{2}$$\overline{3}1\overline{2}$$2\overline{3}\overline{1}$$\overline{3}2\overline{1}$$21\overline{3}$$\overline{3}\overline{2}1$$\overline{1}\overline{3}\overline{2}$$2\overline{1}\overline{3}$$2\overline{3}1$$\overline{3}21$$\overline{1}2\overline{3}$$32\overline{1}$$\overline{2}\overline{3}\overline{1}$$1\overline{3}2$$1\overline{2}\overline{3}$$\overline{1}\overline{3}2$$\overline{2}\overline{1}\overline{3}$$3\overline{2}\overline{1}$$13\overline{2}$$1\overline{2}3$$\overline{1}32$$321$$132$$213$$\overline{3}\overline{2}\overline{1}$$1\overline{3}\overline{2}$$\overline{3}\overline{1}2$$3\overline{1}2$$\overline{2}3\overline{1}$$\overline{2}\overline{1}3$$12\overline{3}$$\overline{3}12$$\overline{2}\overline{3}1$$\overline{1}\overline{2}\overline{3}$$3\overline{1}\overline{2}$$23\overline{1}$$\overline{2}1\overline{3}$$3\overline{2}1$$\overline{1}3\overline{2}$$\overline{2}13$$\overline{2}31$$\overline{1}\overline{2}3$$2\overline{1}3$$\overline{1}23$$31\overline{2}$$312$$231$$123$

Figure 2. A diagram of the Coxeter pop-tsack torsing operator
$\mathsf{Pop}_{T}:B_{3}\to B_{3}$ using the Coxeter element
$c=23\overline{1}$, which is
$(\overline{1}\,\,\overline{2}\,\,\overline{3}\,\,1\,\,2\,\,3)$ in cycle
notation (bars denote negative signs). The root vertex of the tree is the
identity element $e$; the parent of each non-root vertex $w$ is
$\mathsf{Pop}_{T}(w,c)$.

######  Theorem 4.10.

The forward orbit of $c^{-1}$ under $\mathsf{Pop}_{T}(\cdot,c)$ is

$O_{\mathsf{Pop}_{T}}(c^{-1})=\\{c^{-1},c^{-2},\ldots,c^{-(h-1)},e\\}.$

The element $c^{-1}$ has no preimages under $\mathsf{Pop}_{T}$. Moreover, if
$2\leq i\leq h-1$, then the only preimage of $c^{-i}$ under $\mathsf{Pop}_{T}$
is $c^{-(i-1)}$.

Recall that a Coxeter group is called _coincidental_ if it is of type $A$,
type $B$, type $H_{3}$, or dihedral type. We use case-by-case methods to
understand the maximum possible size of a forward orbit of $\mathsf{Pop}_{T}$
in the coincidental types (Section 3 for the dihedral types, Section 5.1 for
type $A$, and Section 5.2 for type $B$; the data for $H_{3}$ is in Figure 3).

######  Theorem 5.1.

Let $W$ be a Coxeter group of coincidental type with a fixed Coxeter element
$c$ and Coxeter number $h$. Then

  * •

$\mathsf{Pop}_{T}^{h-1}(w)=e$ for all $w\in W$,

  * •

$\max\limits_{w\in W}\left|O_{\mathsf{Pop}_{T}}(w)\right|=h$, and

  * •

the only forward orbit of size $h$ is $O_{\mathsf{Pop}_{T}}(c^{-1})$.

Beyond the coincidental types, a few things can go awry. The first unexpected
phenomenon is that there can be more than one forward orbit of size $h$. In
type $D_{n}$, for example, the maximum size of a forward orbit is $h=2n-2$—but
there may be several such orbits ($D_{4}$ has seven forward orbits of size
$6$). In Section 6, we prove the following weaker version of Theorem 5.1.

######  Theorem 6.1.

Let $W$ be a Coxeter group of type $D_{n}$. Then

  * •

$\mathsf{Pop}_{T}^{2n-3}(w)=e$ for all $w\in W$, and

  * •

$\max\limits_{w\in W}\left|O_{\mathsf{Pop}_{T}}(w)\right|=2n-2$.

5.5 and 5.8 give conjectural formulas for the number of elements with forward
orbits of size $h$ or $h-1$ when $W$ is of type $A$ or $B$. 6.6 gives a
conjectural formula for the number of elements whose forward orbits have size
$h$ when $W$ is of type $D$.

Things go even more haywire beyond types $A,B,D,H_{3},$ and $I_{2}(m)$: not
only do the remaining types have forward orbits of size greater than $h$, but
they have periodic orbits that do not contain the identity. This contrasts
with $\mathsf{Pop}_{S}$, which always moves each non-identity element strictly
downward in the right weak order.

Call an element $x\in W$ _periodic_ if $\mathsf{Pop}_{T}^{k}(x)=x$ for some
$k\geq 1$; in this case, we also say the forward orbit
$O_{\mathsf{Pop}_{T}}(x)$ is _periodic_. In general, a finite irreducible
Coxeter group can contain periodic orbits other than that of the identity
element.

######  Theorem 7.1.

The map $\mathsf{Pop}_{T}$ has periodic orbits of size $h$ for $W$ of types
$E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$, or $H_{4}$.

Type | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | $\infty$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{2}$ | 1 | 4 | 1 |  |  |  |  |  |  |  |  |  |  |  |  |  |
$A_{3}$ | 1 | 13 | 9 | 1 |  |  |  |  |  |  |  |  |  |  |  |  |
$A_{4}$ | 1 | 41 | 56 | 21 | 1 |  |  |  |  |  |  |  |  |  |  |  |
$A_{5}$ | 1 | 131 | 305 | 234 | 48 | 1 |  |  |  |  |  |  |  |  |  |  |
$B_{2}$ | 1 | 5 | 1 | 1 |  |  |  |  |  |  |  |  |  |  |  |  |
$B_{3}$ | 1 | 19 | 13 | 10 | 4 | 1 |  |  |  |  |  |  |  |  |  |  |
$B_{4}$ | 1 | 69 | 101 | 91 | 61 | 49 | 11 | 1 |  |  |  |  |  |  |  |  |
$B_{5}$ | 1 | 251 | 646 | 816 | 686 | 761 | 466 | 186 | 26 | 1 |  |  |  |  |  |  |
$D_{4}$ | 1 | 49 | 85 | 34 | 15 | 7 |  |  |  |  |  |  |  |  |  |  |
$D_{5}$ | 1 | 181 | 565 | 523 | 301 | 217 | 107 | 25 |  |  |  |  |  |  |  |  |
$D_{6}$ | 1 | 671 | 3336 | 5396 | 4416 | 3641 | 2946 | 2026 | 536 | 71 |  |  |  |  |  |  |
$D_{7}$ | 1 | 2507 | 18872 | 45274 | 55701 | 50960 | 50835 | 50643 | 32080 | 13193 | 2313 | 181 |  |  |  |  |
$F_{4}$ | 1 | 104 | 171 | 194 | 191 | 119 | 71 | 71 | 71 | 59 | 59 | 11 | 3 | 3 |  |  | 24
$E_{6}$ | 1 | 832 | 4619 | 8214 | 7843 | 8039 | 7307 | 4835 | 3407 | 2687 | 2423 | 1055 | 371 | 107 | 68 | 8 | 24
$H_{3}$ | 1 | 31 | 21 | 16 | 21 | 11 | 6 | 6 | 6 | 1 |  |  |  |  |  |  |

Figure 3. Some data for the map $\mathsf{Pop}_{T}$. Columns labeled by $i$
record the number of elements requiring $i$ iterations of $\mathsf{Pop}_{T}$
to reach the identity. The column labeled by $\infty$ represents elements that
lie in periodic orbits other than the orbit $O_{\mathsf{Pop}_{T}}(e)=\\{e\\}$.

From Theorems 5.1, 6.1 and 7.1, we conclude the following characterization of
coincidental Coxeter groups.

######  Corollary 1.3.

A finite irreducible Coxeter group $W$ with Coxeter number $h$ is coincidental
if and only if $\mathsf{Pop}_{T}:W\to W$ has a unique forward orbit of size
$h$.

###  1.4. Normal forms

Because a presentation of the braid group of a finite Coxeter group is given
by simply forgetting that simple reflections square to the identity, the
Coxeter group embeds canonically into the positive braid monoid by simply
lifting reduced expressions (in simple reflections). It has been an open
problem to find a similar “canonical” embedding in the dual braid monoid.222We
thank T. Gobet for bringing this problem to our attention. The pop-tsack
torsing operator gives a solution to this problem for those finite Coxeter
groups in which the only periodic orbit under $\mathsf{Pop}_{T}$ is that of
the identity—that is, for coincidental Coxeter groups and for type $D$.

######  Theorem 8.1.

Let $W$ be a finite irreducible Coxeter group whose only periodic orbit under
$\mathsf{Pop}_{T}$ is that of the identity. For $w\in W$, write $w_{1}=w$ and
$w_{i}=\mathsf{Pop}_{T}(w_{i-1},c)$ for $i\geq 2$. Let $k=\lvert
O_{\mathsf{Pop}_{T}}(w)\rvert-1$ so that (assuming $w\neq e$) $w_{k}\neq
w_{k+1}=e$. Then the factorization

$w=\pi_{T}(w_{k},c)\pi_{T}(w_{k-1},c)\cdots\pi_{T}(w_{1},c)$

gives a lift from $W$ to the dual braid monoid of $W$.

It would be interesting to study the properties of this lift or to extend it
to complex reflection groups (one potential issue is that for such groups,
some reflections may not be noncrossing partitions).

We conclude with a generalization of stabilized-interval free permutations (as
in [Cal04, Bli]) to finite Coxeter groups using our noncrossing projection
map.

##  2\. Preliminaries

Throughout this article, we assume that $W$ is a finite irreducible Coxeter
group with a set $S$ of simple reflections. The _rank_ of $W$ is $|S|$. Given
$g,w\in W$, we write $w^{g}$ for the conjugate $gwg^{-1}$ of $w$ by $g$. Let
$T=\\{s^{w}:s\in S,w\in W\\}$ denote the set of _reflections_ of $W$. A _$T$
-word_ for an element $w\in W$ is a word $t_{1}\cdots t_{r}$ over the alphabet
$T$ that is equal to $w$ when considered as an element of $W$. The _reflection
length_ of $w$, denoted $\ell_{T}(w)$, is the minimum length of a $T$-word for
$w$. The _absolute order_ on $W$, denoted $\leq_{T}$, is defined by saying
$v\leq_{T}w$ if $\ell_{T}(wv^{-1})=\ell_{T}(w)-\ell_{T}(v)$. Since $T$ is
closed under conjugation by elements of $W$, we have $v\leq_{T}w$ if and only
if $\ell_{T}(v^{-1}w)=\ell_{T}(w)-\ell_{T}(v)$.

It is often useful to consider the _geometric representation_ of $W$ on a
vector space $V\cong\mathbb{R}^{|S|}$. We refer the reader to [Arm09, BB05]
for more thorough discussions of this representation; here, we simply mention
the basic results that we will need. We denote the action of an element $w\in
W$ on a vector $\alpha\in V$ by $w\cdot\alpha$. For every $t\in T$, there is a
hyperplane $H_{t}\subseteq V$ such that $t$ acts via the linear transformation
that reflects through $H_{t}$. Furthermore, $w\cdot H_{t}=H_{t^{w}}$ for every
$w\in W$ and $t\in T$. Define the _moved space_ of $w$ to be
$\mathrm{Mov}(w)=\mathrm{im}(w-1)$, the image of the linear transformation
$w-1:V\to V$ ($1$ is the identity transformation). Brady and Watt [BW02]
showed that if $t\in T$ and $w\in W$, then

(4) $\mathrm{Mov}(t)\subseteq\mathrm{Mov}(w)\text{ if and only if
}t\leq_{T}w.$

The ring $\mathrm{Sym}(V_{\mathbb{C}}^{*})^{W}$ of $W$-invariant polynomials
is a polynomial ring $\mathbb{C}[f_{1},\ldots,f_{n}]$ over a set of invariant
polynomials $f_{i}$ that may be chosen homogeneous and whose degrees
$d_{1}\leq\cdots\leq d_{n}$ are uniquely determined. We write $h=d_{n}$ for
the _Coxeter number_ of $W$.

A _standard Coxeter element_ of $W$ is an element obtained by multiplying the
simple reflections together in some order. A _regular element_ of $W$ is an
element with an eigenvector that lies in the complement of the reflecting
hyperplanes. Following [RRS17], we define a _Coxeter element_ of $W$ to be a
regular element of order $h$. For Weyl groups, Coxeter elements coincide with
conjugates of standard Coxeter elements. More generally, a _reflection
automorphism_ of $W$ is a group automorphism that preserves the set of
reflections of $W$. By [RRS17, Proposition 1.4], the reflection automorphisms
of $W$ act transitively on its Coxeter elements—thus, all Coxeter elements
have the same order $h$.

We assume throughout this article that we have fixed a Coxeter element $c$ of
$W$. A _noncrossing partition_ is an element $v\in W$ such that $v\leq_{T}c$.
The interval $[e,c]$ in the absolute order is called the _noncrossing
partition lattice_ and denoted $\mathrm{NC}(W,c)$; it is indeed a lattice.
This is the only type of lattice we will consider in the rest of the article,
so all meets (denoted by $\bigwedge$) and joins (denoted by $\bigvee$) will be
with respect to this lattice. Furthermore, the definitions of the noncrossing
projection $\pi_{T}$ and the Coxeter pop-tsack torsing operator
$\mathsf{Pop}_{T}$ from (1.3) and Definition 1.2 are made with respect to this
fixed Coxeter element. Since the reflection automorphisms of $W$ act
transitively on the Coxeter elements, different Coxeter elements give rise to
isomorphic noncrossing partition lattices.

##  3\. Dihedral Groups

As the dihedral case will be useful when analyzing the general case, we
immediately dispense with it in this short section. The dihedral group
$I_{2}(h)$ has $2h$ elements: the identity $e$, $h$ reflections, a Coxeter
element $c$, and then the elements $c^{-1},c^{-2},\ldots,c^{-(h-1)}$. The
identity, reflections, and Coxeter element are all noncrossing partitions, and
they all map to the identity under a single application of $\mathsf{Pop}_{T}$.
For each $1\leq i\leq h-1$, the element $c^{-i}$ requires exactly $h-i$
iterations of $\mathsf{Pop}_{T}$ to reach the identity. The right-hand side of
Figure 4 illustrates the hyperplane arrangement for the dihedral group
$I_{2}(4)$.

$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{e}$$(12)(34)$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{s}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{t}$$(23)$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{c}$$(13)(24)$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{c^{3}}$$(14)$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{sts}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{tst}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{c^{2}}$

Figure 4. Left: the permutahedron for $\mathfrak{S}_{4}$ with the eight
vertices on the Coxeter plane for bipartite $c$ labeled in red. Right: the
hyperplane arrangement for the corresponding dihedral group $I_{2}(4)$.

##  4\. General Properties of $\mathsf{Pop}_{T}$

###  4.1. Absolute monotonicity

Given $x\in W$, let $W_{x}$ be the subgroup of $W$ generated by the
reflections lying weakly below $x$ in the absolute order. The subgroups
$W_{x}$ for $x\in\mathrm{NC}(W,c)$ are called the _noncrossing parabolic
subgroups_ of $W$. As discussed in [Rea10], the noncrossing parabolic
subgroups of $W$ form a lattice under the containment order, and the map
$x\mapsto W_{x}$ gives an isomorphism from $\mathrm{NC}(W,c)$ to this lattice.

######  Lemma 4.1.

For every $x\in W$, we have $\pi_{T}(\mathsf{Pop}_{T}(x))\leq_{T}\pi_{T}(x)$.

###### Proof.

The parabolic subgroup generated by the reflections lying weakly below
$\pi_{T}(x)$ in absolute order is $W_{\pi_{T}(x)}$. All of the reflections
lying weakly below $x$ also lie weakly below $\pi_{T}(x)$, so $x$ is an
element of $W_{\pi_{T}(x)}$. Of course, $\pi_{T}(x)$ also belongs to
$W_{\pi_{T}(x)}$. This means that $\mathsf{Pop}_{T}(x)=x\pi_{T}(x)^{-1}$ is in
$W_{\pi_{T}(x)}$, so all of the reflections weakly below $\mathsf{Pop}_{T}(x)$
are in $W_{\pi_{T}(x)}$. It follows that $W_{\pi_{T}(\mathsf{Pop}_{T}(x))}$ is
contained in $W_{\pi_{T}(x)}$, and this proves the result. ∎

The following proposition gives a condition for elements to have an “isolated”
forward orbit under $\mathsf{Pop}_{T}$. It states that the only possible
preimage under $\mathsf{Pop}_{T}$ of an element $w\in W$ with $\pi_{T}(w)=c$
is $wc$.

######  Proposition 4.2.

If $x,w\in W$ are such that $\mathsf{Pop}_{T}(x)=w$ and $\pi_{T}(w)=c$, then
$x=wc$.

###### Proof.

Suppose $x$ is such that $\mathsf{Pop}_{T}(x)=w$. By Lemma 4.1,
$c\leq_{T}\pi_{T}(x)$, so we must have $\pi_{T}(x)=c$. However, this implies
that $x=\mathsf{Pop}_{T}(x)\pi_{T}(x)=wc$. ∎

###  4.2. Equivariance under reflection automorphisms

In this section, we make a point of decorating $\pi_{T}$ and
$\mathsf{Pop}_{T}$ by the Coxeter element $c$ in order to differentiate
different choices of Coxeter elements. We show a sort of equivariance of
$\mathsf{Pop}_{T}$ with respect to reflection automorphisms, which establishes
that the orbit structure of $\mathsf{Pop}_{T}$ does not depend on the choice
of Coxeter element $c$. Thus, the Coxeter pop-tsack torsing operators defined
with respect to different Coxeter elements are dynamically equivalent to each
other, so none of our results about $\mathsf{Pop}_{T}$ depend on the choice of
$c$. This means that we have the freedom to choose any Coxeter element we want
when we work with specific Coxeter groups, and it will often be helpful to use
a specific choice that makes the resulting combinatorics easiest to describe.

######  Proposition 4.3.

Let $\phi$ be a reflection automorphism of $W$. For $w\in W$, we have

  * •

$\phi(\pi_{T}(w,\phi^{-1}(c)))=\pi_{T}(\phi(w),c)$, and

  * •

$\phi(\mathsf{Pop}_{T}(w,\phi^{-1}(c)))=\mathsf{Pop}_{T}(\phi(w),c)$.

###### Proof.

For the first point, observe that the map $w\mapsto\phi(w)$ is a lattice
isomorphism from $\mathrm{NC}(W,c)$ to $\mathrm{NC}(W,\phi(c))$. Furthermore,
for each reflection $t$, we have $t\leq_{T}w$ if and only if
$\phi(t)\leq_{T}\phi(w)$. Therefore,

$\phi\left(\pi_{T}(w,\phi^{-1}(c))\right)=\phi\left(\bigvee_{t\leq_{T}w}^{\mathrm{NC}(W,\phi^{-1}(c))}t\right)=\bigvee_{t\leq_{T}w}^{\mathrm{NC}(W,c)}\phi(t)=\bigvee_{t\leq_{T}\phi(w)}^{\mathrm{NC}(W,c)}t=\pi_{T}(\phi(w),c).$

For the second point, we use the first point to check that

$\displaystyle\mathsf{Pop}_{T}(\phi(w),c)$
$\displaystyle=\phi(w)\pi_{T}(\phi(w),c)^{-1}=\phi(w)\left(\phi\left(\pi_{T}(w,\phi^{-1}(c))\right)\right)^{-1}$
$\displaystyle=\phi\left(w\pi_{T}(w,\phi^{-1}(c))^{-1}\right)=\phi\left(\mathsf{Pop}_{T}(w,\phi^{-1}(c))\right).\qed$

As conjugation by $c$ is a special kind of reflection automorphism that
restricts to a lattice isomorphism of $\mathrm{NC}(W,c)$, we can specialize
the preceding result to conclude the following.

######  Corollary 4.4.

For $w\in W$, we have

  * •

$\pi_{T}(w,c)^{c}=\pi_{T}(w^{c},c)$, and

  * •

$\mathsf{Pop}_{T}(w,c)^{c}=\mathsf{Pop}_{T}(w^{c},c)$.

Since reflection automorphisms act transitively on the set of Coxeter
elements, we further obtain:

######  Corollary 4.5.

The orbit structure of $\mathsf{Pop}_{T}$ does not depend on the choice of
Coxeter element.

###  4.3. Folding and the Coxeter plane

######  Definition 4.6.

Let $(W,S)$ and $({W^{\prime}},S^{\prime})$ be Coxeter systems such that $W$
and $W^{\prime}$ are finite and irreducible. Suppose there is a surjective map
$\mathrm{fold}:S\to S^{\prime}$ such that for each $s^{\prime}\in S^{\prime}$,
the elements of $\mathrm{fold}^{-1}(s^{\prime})$ all commute with each other.
If there is an injective homomorphism $\mathrm{unfold}:{W^{\prime}}\to W$ with
$\mathrm{unfold}(s^{\prime})=\prod_{s\in\mathrm{fold}^{-1}(s^{\prime})}s$ for
all $s^{\prime}\in S^{\prime}$, then we say that $W$ _folds_ to $W^{\prime}$
and that $W^{\prime}$ _unfolds_ into $W$.

Figure 5 illustrates several examples of folding. If $W$ folds to
${W^{\prime}}$ and $c^{\prime}$ is a Coxeter element of ${W^{\prime}}$, then
$\mathrm{unfold}(c^{\prime})$ is a Coxeter element of $W$. Hence, the Coxeter
number is preserved by folding.

A useful example of folding goes through the Coxeter plane. Any finite
irreducible Coxeter group $W$ with Coxeter number $h$ folds to the dihedral
group $I_{2}(h)$ of order $2h$, as we now explain. Since the Coxeter-Dynkin
diagram of $W$ is a tree (hence, a bipartite graph), the set $S$ of simple
reflections of $W$ naturally separates into two sets $S_{+}$ and $S_{-}$,
where the simple reflections within each set all commute with each other.
Writing $c_{\pm}$ for the product of the reflections in $S_{\pm}$ so that
$c=c_{+}c_{-}$ is a Coxeter element of $W$, $\langle c_{+},c_{-}\rangle$ is
the dihedral group $I_{2}(h)$. Each of $c_{+}$ and $c_{-}$ acts as a
reflection on the _Coxeter plane_ , a plane spanned by the real and imaginary
parts of an eigenvector for $c$ with eigenvalue $e^{2\pi i/h}$. Figure 4
illustrates this for the symmetric group $\mathfrak{S}_{4}$ with Coxeter plane
the dihedral group $I_{2}(4)$.

$W$ | ${W^{\prime}}$ | $h$ | Diagrammatic folding
---|---|---|---
$A_{2n-1}$ | $B_{n}$ | $2n$ |
$D_{n}$ | $B_{n-1}$ | $2n-2$ |
$E_{6}$ | $F_{4}$ | $12$ |
$E_{8}$ | $H_{4}$ | 30 | 5
Figure 5. Examples of folding of finite Coxeter groups.

In this subsection, we write $T$ and $T^{\prime}$ for the sets of reflections
in $W$ and $W^{\prime}$, respectively.

######  Lemma 4.7.

Let ${W^{\prime}}$ be a finite irreducible Coxeter group that unfolds into
$W$. Let $c^{\prime}$ be a bipartite Coxeter element of ${W^{\prime}}$, and
let $c=\mathrm{unfold}(c^{\prime})$ be the corresponding Coxeter element of
$W$. Then $\mathrm{unfold}(\mathrm{NC}({W^{\prime}}),c^{\prime})$ is a
sublattice of $\mathrm{NC}(W,c)$.

###### Proof.

Suppose that $w^{\prime}\in W$ and $w^{\prime}\leq_{T^{\prime}}c^{\prime}$ so
that ${w^{\prime}}$ can be found as a prefix of $c^{\prime}$. Then
$\mathrm{unfold}({w^{\prime}})$ can be found as a prefix of $c$, so
$\mathrm{unfold}({w^{\prime}})\leq_{T}c$. By the same reasoning (replacing now
$c^{\prime}$ by any element of $\mathrm{NC}({W^{\prime}},c^{\prime})$), we
have that $\mathrm{unfold}(\mathrm{NC}({W^{\prime}},c^{\prime}))$ is a
subposet of $\mathrm{NC}(W,c)$.

We now check that $\mathrm{unfold}(\mathrm{NC}({W^{\prime}},c^{\prime}))$ is a
sublattice of $\mathrm{NC}(W,c)$ (that is, that
$\mathrm{unfold}(\mathrm{NC}({W^{\prime}},c^{\prime}))$ is closed under the
meet and join operations of $\mathrm{NC}(W,c)$). We know by [BW08] that a
noncrossing partition $w$ is characterized by the set of reflections that lie
below it: $w=\bigvee_{t\leq_{T}w}t$. Since $w\wedge
v=\bigvee_{t:t\leq_{T}w\text{ and }t\leq_{T}v}t$, it follows that
$\mathrm{unfold}(\mathrm{NC}({W^{\prime}}))$ is a meet-sublattice. Indeed, for
fixed $t^{\prime}\in T^{\prime}$ and $r\in\mathrm{fold}^{-1}(t^{\prime})$, we
have $r\leq\mathrm{unfold}(w^{\prime})\wedge\mathrm{unfold}(v^{\prime})$ if
and only if $r\leq_{T}\mathrm{unfold}(w^{\prime})$ and
$r\leq_{T}\mathrm{unfold}(v^{\prime})$. This occurs if and only if
$t^{\prime}\leq_{T^{\prime}}w^{\prime}$ and
$t^{\prime}\leq_{T^{\prime}}v^{\prime}$, which occurs if and only if
$t^{\prime}\leq_{T^{\prime}}w^{\prime}\wedge v^{\prime}$, which occurs if and
only if $r\leq_{T}\mathrm{unfold}(w^{\prime}\wedge v^{\prime})$.

The Kreweras complement maps $K:\mathrm{NC}(W,c)\to\mathrm{NC}(W,c)$ and
$K^{\prime}:\mathrm{NC}(W^{\prime},c^{\prime})\to\mathrm{NC}(W^{\prime},c^{\prime})$,
defined by $K(w)=cw^{-1}$ and
$K^{\prime}(w^{\prime})=c^{\prime}(w^{\prime})^{-1}$, are lattice anti-
isomorphisms [Arm09]. We have $K\circ\mathrm{unfold}=\mathrm{unfold}\circ
K^{\prime}$ because $\mathrm{unfold}$ is a homomorphism that sends
$c^{\prime}$ to $c$. This shows that $K$ preserves the set
$\mathrm{unfold}(\mathrm{NC}({W^{\prime}},c^{\prime}))$, so
$\mathrm{unfold}(\mathrm{NC}({W^{\prime}},c^{\prime}))$ is also a join-
sublattice of $\mathrm{NC}(W,c)$. ∎

######  Theorem 4.8.

If a finite irreducible Coxeter group $W^{\prime}$ unfolds into $W$, then for
$w^{\prime}\in W^{\prime}$,

$\mathrm{unfold}(\mathsf{Pop}_{T^{\prime}}^{k}(w^{\prime}))=\mathsf{Pop}_{T}^{k}(\mathrm{unfold}(w^{\prime})).$

###### Proof.

It suffices to prove the case when $k=1$ since the general result then follows
by induction. Choose $w^{\prime}\in{W^{\prime}}$. If $t^{\prime}\in
T^{\prime}$ is such that $t^{\prime}\leq_{T^{\prime}}w^{\prime}$, then
$\mathrm{unfold}(t^{\prime})\leq_{T^{\prime}}\mathrm{unfold}(w^{\prime})$. By
the lattice embedding of Lemma 4.7, we have
$\pi_{T}({W^{\prime}})=\mathrm{unfold}(\bigvee_{t^{\prime}\leq_{W^{\prime}}{w^{\prime}}}t^{\prime})=\bigvee_{t^{\prime}\leq_{W^{\prime}}{w^{\prime}}}\mathrm{unfold}(t^{\prime})=\pi_{T}(\mathrm{unfold}({w^{\prime}}))$.
Because $\mathrm{unfold}$ is a group homomorphism, we have

$\mathrm{unfold}(\mathsf{Pop}_{T^{\prime}}({w^{\prime}}))=\mathrm{unfold}({w^{\prime}}\pi_{T^{\prime}}({w^{\prime}})^{-1})=\mathrm{unfold}({w^{\prime}})\mathrm{unfold}(\pi_{T^{\prime}}({w^{\prime}}))^{-1}=\mathrm{unfold}({w^{\prime}})\pi_{T}(\mathrm{unfold}({w^{\prime}}))^{-1}$
$=\mathsf{Pop}_{T}(\mathrm{unfold}({w^{\prime}})).\qed$

We can now prove the part of Theorem 4.10 stating that $c^{-1}$ requires $h-1$
iterations of $\mathsf{Pop}_{T}$ to reach the identity.

######  Corollary 4.9.

We have $\mathsf{Pop}_{T}^{h-1}(c^{-1})=e$. If $W$ is not the trivial group,
then $\mathsf{Pop}_{T}^{h-2}(c^{-1})\neq e$.

###### Proof.

If $W$ is trivial, then so is the result, so we may assume $|W|\geq 2$. Let
$W^{\prime}=I_{2}(h)$, and let $c^{\prime}$ be a Coxeter element of
$W^{\prime}$. We have seen that $W$ folds to $W^{\prime}$ via the Coxeter
plane and that $\mathrm{unfold}(c^{\prime})$ is a Coxeter element of $W$. By
Proposition 4.3, we may assume $c=\mathrm{unfold}(c^{\prime})$. As discussed
in Section 3, we have
$\mathsf{Pop}_{T^{\prime}}^{h-2}(c^{-1})\neq\mathsf{Pop}_{T^{\prime}}(c^{-1})=e$.
The result now follows from Theorem 4.8 and the fact that
$\mathrm{unfold}:W^{\prime}\to W$ is injective. ∎

###  4.4. The inverse Coxeter element orbit

In this section, we prove the following theorem regarding the forward orbit of
$c^{-1}$ under $\mathsf{Pop}_{T}(\cdot,c)$.

######  Theorem 4.10.

The forward orbit of $c^{-1}$ under $\mathsf{Pop}_{T}(\cdot,c)$ is

$O_{\mathsf{Pop}_{T}}(c^{-1})=\\{c^{-1},c^{-2},\ldots,c^{-(h-1)},e\\}.$

The element $c^{-1}$ has no preimages under $\mathsf{Pop}_{T}$. Moreover, if
$2\leq i\leq h-1$, then the only preimage of $c^{-i}$ under $\mathsf{Pop}_{T}$
is $c^{-(i-1)}$.

Using the fact that $W$ folds to $I_{2}(h)$ via the Coxeter plane, we proved
in Corollary 4.9 that $c^{-1}$ requires $(h-1)$ iterations of
$\mathsf{Pop}_{T}$ to reach the identity. Let us provide an alternative proof
of this fact. We require a lemma, which states that the join in the
noncrossing partition lattice of any orbit of a reflection under the
conjugation action of $c$ is just $c$.

######  Lemma 4.11.

If $t\in T$ is a reflection, then $c=\bigvee\limits_{k=0}^{h-1}t^{c^{k}}$.

###### Proof.

Since $W$ folds to $I_{2}(h)$ via the Coxeter plane, we can consider the
Coxeter element $\mathrm{unfold}(c^{\prime})$ of $W$, where $c^{\prime}$ is a
Coxeter element of $I_{2}(h)$. Each Coxeter element of $W$ can be obtained by
applying a reflection automorphism to $\mathrm{unfold}(c^{\prime})$, so it
suffices to prove the lemma when $c=\mathrm{unfold}(c^{\prime})$.

Conjugation by $c$ induces an automorphism of $\mathrm{NC}(W,c)$; therefore,
if a set $O\subseteq\mathrm{NC}(W,c)$ is invariant under conjugation by $c$,
then the element $x=\bigvee_{w\in O}w$ is invariant under conjugation by $c$:

$x^{c}=\left(\bigvee_{w\in O}w\right)^{c}=\bigvee_{w\in O}w^{c}=\bigvee_{w\in
O}w=x.$

It is a classical result that the centralizer of $c$ is the cyclic group
generated by $c$ (see [Car72, Proposition 30]), so we must have $x=c^{i}$ for
some integer $i$. Therefore, we are left to prove that the only powers of $c$
in $\mathrm{NC}(W,c)$ are $c$ and $e$. This will follow if we can show that
the only powers of $c^{\prime}$ in $\mathrm{NC}(I_{2}(h),c^{\prime})$ are
$c^{\prime}$ and $e$. But this is clear since the absolute length of any power
of $c^{\prime}$ (a rotation) is $0$ or $2$ (see also [Bes15, Lemma 12.2]).∎

######  Proposition 4.12.

For each integer $1\leq i\leq h-1$, we have
$\mathsf{Pop}_{T}(c^{-i})=c^{-(i+1)}$. Thus, the forward orbit of $c^{-1}$
under $\mathsf{Pop}_{T}$ is $\\{c^{-1},c^{-2},\ldots,c^{-(h-1)},e\\}$, which
has size $h$.

###### Proof.

Fix $1\leq i\leq h-1$. We claim that if $t\in T$ is such that
$t\leq_{T}c^{-i}$, then $t^{c}\leq_{T}c^{-i}$. According to (4), we have
$\mathrm{Mov}(t)\subseteq\mathrm{Mov}(c^{-i})$. Take $\alpha_{t}$ to be a
vector orthogonal to the hyperplane $H_{t}$ so that
$\mathrm{Mov}(t)=\mathrm{span}\\{\alpha_{t}\\}$. Then
$\alpha_{t}\in\mathrm{Mov}(c^{-i})$, so there exists a vector $\beta\in V$
such that $(c^{-i}-1)\cdot\beta=\alpha_{t}$. Applying $c$ to each side of this
equation shows that

$(c^{-i}-1)\cdot(c\cdot\beta)=c\cdot((c^{-i}-1)\cdot\beta)=c\cdot\alpha_{t}.$

Thus, $\mathrm{span}\\{c\cdot\alpha_{t}\\}\subseteq\mathrm{Mov}(c^{-i})$.
Since $c\cdot H_{t}=H_{t^{c}}$, we know that $c\cdot\alpha_{t}$ is orthogonal
to $H_{t^{c}}$. Thus,
$\mathrm{Mov}(t^{c})=\mathrm{span}\\{c\cdot\alpha_{t}\\}$. Appealing to (4)
again, we find that $t^{c}\leq_{T}c^{-i}$.

Note that there must exist some $t\in T$ with $t\leq_{T}c^{-i}$ since
$c^{-i}\neq e$. It follows from the preceding paragraph that
$t^{c^{k}}\leq_{T}c^{-i}$ for all $0\leq k\leq h-1$. Therefore, we can use
Lemma 4.11 to see that

$c=\bigvee\limits_{k=0}^{h-1}t^{c^{k}}\leq_{T}\bigvee_{t^{\prime}\leq_{T}c^{-i}}t^{\prime}=\pi_{T}(c^{-i}).$

Consequently, $\pi_{T}(c^{-i})=c$, so
$\mathsf{Pop}_{T}(c^{-i})=c^{-i}c^{-1}=c^{-(i+1)}$. ∎

######  Corollary 4.13.

The element $c^{-1}$ has no preimages under $\mathsf{Pop}_{T}$. Moreover, if
$2\leq i\leq h-1$, then the only preimage of $c^{-1}$ under $\mathsf{Pop}_{T}$
is $c^{-(i-1)}$.

###### Proof.

By Proposition 4.2 and Proposition 4.12, the only possible preimage of
$c^{-i}$ under $\mathsf{Pop}_{T}$ is $c^{-(i-1)}$. Furthermore, $c^{-(i-1)}$
is actually a preimage of $c^{-i}$ unless $i=1$. ∎

This completes the proof of Theorem 4.10.

##  5\. Types $A$ and $B$

In this section, we prove the following theorem using type-by-type analysis of
the dynamics of $\mathsf{Pop}_{T}$ on Coxeter groups of types $A$ and $B$.
(The dihedral case was addressed in Section 3, and $H_{3}$ was checked by
computer).

######  Theorem 5.1.

Let $W$ be a Coxeter group of coincidental type with a fixed Coxeter element
$c$ and Coxeter number $h$. Then

  * •

$\mathsf{Pop}_{T}^{h-1}(w)=e$ for all $w\in W$,

  * •

$\max\limits_{w\in W}\left|O_{\mathsf{Pop}_{T}}(w)\right|=h$, and

  * •

the only forward orbit of $\mathsf{Pop}_{T}$ of size $h$ is
$O_{\mathsf{Pop}_{T}}(c^{-1})$.

###  5.1. Type $A$

Recall that the Coxeter group of type $A_{n-1}$ is the symmetric group
$\mathfrak{S}_{n}$, the group of permutations of $[n]$. We write permutations
in disjoint cycle notation. Throughout this section, we fix the Coxeter
element $c=(12\cdots n)$.

There is a simple combinatorial way to describe the noncrossing partitions in
$\mathrm{NC}(\mathfrak{S}_{n},c)$. Begin by labeling $n$ equally-spaced points
on a circle with the numbers $1,\ldots,n$ in clockwise order. Now choose a set
partition $\rho$ of $[n]$. For each block of $\rho$, draw the convex hull of
the points on the circle labeled with the elements of that block. We say
$\rho$ is a _noncrossing set partition_ if none of the convex hulls of
different blocks intersect each other. Suppose $\rho$ is a noncrossing set
partition. For each block of $\rho$, form a cycle by ordering the elements of
the block cyclically in clockwise order around the circle. The product of
these cycles is a noncrossing partition in $\mathrm{NC}(\mathfrak{S}_{n},c)$,
and every noncrossing partition in $\mathrm{NC}(\mathfrak{S}_{n},c)$ arises
uniquely in this way. See Figure 6.

Figure 6. A noncrossing set partition. The corresponding noncrossing partition
in $\mathrm{NC}(\mathfrak{S}_{n},c)$ is
$(1\,\,3\,\,6\,\,7)(2)(4\,\,5)(8\,\,9\,\,10)$.

We will find it convenient to refer to a cycle of a permutation and to the set
of entries in the cycle interchangeably. For example, we sometimes write
$i\in\mathcal{C}$ to mean that $i$ is an entry in a cycle $\mathcal{C}$. Let
$\mathrm{cyc}_{>1}(w)$ denote the number of non-singleton cycles of a
permutation $w$.

The partial order on $\mathrm{NC}(\mathfrak{S}_{n},c)$, which is the
restriction of the absolute order on $\mathfrak{S}_{n}$ to $[e,c]$, is easy to
describe combinatorially because it corresponds to the reverse refinement
order on noncrossing set partitions. In other words, if
$u,v\in\mathrm{NC}(\mathfrak{S}_{n},c)$, then $u\leq_{T}v$ if and only if
every cycle in $u$ is contained (as a set) in a cycle in $v$. The noncrossing
projection of a permutation $w\in\mathfrak{S}_{n}$ is the smallest noncrossing
partition $v\in\mathrm{NC}(\mathfrak{S}_{n},c)$ such that every cycle in $w$
is contained in a cycle in $v$.

An _antiexceedance_ of a permutation $w\in\mathfrak{S}_{n}$ is an element
$i\in[n]$ such that $i<w^{-1}(i)$. Write $\mathrm{Aexc}(w)$ for the set of
antiexceedances of $w$, and let $\mathrm{aexc}=\lvert\mathrm{Aexc}(w)\rvert$.

######  Theorem 5.2.

The following properties relating antiexceedances and $\mathsf{Pop}_{T}$ hold:

  * •

The element $c^{-1}$ has $h-1=n-1$ antiexceedances.

  * •

Every element of $\mathfrak{S}_{n}$ other than $c^{-1}$ has at most $n-2$
antiexceedances.

  * •

For every $w\in\mathfrak{S}_{n}$, we have
$\mathrm{aexc}(\mathsf{Pop}_{T}(w))=\mathrm{aexc}(w)-\mathrm{cyc}_{>1}(\pi_{T}(w))$.

###### Proof.

The set of antiexceedances of a permutation $w\in\mathfrak{S}_{n}$ is a subset
of $[n-1]$. It is straightforward to see that this set of antiexceedances is
equal to $[n-1]$ if and only if $w=c^{-1}=(n\cdots 21)$. This proves the first
two bulleted items. Now fix $w\in\mathfrak{S}_{n}$, and let $v=\pi_{T}(w)$ so
that $\mathsf{Pop}_{T}(w)=wv^{-1}$.

Suppose $i\in[n]$ is not an antiexceedance of $w$. This means that $i\geq
w^{-1}(i)$. Let $\mathcal{C}$ be the cycle of $v$ that contains $i$. Since
every cycle in $w$ is contained in a cycle in $v$, we know that $\mathcal{C}$
also contains $w^{-1}(i)$. The cycle $\mathcal{C}$ is cyclically ordered
clockwise, so $v(w^{-1}(i))$, which is the entry immediately following
$w^{-1}(i)$ in $\mathcal{C}$, is less than or equal to $i$. Because
$v(w^{-1}(i))=(\mathsf{Pop}_{T}(w))^{-1}(i)$, this shows that $i$ is not an
antiexceedance of $\mathsf{Pop}_{T}(w)$. As $i$ was arbitrary, this proves
that $\mathrm{Aexc}(\mathsf{Pop}_{T}(w))\subseteq\mathrm{Aexc}(w)$.

We claim $j\in\mathrm{Aexc}(w)\setminus\mathrm{Aexc}(\mathsf{Pop}_{T}(w))$ if
and only if $v^{-1}(j)$ is the largest entry in a non-singleton cycle. Note
that this claim immediately implies the third bulleted item in the statement
of the theorem. To prove this claim, suppose first that
$j\in\mathrm{Aexc}(w)\setminus\mathrm{Aexc}(\mathsf{Pop}_{T}(w))$. The cycle
of $w$ containing $j$ and $w^{-1}(j)$ is contained in a cycle $\mathcal{C}$ of
$v$; note that $\mathcal{C}$ is not a singleton cycle because $w^{-1}(j)\neq
j$. If $w^{-1}(j)$ is not the largest entry in $\mathcal{C}$, then
$v(w^{-1}(j))$, which is the entry immediately following $w^{-1}(j)$ in
$\mathcal{C}$, is larger than $w^{-1}(j)$. However, this implies that
$\mathsf{Pop}_{T}(w)^{-1}(j)=v(w^{-1}(j))>w^{-1}(j)>j$, contradicting the fact
that $j\not\in\mathrm{Aexc}(\mathsf{Pop}_{T}(w))$. This proves that
$w^{-1}(j)$ must be the largest entry in a non-singleton cycle in $v$.

To prove the reverse containment, suppose $w^{-1}(j)$ is the largest entry in
a non-singleton cycle $\mathcal{C}$ in $v$. Note that $j\in\mathcal{C}$. If
$j$ were a fixed point of $w$, then it would also be a fixed point of $v$,
contradicting the fact that $\mathcal{C}$ is not a singleton cycle. Therefore,
$w^{-1}(j)>j$, so $j\in\mathrm{Aexc}(w)$. Observe that $v(w^{-1}(j))$, which
is the entry immediately following $w^{-1}(j)$ in $\mathcal{C}$, is the
smallest entry in $\mathcal{C}$. Therefore, $v(w^{-1}(j))\leq j$. This proves
that $j\not\in\mathrm{Aexc}(\mathsf{Pop}_{T}(w))$, as desired. ∎

######  Example 5.3.

Consider the permutation $w=(1\,\,2\,\,4\,\,6\,\,5)(3)(7\,\,9)(8\,\,10)$ in
$\mathfrak{S}_{10}$. This permutation is represented pictorially on the left
of Figure 7; its noncrossing projection $\pi_{T}(w)$ is
$(1\,\,2\,\,4\,\,5\,\,6)(3)(7\,\,8\,\,9\,\,10)$, which is drawn in the middle
of Figure 7. Finally, $\mathsf{Pop}_{T}(w)=w\pi_{T}(w)^{-1}$ appears on the
right side of the figure. The $4$ anti-exceedances of $w$ (shown in red in the
figure) are $1,5,7,8$. Since $\pi_{T}(w)$ has $2$ non-singleton cycles, the
permutation $\mathsf{Pop}_{T}(w)$ has $4-2=2$ anti-exceedances—namely, $1$ and
$7$.

Figure 7. A permutation (left), its noncrossing projection (middle), and its
image under $\mathsf{Pop}_{T}$ (right).

The Coxeter number of $\mathfrak{S}_{n}$ is $n$, so the following corollary
completes the proof of Theorem 5.1 in type $A$.

######  Corollary 5.4.

Every forward orbit of the map
$\mathsf{Pop}_{T}:\mathfrak{S}_{n}\to\mathfrak{S}_{n}$ has size at most $n$,
and the only forward orbit that has size equal to $n$ is
$O_{\mathsf{Pop}_{T}}(c^{-1})$.

###### Proof.

We know that $O_{\mathsf{Pop}_{T}}(c^{-1})$ has size $n$ by Proposition 4.12.
The only element of $\mathfrak{S}_{n}$ whose noncrossing projection has no
non-singleton cycles is the identity element, so it follows from the third
bulleted item in Theorem 5.2 that
$\mathrm{aexc}(\mathsf{Pop}_{T}(w))\leq\mathrm{aexc}(w)-1$ for all
$w\in\mathfrak{S}_{n}\setminus\\{e\\}$. The only element of $\mathfrak{S}_{n}$
with $0$ antiexceedances is the identity element, so it follows from the
second bulleted item in Theorem 5.2 that $\mathsf{Pop}_{T}^{n-2}(w)=e$ for all
$w\in\mathfrak{S}_{n}\setminus\\{c^{-1}\\}$. ∎

######  Conjecture 5.5.

In type $A_{n-1}$, the number of permutations that require exactly $n-2$ or
$n-1$ iterations of $\mathsf{Pop}_{T}$ to reach the identity is
$2^{n}-\binom{n}{2}$.

###  5.2. Type $B$

The previous section established Theorem 5.1 in type $A$. The group $A_{2n-1}$
folds to $B_{n}$, so it follows immediately from Theorem 4.8 that Theorem 5.1
must also hold in type $B$. In this section, we will verify Theorem 5.1 in
type $B$ directly using the combinatorics of the hyperoctahedral groups; this
will also serve as a useful warm-up for our analysis of type $D$ in the next
section.

For the sake of convenience, we replace negative signs with bars, writing
$\overline{i}$ instead of $-i$. Let
$\pm[n]=\\{\overline{n},\ldots,\overline{1},1,\ldots,n\\}$. The
_hyperoctahedral group_ $B_{n}$ is the group of permutations $w$ of the set
$\pm[n]$ such that $w(\overline{i})=\overline{w(i)}$ for all $i\in[n]$.
Throughout this section, we fix the Coxeter element
$c=(\overline{1}\,\,\overline{2}\,\cdots\overline{n}\,\,1\,\,2\cdots n)$ of
$B_{n}$.

Label $2n$ equally-spaced points on a circle with the numbers
$\overline{1},\overline{2},\ldots,\overline{n},1,2,$ $\ldots,n$ in clockwise
order. Now choose a set partition $\rho$ of $\pm[n]$. For each block of
$\rho$, draw the convex hull of the points on the circle labeled with the
elements of that block. We say $\rho$ is a _type- $B_{n}$ noncrossing set
partition_ if the resulting diagram is invariant under a $180^{\circ}$
rotation and none of the convex hulls of different blocks intersect each
other. Suppose $\rho$ is a type-$B_{n}$ noncrossing set partition. For each
block of $\rho$, form a cycle by ordering the elements of the block cyclically
in clockwise order around the circle. The product of these cycles is a
noncrossing partition in $\mathrm{NC}(B_{n},c)$, and every noncrossing
partition in $\mathrm{NC}(B_{n},c)$ arises uniquely in this way.

Figure 8. A type-$B_{6}$ noncrossing set partition. The corresponding
noncrossing partition in $\mathrm{NC}(B_{n},c)$ is
$(\overline{1}\,\,\overline{3}\,\,6)(1\,\,3\,\,\overline{6})(\overline{2})(2)(\overline{4}\,\,\overline{5}\,\,4\,\,5)$.

As in the previous section, we refer to a cycle of an element of $B_{n}$ and
to the set of entries in the cycle interchangeably. Let $\mathrm{cyc}_{>1}(w)$
be the number of non-singleton cycles in $w$.

The partial order on $\mathrm{NC}(B_{n},c)$ corresponds to the reverse
refinement order on type-$B_{n}$ noncrossing set partitions. That is, if
$u,v\in\mathrm{NC}(B_{n},c)$, then $u\leq_{T}v$ if and only if every cycle in
$u$ is contained in a cycle in $v$. The noncrossing projection of $w\in B_{n}$
is the smallest noncrossing partition $v\in\mathrm{NC}(B_{n},c)$ such that
every cycle in $w$ is contained in a cycle in $v$.

Define a total order $\prec$ on the set $\pm[n]$ by

$\overline{1}\prec\overline{2}\prec\cdots\prec\overline{n}\prec 1\prec
2\prec\cdots\prec n.$

An _antiexceedance_ of $w\in B_{n}$ is an element $i\in[n]$ such that $i\prec
w^{-1}(i)$. Write $\mathrm{Aexc}(w)$ for the set of antiexceedances of $w$,
and let $\mathrm{aexc}=\lvert\mathrm{Aexc}(w)\rvert$.

######  Theorem 5.6.

The following properties relating antiexceedances and $\mathsf{Pop}_{T}$ hold:

  * •

The element $c^{-1}$ has $h-1=2n-1$ antiexceedances.

  * •

Every element of $B_{n}$ other than $c^{-1}$ has at most $2n-2$
antiexceedances.

  * •

For every $w\in B_{n}$, we have
$\mathrm{aexc}(\mathsf{Pop}_{T}(w))=\mathrm{aexc}(w)-\mathrm{cyc}_{>1}(w)$.

###### Proof.

The proof is virtually identical to that of Theorem 5.2. As before, the first
two bulleted items are easy to check directly. For the third bulleted item,
fix $w\in B_{n}$, and let $v=\pi_{T}(w)$. Mimicking the proof of Theorem 5.2,
one can show that
$\mathrm{Aexc}(\mathsf{Pop}_{T}(w))\subseteq\mathrm{Aexc}(w)$. Furthermore, we
have $j\in\mathrm{Aexc}(w)\setminus\mathrm{Aexc}(\mathsf{Pop}_{T}(w))$ if and
only if $v^{-1}(j)$ is the largest entry in a non-singleton cycle, where the
word “largest” refers to the total order $\prec$. ∎

The Coxeter number of $B_{n}$ is $2n$, so the following corollary completes
the proof of Theorem 5.1 in type $B$.

######  Corollary 5.7.

Every forward orbit of the map $\mathsf{Pop}_{T}:B_{n}\to B_{n}$ has size at
most $2n$, and the only forward orbit that has size equal to $2n$ is
$O_{\mathsf{Pop}_{T}}(c^{-1})$.

###### Proof.

We know that $O_{\mathsf{Pop}_{T}}(c^{-1})$ has size $2n$ by Proposition 4.12.
The only element of $B_{n}$ whose noncrossing projection has no non-singleton
cycles is the identity element, so it follows from the third bulleted item in
Theorem 5.6 that $\mathrm{aexc}(\mathsf{Pop}_{T}(w))\leq\mathrm{aexc}(w)-1$
for all $w\in\mathfrak{S}_{n}\setminus\\{e\\}$. The only element of $B_{n}$
with $0$ antiexceedances is the identity element, so it follows from the
second bulleted item in Theorem 5.6 that $\mathsf{Pop}_{T}^{2n-2}(w)=e$ for
all $w\in B_{n}\setminus\\{c^{-1}\\}$. ∎

######  Conjecture 5.8.

The number of elements of $B_{n}$ that require exactly $2n-2$ or $2n-1$
iterations of $\mathsf{Pop}_{T}$ to reach the identity is $2^{n}-n$.

##  6\. Type $D$

In this section, we prove the following theorem.

######  Theorem 6.1.

Let $W$ be a Coxeter group of type $D_{n}$. Then

  * •

$\mathsf{Pop}_{T}^{2n-3}(w)=e$ for all $w\in W$, and

  * •

$\max\limits_{w\in W}\left|O_{\mathsf{Pop}_{T}}(w)\right|=2n-2$.

As in the previous section, let
$\pm[n]=\\{\overline{n},\ldots,\overline{1},1,\ldots,n\\}$ (with bars
replacing negative signs). The Coxeter group $D_{n}$ is the group of
permutations $w:\pm[n]\to\pm[n]$ such that $w(\overline{i})=\overline{w(i)}$
for all $i\in\pm[n]$ and such that $\lvert\\{i\in[n]:w(i)<0\\}\rvert$ is even.
The reflections in $D_{n}$ are the elements of the form
$(i\,\,j)(\overline{i}\,\,\overline{j})$, where $i,j\in\pm[n]$ and
$i\neq\overline{j}$. Throughout this section, we fix the Coxeter element
$c=(\overline{1}\,\,\overline{2}\,\cdots\overline{n-1}\,\,1\,\,2\cdots
n-1)(\overline{n}\,\,n)$. Given a cycle $\mathcal{C}=(i_{1}i_{2}\cdots i_{k})$
in an element $w\in D_{n}$, let $\overline{\mathcal{C}}$ denote the cycle
$(\overline{i_{1}}\,\,\overline{i_{2}}\cdots\overline{i_{k}})$, and note that
$\overline{\mathcal{C}}$ is also a cycle in $w$. We say the cycle
$\mathcal{C}$ is _balanced_ if $\mathcal{C}=\overline{\mathcal{C}}$. It
follows from the definition of $D_{n}$ that $\mathcal{C}$ is balanced if and
only if there exists an entry $k$ such that $k$ and $\overline{k}$ are both in
$\mathcal{C}$. A cycle that is not balanced is _unbalanced_. Each element of
$D_{n}$ has an even number of balanced cycles.

Every element $w\in D_{n}$ can be written uniquely as

$w=\mathcal{C}_{1}\overline{\mathcal{C}_{1}}\cdots\mathcal{C}_{r}\overline{\mathcal{C}_{r}}\mathcal{D}_{1}\cdots\mathcal{D}_{s},$

where $\mathcal{C}_{1},\ldots,\mathcal{C}_{r}$ are unbalanced cycles,
$\mathcal{D}_{1},\ldots,\mathcal{D}_{s}$ are balanced cycles, and the cycles
appearing in the factorization are pairwise disjoint. The reflection length of
$w$ can be computed as

$\ell_{T}(w)=\sum_{i=1}^{r}(k_{i}-1)+\sum_{j=1}^{s}m_{j},$

where $k_{i}$ is the number of entries in $\mathcal{C}_{i}$ (and, therefore,
also $\overline{\mathcal{C}_{i}}$) and $m_{j}$ is the number of entries in
$\mathcal{D}_{j}$ (see [BW02, Section 3]). The following useful lemma is a
consequence of the computations in [BW02, Example 3.6] (see also [AR04,
Section 3]).

######  Lemma 6.2 ([BW02]).

Let $w\in D_{n}$, and consider $i,j\in\pm[n]$ with
$i\not\in\\{\overline{j},j\\}$. We have
$(i\,\,j)(\overline{i}\,\,\overline{j})\leq_{T}w$ if and only if $i$ and $j$
are in the same cycle of $w$ or $i$ and $j$ are in different balanced cycles
of $w$.

Our combinatorial description of noncrossing partitions in type $D$ is based
on the work of Athanasiadis and Reiner [AR04]. Begin by labeling $2n{-}2$
equally-spaced points on a circle with the numbers
$\overline{1},\overline{2},\ldots,$ $\overline{n-1},1,2,\ldots,n-1$ in
clockwise order. Now place a single extra point at the center of the circle,
and label it with both the number $n$ and the number $\overline{n}$. Consider
a set partition $\rho$ of $\pm[n]$; we associate each block of $\rho$ with the
set of points labeled by the elements of that block. We say $\rho$ is a _type-
$D_{n}$ noncrossing set partition_ if it satisfies the following conditions:

  1. (1)

For every block $B$ of $\rho$, the set $\overline{B}=\\{\overline{i}:i\in
B\\}$ is also a block of $\rho$.

  2. (2)

If $B$ and $B^{\prime}$ are distinct blocks of $\rho$, then the relative
interior of the convex hull of $B$ does not contain a point of the convex hull
of $B^{\prime}$.

  3. (3)

The set $\\{\overline{n},n\\}$ is not a block of $\rho$.

We associate each type-$D_{n}$ noncrossing set partition $\rho$ with the
diagram showing the labeled points on the circle and at the center of the
circle together with the convex hulls of the blocks of $\rho$. A _zero block_
of $\rho$ is a block $Z$ of $\rho$ such that $Z=\overline{Z}$. Note that
$\rho$ has at most $1$ zero block; moreover, if $\rho$ does have a zero block
$Z$, then $Z$ must properly contain the set $\\{\overline{n},n\\}$.

Suppose $\rho$ is a type-$D_{n}$ noncrossing set partition. For each block $B$
of $\rho$ that is not a zero block, form a cycle by reading the elements of
$B$ in clockwise order around the boundary of the convex hull of $B$. If
$\rho$ has a zero block $Z$, then form the cycle $(\overline{n}\,\,n)$, and
form another cycle by reading the elements of $Z\setminus\\{\overline{n},n\\}$
in clockwise order around the boundary of the convex hull of $Z$. Multiplying
all of these cycles together forms the disjoint cycle decomposition of a
permutation $g(\rho)$ of $\pm[n]$. Athanasiadis and Reiner [AR04] proved that
$g(\rho)$ is actually a noncrossing partition in $\mathrm{NC}(D_{n},c)$ and
that the map $g$ provides a bijection from the set of type-$D_{n}$ noncrossing
set partitions to the set $\mathrm{NC}(D_{n},c)$.

Figure 9. Two type-$D_{7}$ noncrossing set partitions. Technically,
$\overline{7}$ and $7$ should both label a single point at the center of the
circle, but we have drawn two points very close to each other in order to
distinguish between set partitions whose diagrams would otherwise look
identical. The type-$D_{7}$ noncrossing set partition on the left corresponds
to the noncrossing partition
$(\overline{1}\,\,\overline{3}\,\,6)(1\,\,3\,\,\overline{6})(\overline{2})(2)(\overline{4}\,\,\overline{5}\,\,4\,\,5)(\overline{7}\,\,7)$,
while the one on the right corresponds to
$(\overline{1})(1)(\overline{2}\,\,\overline{7}\,\,6)(2\,\,7\,\,\overline{6})(\overline{3}\,\,\overline{4}\,\,\overline{5})(3\,\,4\,\,5)$.

######  Lemma 6.3.

Suppose $v\in\mathrm{NC}(D_{n},c)$ and $k\in\pm[n-1]$ are such that
$v(k)\in\\{\overline{n},n\\}$. Let $Y$ be the set consisting of the next $n-1$
numbers after $k$ in clockwise order around the circle. Then none of the
elements of $Y$ lie in the same cycle as $k$ in $v$.

###### Proof.

Let $\mathcal{C}$ be the cycle in $v$ containing $k$. We can write $v=g(\rho)$
for some type-$D_{n}$ noncrossing set partition $\rho$. The entries in
$\mathcal{C}$ form a block $B$ of $\rho$ that is not a zero block. By the
definition of $g$, the cycle $\mathcal{C}$ is formed by reading the elements
of $B$ in clockwise order around the boundary of the convex hull of $B$. This
immediately implies that $\mathcal{C}\cap Y=\emptyset$. ∎

######  Lemma 6.4.

Let $u\in D_{n}$, and let $v=\pi_{T}(u)$. Consider distinct $i,j\in\pm[n-1]$.
If $i$ and $j$ are in the same cycle of $u$, then they are in the same cycle
of $v$.

###### Proof.

Suppose first that $i\neq\overline{j}$. It follows from Lemma 6.2 that
$(i\,\,j)(\overline{i}\,\,\overline{j})\leq_{T}u$, so
$(i\,\,j)(\overline{i}\,\,\overline{j})\leq_{T}v$. Since
$v\in\mathrm{NC}(D_{n},c)$ is noncrossing, it has at most $1$ balanced cycle
other than $(\overline{n}\,\,n)$. This means that $i$ and $j$ cannot belong to
different balanced cycles of $v$, so they must belong to the same cycle of $v$
by Lemma 6.2.

Now suppose $i=\overline{j}$. This means $i$ is in a balanced cycle in $u$. We
want to show that $i$ is in a balanced cycle of $v$. Suppose, by way of
contradiction, that $i$ is in an unbalanced cycle of $v$. Each element of
$D_{n}$ has an even number of balanced cycles, so there is some balanced cycle
in $u$ that does not contain $i$; let $k$ be one of the numbers in this other
balanced cycle. Since $i$ and $k$ are in different balanced cycles in $u$, we
have $(i\,\,k)(\overline{i}\,\,\overline{k})\leq_{T}u$ by Lemma 6.2.
Therefore, $(i\,\,k)(\overline{i}\,\,\overline{k})\leq_{T}v$. Because $i$ is
in an unbalanced cycle in $v$, Lemma 6.2 tells us that $i$ and $k$ are in the
same cycle in $v$. Since $\overline{k}$ is in the same balanced cycle in $u$
as $k$, the same argument we have just given shows that $i$ and $\overline{k}$
are in the same cycle in $v$. However, this means that the cycle in $v$
containing $i$ contains both $k$ and $\overline{k}$, so it is balanced. This
is a contradiction. ∎

Our goal in the remainder of this section is to prove Theorem 6.1. The Coxeter
number of $D_{n}$ is $2n{-}2$. We already know by Proposition 4.12 that
$O_{\mathsf{Pop}_{T}}(c^{-1})$ is a forward orbit of
$\mathsf{Pop}_{T}:D_{n}\to D_{n}$ of size $2n{-}2$. Thus, we need to prove
that $\mathsf{Pop}_{T}^{2n-3}(u_{0})=e$ for all $u_{0}\in D_{n}$.

Fix $u_{0}\in D_{n}$. For $k\geq 0$, let $u_{k}=\mathsf{Pop}_{T}^{k}(u_{0})$
and $v_{k}=\pi_{T}(u_{k})$. Thus, $u_{k+1}=u_{k}v_{k}^{-1}$. Our goal is to
show that $u_{2n-3}=e$. Since $\mathsf{Pop}_{T}^{-1}(e)=\mathrm{NC}(D_{n},c)$,
it suffices to show that $u_{2n-4}\in\mathrm{NC}(D_{n},c)$. The following
lemma will do all of the heavy lifting. When we refer to “the circle,” we mean
the circle that has $2n{-}2$ equally-spaced points labeled with the numbers
$\overline{1},\overline{2},\ldots,\overline{n-1},1,2,\ldots,n-1$ in clockwise
order and that has its center labeled with the numbers $\overline{n}$ and $n$.
Define the _predecessor_ of an element $j\in\pm[n-1]$ to the be number that
appears immediately before $j$ in the clockwise order of the circle. For
example, the predecessor of $2$ is $1$, and the predecessor of $\overline{1}$
is $n-1$.

######  Lemma 6.5.

Suppose $j\in\pm[n-1]$ is such that
$u_{0}^{-1}(j)\not\in\\{\overline{n},n\\}$. Then $u_{2n-4}^{-1}(j)$ is either
$j$ or the predecessor of $j$.

###### Proof.

Define a total order $\prec$ on $\pm[n-1]$ by
$\overline{1}\prec\cdots\prec\overline{n-1}\prec 1\prec\cdots\prec n-1$.
Define $\prec_{j}$ to be the total order on $\pm[n-1]$ obtained by cyclically
shifting the total order $\prec$ so that the maximum element in the order
$\prec_{j}$ is $j$. In other words, $\prec_{j}$ is given by reading the
numbers clockwise around the circle so that the last number read is $j$. Given
a set $Q\subseteq\pm[n-1]$, we write $i\prec_{j}Q$ (respectively,
$Q\prec_{J}i$) to mean that $i\prec_{j}q$ (respectively, $q\prec_{j}i$) for
all $q\in Q$.

If $u_{k}^{-1}(j)=j$ for some $0\leq k\leq 2n-4$, then
$u_{k^{\prime}}^{-1}(j)=j$ for all $k^{\prime}\geq k$. In particular,
$u_{2n-4}^{-1}(j)=j$ in this case. Thus, we may assume $u_{k}^{-1}(j)\neq j$
for all $0\leq k\leq 2n-4$.

Suppose $k\in\\{0,\ldots,2n-4\\}$ is such that
$u_{k}^{-1}(j)\not\in\\{\overline{n},n\\}$ and
$u_{k+1}^{-1}(j)=v_{k}(u_{k}^{-1}(j))\not\in\\{\overline{n},n\\}$. Let
$\mathcal{C}$ be the cycle in $v_{k}$ containing $j$. Since $u_{k}^{-1}(j)$
and $j$ are in the same cycle in $u_{k}$, Lemma 6.4 tells us that
$u_{k}^{-1}(j)\in\mathcal{C}$. Because $v_{k}$ is a noncrossing partition, the
cycle $\mathcal{C}$ is ordered clockwise. Consequently, $u_{k+1}^{-1}(j)$ is
the element of $\mathcal{C}$ that occurs right after $u_{k}^{-1}(j)$ in the
clockwise cyclic order. Since $u_{k}^{-1}(j)\neq j$, we must have
$u_{k}^{-1}(j)\prec_{j}u_{k+1}^{-1}(j)$.

If there does not exist an integer $i\in\\{0,\ldots,2n-4\\}$ such that
$u_{i}^{-1}(j)\in\\{\overline{n},n\\}$, then it follows from the preceding
paragraph that
$u_{0}^{-1}(j)\prec_{j}u_{1}^{-1}(j)\prec_{j}\cdots\prec_{j}u_{2n-4}^{-1}(j)$.
We have assumed $u_{2n-4}^{-1}(j)\neq j$, so it follows that
$u_{2n-4}^{-1}(j)$ is the predecessor of $j$, as desired.

We may now assume there exists $i\in\\{0,\ldots,2n-4\\}$ with
$u_{i}^{-1}(j)\in\\{\overline{n},n\\}$. We will actually show that this case
cannot occur because it contradicts our assumption that $u_{k}^{-1}(j)\neq j$
for all $0\leq k\leq 2n-4$. Since $u_{0}^{-1}(j)\not\in\\{\overline{n},n\\}$
by the hypothesis of the lemma, we must have $i\geq 1$ and
$u_{i-1}^{-1}(j)\not\in\\{\overline{n},n\\}$. Let $\mathcal{C}_{i-1}$ and
$\mathcal{C}_{i}$ be the cycles in $v_{i-1}$ and $v_{i}$, respectively, that
contain $j$.

Let $Y$ be the set consisting of the next $n-1$ numbers after
$u_{i-1}^{-1}(j)$ in clockwise order around the circle. Because
$v_{i-1}(u_{i-1}^{-1}(j))=u_{i}^{-1}(j)\in\\{\overline{n},n\\}$, we can use
Lemma 6.3 to see that none of the elements of $Y$ lie in the same cycle of
$v_{i-1}$ as $u_{i-1}^{-1}(j)$. Since $j$ and $u_{i-1}^{-1}(j)$ are in the
same cycle of $u_{i-1}$, we know by Lemma 6.4 that $j$ and $u_{i-1}^{-1}(j)$
are in the same cycle of $v_{i-1}$. In other words,
$u_{i-1}^{-1}(j)\in\mathcal{C}_{i-1}$, and $\mathcal{C}_{i-1}\cap
Y=\emptyset$. Hence, $j\not\in Y$. This implies that
$u_{i-1}^{-1}(j)\prec_{j}Y\prec_{j}j$. In particular, $u_{i-1}^{-1}(j)$ must
be among the smallest $n-2$ numbers in the order $\prec_{j}$. We also know
that $u_{i}^{-1}(j)\in\mathcal{C}_{i-1}$ because
$u_{i}^{-1}(j)=v_{i-1}(u_{i-1}^{-1}(j))$. Since
$u_{i}^{-1}(j)\in\\{\overline{n},n\\}$, this implies that $\mathcal{C}_{i-1}$
is an unbalanced cycle in $v_{i-1}$. In fact, this shows that $v_{i-1}$ has no
balanced cycles (if it did, then by the definition of the bijection $g$, one
of the balanced cycles of $v_{i-1}$ would be $(\overline{n}\,\,n)$).

Athanasiadis and Reiner [AR04] gave an explicit combinatorial description of
the covering relations in $\mathrm{NC}(D_{n},c)$; it follows from that
description that if $x,y\in\mathrm{NC}(D_{n},c)$ are such that $x\leq_{T}y$,
then every unbalanced cycle in $y$ is a union (as a set) of unbalanced cycles
in $x$. According to Lemma 4.1, we have $v_{i}\leq_{T}v_{i-1}$. Because
$v_{i-1}$ has no balanced cycles, neither does $v_{i}$. Furthermore,
$\mathcal{C}_{i}\subseteq\mathcal{C}_{i-1}$. Since $j$ and $u_{i}^{-1}(j)$ are
in the same cycle in $u_{i}$, we have
$(j\,\,u_{i}^{-1}(j))(\overline{j}\,\,\overline{u_{i}^{-1}(j)})\leq_{T}u_{i}$
by Lemma 6.2. Therefore,
$(j\,\,u_{i}^{-1}(j))(\overline{j}\,\,\overline{u_{i}^{-1}(j)})\leq_{T}v_{i}$.
Since $v_{i}$ has no balanced cycles, we can use Lemma 6.2 again to see that
$j$ and $u_{i}^{-1}(j)$ are in the same unbalanced cycle in $v_{i}$. That is,
$u_{i}^{-1}(j)\in\mathcal{C}_{i}$. Now, $u_{i+1}^{-1}(j)=v_{i}(u_{i}^{-1}(j))$
is the element of $\mathcal{C}_{i}$ that occurs immediately after
$u_{i}^{-1}(j)$ in the clockwise cyclic order, so $Y\prec_{j}u_{i+1}^{-1}(j)$.
Hence, $u_{i+1}^{-1}(j)$ is among the largest $n-2$ elements of $\pm[n-1]$ in
the order $\prec_{j}$.

We have shown that if $i\in\\{1,\ldots,2n-4\\}$ is such that
$u_{i}^{-1}(j)\in\\{\overline{n},n\\}$ and
$u_{i-1}^{-1}(j)\not\in\\{\overline{n},n\\}$, then $u_{i-1}^{-1}(j)$ is among
the smallest $n-2$ elements of $\pm[n-1]$ in the order $\prec_{j}$ while
$u_{i+1}^{-1}(j)$ is among the largest $n-2$ elements. We have also seen that
if $k\in\\{0,\ldots,2n-4\\}$ is such that
$u_{k}^{-1}(j)\not\in\\{\overline{n},n\\}$, then
$u_{k}^{-1}(j)\prec_{j}u_{k+1}^{-1}(j)$. We assumed in the hypothesis of the
lemma that $u_{0}^{-1}(j)\not\in\\{\overline{n},n\\}$. It follows from these
facts that $u_{k}^{-1}(j)\not\in\\{\overline{n},n\\}$ for every
$k\in\\{0,\ldots,2n-4\\}\setminus\\{i\\}$. Therefore, the sequence
$u_{0}^{-1}(j),u_{1}^{-1}(j),\ldots,u_{i-1}^{-1}(j),u_{i+1}^{-1}(j),\ldots,u_{2n-4}^{-1}(j)$
is strictly increasing in the order $\prec_{j}$. However, this sequence does
not include any of the $n-1$ elements of $Y$ because
$u_{i-1}^{-1}(j)\prec_{j}Y\prec_{j}u_{i+1}^{-1}(j)$. This is our desired
contradiction. ∎

###### Proof of Theorem 6.1.

Earlier, we chose an arbitrary $u_{0}\in D_{n}$ and let
$u_{k}=\mathsf{Pop}_{T}^{k}(u_{0})$. Our goal is to prove that $u_{2n-3}=e$;
in order to do so, it suffices to show that $u_{2n-4}\in\mathrm{NC}(D_{n},c)$.
Let $m=u_{0}(n)$. Lemma 6.5 tells us that if
$j\in\pm[n-1]\setminus\\{\overline{m},m\\}$, then $u_{2n-4}^{-1}(j)$ is either
$j$ or the predecessor of $j$. We now consider several cases.

Case 1. Suppose $u_{2n-4}^{-1}(m)=m$. Then
$u_{2n-4}^{-1}(\overline{m})=\overline{m}$. If $m\not\in\\{\overline{n},n\\}$,
then one can straightforwardly deduce from Lemma 6.5 that
$u_{2n-4}=e\in\mathrm{NC}(D_{n},c)$. If $m\in\\{\overline{n},n\\}$, then one
can use Lemma 6.5 to see that $u_{2n-4}$ is either $e$ or $c$; in either case,
$u_{2n-4}\in\mathrm{NC}(D_{n},c)$.

Case 2. Suppose that $u_{2n-4}^{-1}(m)\not\in\\{\overline{n},n,m\\}$ and that
$\overline{m},u_{2n-4}^{-1}(m),m$ appear in this (cyclic) order when we read
the numbers on the circle in clockwise order. Let $Z$ be the set of numbers
that either appear between $\overline{m}$ and $u_{2n-4}^{-1}(m)$ (including
$\overline{m}$ and $u_{2n-4}^{-1}(m)$) or appear between $m$ and
$\overline{u_{2n-4}^{-1}(m)}$ (including $m$ and
$\overline{u_{2n-4}^{-1}(m)}$) in the clockwise order on the circle. Using
Lemma 6.5, it is straightforward to check that $u_{2n-4}$ cyclically permutes
the elements of $Z$ in clockwise order and fixes each element of
$\pm[n-1]\setminus Z$ (see Figure 10). Since $u_{2n-4}\in D_{n}$, we must have
$u_{2n-4}(n)=\overline{n}$ and $u_{2n-4}(\overline{n})=n$. Therefore,
$u_{2n-4}=g(\rho)$, where $\rho$ is the type-$D_{n}$ noncrossing set partition
whose blocks are $Z\cup\\{\overline{n},n\\}$ and all of the singleton subsets
of $\pm[n-1]\setminus Z$. Thus, $u_{2n-4}\in\mathrm{NC}(D_{n},c)$.

Case 3. Suppose that $u_{2n-4}^{-1}(m)\not\in\\{\overline{n},n,m\\}$ and that
$m,u_{2n-4}^{-1}(m),\overline{m}$ appear in this (cyclic) order when we read
the numbers on the circle in clockwise order. Let $B$ be the set of numbers
that appear between $m$ and $u_{2n-4}^{-1}(m)$ (including $m$ and
$u_{2n-4}^{-1}(m)$) in the clockwise order on the circle. Using Lemma 6.5, it
is straightforward to check that $u_{2n-4}$ cyclically permutes the elements
of $B$ in clockwise order, cyclically permutes the elements of $\overline{B}$
in clockwise order, and fixes each element of
$\pm[n-1]\setminus(B\cup(\overline{B}))$ (see Figure 10). Since $u_{2n-4}$ is
in $D_{n}$, it must fix $n$ and $\overline{n}$. Therefore, $u_{2n-4}=g(\rho)$,
where $\rho$ is the type-$D_{n}$ noncrossing set partition whose blocks are
$B$, $\overline{B}$, and all of the singleton subsets of
$\pm[n]\setminus(B\cup(\overline{B}))$. Thus,
$u_{2n-4}\in\mathrm{NC}(D_{n},c)$.

Case 4. Suppose that $u_{2n-4}^{-1}(m)\in\\{\overline{n},n\\}$ and either that
$u_{2n-4}^{-2}(m)=m$ or that the numbers $m,u_{2n-4}^{-2}(m),\overline{m}$ are
distinct and appear in this (cyclic) order when we read the numbers on the
circle in clockwise order. Let $B$ be the set of numbers that appear between
$m$ and $u_{2n-4}^{-2}(m)$ (including $m$ and $u_{2n-4}^{-1}(m)$) in the
clockwise order on the circle. Using Lemma 6.5, it is straightforward to check
that $u_{2n-4}$ cyclically permutes the elements of
$B\cup\\{u_{2n-4}^{-1}(m)\\}$ in clockwise order, cyclically permutes the
elements of $\overline{B}\cup\\{\overline{u}_{2n-4}^{-1}(m)\\}$ in clockwise
order, and fixes each element of $\pm[n-1]\setminus(B\cup(\overline{B}))$ (see
Figure 10). Therefore, $u_{2n-4}=g(\rho)$, where $\rho$ is the type-$D_{n}$
noncrossing set partition whose blocks are $B\cup\\{u_{2n-4}^{-1}(m)\\}$,
$\overline{B}\cup\\{\overline{u_{2n-4}^{-1}(m)}\\}$, and all of the singleton
subsets of $\pm[n-1]\setminus(B\cup(\overline{B}))$. Thus,
$u_{2n-4}\in\mathrm{NC}(D_{n},c)$.

Case 5. Suppose that $u_{2n-4}^{-1}(m)\in\\{\overline{n},n\\}$ and either that
$u_{2n-4}^{-2}(m)=\overline{m}$ or that the numbers
$\overline{m},u_{2n-4}^{-2}(m),m$ are distinct and appear in this (cyclic)
order when we read the numbers on the circle in clockwise order. Let
$r_{1},r_{2},\ldots,r_{q}$ be the numbers that appear between $\overline{m}$
and $u_{2n-4}^{-2}(m)$ (including $\overline{m}$ and $u_{2n-4}^{-2}(m)$) in
the clockwise order on the circle. In particular, $r_{1}=\overline{m}$ and
$r_{q}=u_{2n-4}^{-2}(m)$. In this case, one can use Lemma 6.5 to show that
$u_{2n-4}$ is the single cycle

$(r_{1}\,\,r_{2}\cdots
r_{q}\,\,u_{2n-4}^{-1}(m)\,\,\overline{r_{1}}\,\,\overline{r_{2}}\cdots\overline{r_{q}}\,\,\overline{u_{2n-4}^{-1}(m)})$

(meaning $u_{2n-4}$ fixes every element of $\pm[n]$ that is not in this
cycle). However, this means that $u_{2n-4}$ has a single balanced cycle,
contradicting the fact that every element of $D_{n}$ has an even number of
balanced cycles. Therefore, this case cannot even occur. ∎

Figure 10. Diagrams illustrating Case 2 (upper left), Case 3 (upper right),
and Case 4 (bottom middle) from the proof of Theorem 6.1. In each example, we
have $n=7$ and $m=5$.

######  Conjecture 6.6.

The number of elements of $D_{n}$ that require exactly $2n-3$ iterations of
$\mathsf{Pop}_{T}$ to reach the identity is $n(2^{n-1}-2)+1$.

##  7\. Exceptional Types

In this section, we conclude our study of $\mathsf{Pop}_{T}$ on finite
irreducible Coxeter groups by considering the exceptional types $E_{6}$,
$E_{7}$, $E_{8}$, $F_{4}$, and $H_{4}$. Recall that an orbit
$O_{\mathsf{Pop}_{T}}(x)$ is called _periodic_ if $\mathsf{Pop}_{T}^{i}(x)=x$
for some $i>0$, and that for a Weyl group, $c$ is conjugate to $c^{k}$ for any
$k$ relatively prime to $h$ [Spr74].

######  Theorem 7.1.

The map $\mathsf{Pop}_{T}$ has periodic orbits of size $h$ for $W$ of type
$E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$, or $H_{4}$. Furthermore, for any
$k\not\equiv\pm 1\pmod{h}$ coprime to $h$ such that $c$ is conjugate to
$c^{k}$ in $W$, the set

$O_{k}=\\{w\in W:c^{w}=c^{k}\\}$

is a periodic orbit of size $h$, where $c^{w}=w^{-1}cw$.

###### Proof.

We computed the entire orbit structure of $\mathsf{Pop}_{T}$ on groups of
types $F_{4}$, $H_{4}$, and $E_{6}$ using a combination of Sage [The21] and
GAP3 with CHEVIE [S+97, GHL+96]. This data is summarized in Figure 3. We
confirmed by computer that the two periodic orbits of $F_{4}$ (each of size
$h=12$) lift via Section 4.3 to the two periodic orbits of $E_{6}$ (each again
of size $h=12$). In $F_{4}$ and $E_{6}$, the two periodic orbits are $O_{5}$
and $O_{7}$—and there are no others.

The number of elements requiring $i$ iterations of $\mathsf{Pop}_{T}$ to reach
the identity in $H_{4}$ is not included in Figure 3 for reasons of space; but
for $i=0,1,\ldots,34$ these numbers are 1, 279, 467, 465, 663, 690, 675, 660,
750, 660, 495, 390, 390, 390, 390, 390, 390, 390, 390, 390, 375, 375, 375,
375, 375, 300, 300, 300, 225, 75, 66, 66, 33, 30, 15, with an additional
$1800$ elements in 60 periodic orbits, each of size $h=30$. Both $O_{11}$ and
$O_{19}$ are periodic orbits in $H_{4}$ (and $c$ is not conjugate to $c^{k}$
for $k=7,13,17,23$).

In $E_{7}$ with $h=30$, we checked that $O_{k}$ for $k=5,7,11,13$ all form
periodic orbits, but we did not compute the full orbit structure of
$\mathsf{Pop}_{T}$.

To show that $E_{8}$ also has periodic orbits, we can use the unfolding of
$H_{4}$ to $E_{8}$, which necessarily takes periodic orbits to periodic orbits
by Theorem 4.8. Write $t_{1^{a_{1}}\cdots 8^{a_{8}}}$ for the reflection
orthogonal to the root $a_{1}\alpha_{1}+\cdots+a_{8}\alpha_{8}$. For example,
$t_{123^{2}4^{2}5^{2}678}$ is the reflection orthogonal to
$\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}+2\alpha_{5}+\alpha_{6}+\alpha_{7}+\alpha_{8}$.
The $E_{8}$ element

$t_{2}\cdot t_{123456}\cdot t_{3}\cdot t_{1234^{2}5678}\cdot
t_{123^{2}4^{2}5^{2}678}\cdot t_{134567}$

lies in a periodic orbit of size $h=30$ and is the unfolding of the $H_{4}$
element with reduced word (in simple reflections)

$s_{3}s_{4}s_{3}s_{2}s_{1}s_{2}s_{1}s_{3}s_{2}s_{1}s_{4}s_{3}s_{2}s_{1}s_{2}s_{3}s_{4}.$

With computational assistance from C. Stump, we confirmed that the sets
$O_{k}$ for $k\in\\{7,11,13,17,19,23\\}$ all form periodic orbits under
$\mathsf{Pop}_{T}$.∎

It would be interesting to fully characterize which elements of $W$ give rise
to periodic orbits.

##  8\. Normal Forms

###  8.1. Dual braid lifts

For this subsection, fix $W$ to be a finite irreducible Coxeter group with no
periodic orbits under $\mathsf{Pop}_{T}$ other than that of the identity
element. We would like to provide a canonical lift of $W$ to Bessis’ dual
braid monoid, just as $W$ has a standard lift (using reduced words in simple
reflections) to its positive braid monoid.

Recall that we introduced $\mathsf{Pop}_{T}$ as the dual version of
$\mathsf{Pop}_{S}$. As $\mathsf{Pop}_{S}$ is closely related to Brieskorn
normal form, we consider the analogous factorization coming from
$\mathsf{Pop}_{T}$—this succeeds because each image under $\pi_{T}$ is a
noncrossing partition and hence has a canonical lift to the dual braid monoid.

######  Theorem 8.1.

Let $W$ be a finite irreducible Coxeter group whose only periodic orbit under
$\mathsf{Pop}_{T}$ is that of the identity. For $w\in W$, write $w_{1}=w$ and
$w_{i}=\mathsf{Pop}_{T}(w_{i-1},c)$ for $i\geq 2$. Let $k=\lvert
O_{\mathsf{Pop}_{T}}(w)\rvert-1$ so that (assuming $w\neq e$) $w_{k}\neq
w_{k+1}=e$. Then the factorization

$w=\pi_{T}(w_{k},c)\pi_{T}(w_{k-1},c)\cdots\pi_{T}(w_{1},c)$

gives a lift from $W$ to the dual braid monoid of $W$.

######  Example 8.2.

In type $A_{5}$ with $c=(123456)$ and $w=w_{1}=(135642)$, we have

$\displaystyle\pi_{T}(w_{1})$ $\displaystyle=(123456)$ and $\displaystyle
w_{2}=\mathsf{Pop}_{T}(w_{1})$ $\displaystyle=(12634),$
$\displaystyle\pi_{T}(w_{2})$ $\displaystyle=(12346)$ and $\displaystyle
w_{3}=\mathsf{Pop}_{T}(w_{2})$ $\displaystyle=(246),$
$\displaystyle\pi_{T}(w_{3})$ $\displaystyle=(246)$ and $\displaystyle
w_{4}=\mathsf{Pop}_{T}(w_{3})$ $\displaystyle=e,$

so that

$w=(246)\cdot(12346)\cdot(123456),$

which may be interpreted as a product of simple elements in the dual braid
monoid.

######  Remark 8.3.

T. Douvropoulos has suggested that one could extend our construction to a
general finite complex reflection group $W$ by using noncrossing reflections
as a generating set for $W$, with a noncrossing projection of $w\in W$ defined
as the join in the noncrossing partition lattice of those noncrossing
reflections appearing in a shortest length word for $w$. Explicit computations
show that $G_{5}$ has periodic orbits not containing the identity under these
definitions.

###  8.2. Stabilized-interval-free elements

A permutation $w\in\mathfrak{S}_{n}$ is called _stabilized-interval-free_
(SIF) if it does not stabilize a proper subinterval of $[n]$; that is, there
do not exist $i,j\in[n]$ with $1\leq i\leq j\leq n$ satisfying
$w(\\{i,\ldots,j\\})=\\{i,\ldots,j\\}$ and $\\{i,\ldots,j\\}\neq[n]$. These
permutations were initially investigated from an enumerative point of view in
[Cal04] (see also [Bli]). They are also important in the theory of the totally
nonnegative Grassmannian because they parameterize connected positroids
[ARW16]. Using the combinatorial description of noncrossing partitions in type
$A$ discussed in Section 5, it follows that $w$ is SIF if and only if
$\pi_{T}(w,c)=c$, where $c$ is the Coxeter element $(12\cdots n)$.

Now consider an arbitrary finite irreducible Coxeter group $W$ with a fixed
Coxeter element $c$. The following construction naturally generalizes the
notion of a stabilized-interval-free permutation.

######  Definition 8.4.

An element $w\in W$ is _stabilized-interval-free_ (SIF) if $\pi_{T}(w,c)=c$.

For $w\in W$, let $P$ be the smallest parabolic subgroup containing $w$. We
call the irreducible components $P_{1},\ldots,P_{k}$ of $P$ the _blocks_ of
$w$. Note that $P=P_{1}\times P_{2}\times\cdots\times P_{k}$. The following is
immediate.

######  Proposition 8.5.

Every $w\in W$ can be written uniquely as a product of SIF elements in the
blocks of $\pi_{T}(w,c)$.

rank $n$ | 2 | 3 | 4 | 5 | 6 | OEIS
---|---|---|---|---|---|---
$A_{n}$ | 2 | 7 | 34 | 206 | 1476 | A075834
$B_{n}$ | 3 | 20 | 179 | 1944 | 24674 |
$D_{n}$ |  | 7 | 74 | 891 | 12004 |
$E_{6}$ |  |  |  |  | 33610 |
$F_{4}$ |  |  | 762 |  |  |
$G_{2}$ | 5 |  |  |  |  |
$H_{3}$ |  | 69 |  |  |  |
$H_{4}$ |  |  | 12802 |  |  |
$I_{2}(m)$ | $m-1$ |  |  |  |  |
Figure 11. Numbers of SIF elements in finite irreducible Coxeter groups of
small rank.

## Acknowledgements

N.W. was partially supported by a Simons Foundation Collaboration Grant. C.D.
was supported by a Fannie and John Hertz Foundation Fellowship and an NSF
Graduate Research Fellowship (grant number DGE-1656466). We thank C. Stump for
his valuable computational assistance in Theorem 7.1 (especially for $E_{8}$)
and T. Douvropoulos for many insightful comments on an early draft. N.W.
thanks T. Gobet for a fruitful collaboration and conversation in 2012 which
led to the discussion in Section 8.1.

## References

  * [ABB+19] A Asinowski, C. Banerier, S. Billey, B. Hackl, and S. Linusson, _Pop-stack sorting and its image: permutations with overlapping runs_ , Acta. Math. Univ. Comenian. 88 (2019), 395–402.
  * [ABH21] A Asinowski, C. Banerier, and B. Hackl, _Flip-sort and combinatorial aspects of pop-stack sorting_ , Discrete Math. Theor. Comput. Sci. 22 (2021).
  * [AR04] C. A. Athanasiadis and V. Reiner, _Noncrossing partitions for the group ${D}_{n}$_, _SIAM J. Discrete Math._ 18 (2004), 397–417.
  * [Arm09] D. Armstrong, _Generalized noncrossing partitions and combinatorics of Coxeter groups_ , American Mathematical Soc., 2009.
  * [ARW16] F Ardila, F. Rincón, and L. Williams, _Positroids and non-crossing partitions_ , Trans. Amer. Math. Soc. 368 (2016), 337–363.
  * [AV21] M. Albert and V. Vatter, _How many pop-stacks does it take to sort a permutation?_ , arXiv preprint arXiv:2012.05275 (2021).
  * [BB05] A. Björner and F. Brenti, _Combinatorics of Coxeter groups_ , Springer, 2005\.
  * [Bes03] D. Bessis, _The dual braid monoid_ , Ann. Sci. Éc. Norm. Supér. 36 (2003), 647–683.
  * [Bes15] by same author, _Finite complex reflection arrangements are $K(\pi,1)$_, Ann. of Math. 181 (2015), 809–904.
  * [Bli] N Blitvic, _Stabilized-interval-free permutations and chord-connected permutations_ , 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), pp. 801–813.
  * [Bri71] E Brieskorn, _Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe_ , Invent. Math. 12 (1971), 57–61.
  * [BS72] E. Brieskorn and K. Saito, _Artin-gruppen und Coxeter-gruppen_ , Invent. Math. 17 (1972), 245–271.
  * [BW02] T. Brady and C. Watt, _${K}(\pi,1)$ ’s for artin groups of finite type_, Proceedings of the conference on geometric and combinatorial group theory, Part I (Haifa 2000), Geom. Dedicata 94 (2002), 225–250.
  * [BW08] by same author, _Non-crossing partition lattices in finite real reflection groups_ , Trans. Amer. Math. Soc. 360 (2008), 1983–2005.
  * [Cal04] D. Callan, _Counting stabilized-interval-free permutations_ , J. Integer Seq. 7 (2004), no. 2, 3.
  * [Car72] R. W. Carter, _Conjugacy classes in the Weyl group_ , Compositio Math. 25 (1972), 1–59.
  * [CG19] A. Claesson and B. Á Guðmundsson, _Enumerating permutations sortable by $k$ passes through a pop-stack_, Adv. Appl. Math. 108 (2019), 79–96.
  * [CGP19] A. Claesson, B. Á Guðmundsson, and J. Pantone, _Counting pop-stacked permutations in polynomial time_ , arXiv preprint arXiv:1908.08910 (2019).
  * [Def21] C. Defant, _Pop-stack-sorting for Coxeter groups_ , arXiv preprint arXiv:2104.02675 (2021).
  * [EG21] M. Elder and Y. K. Goh, _$k$ -pop stack sortable permutations and $2$-avoidance_, Electron. J. Combin. 28 (2021).
  * [GHL+96] M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, _CHEVIE – A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras_, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210.
  * [PS19] L. Pudwell and R. Smith, _Two-stack-sorting with pop stacks_ , Australas. J. Combin. 74 (2019), 179–195.
  * [Rea10] N. Reading, _Noncrossing partitions, clusters and the Coxeter plane_ , Sém. Lothar. Combin. 63 (2010).
  * [RRS17] V. Reiner, V. Ripoll, and C. Stump, _On non-conjugate Coxeter elements in well-generated reflection groups_ , Math. Z. 285 (2017), 1041–1062.
  * [S+97] Martin Schönert et al., _GAP – Groups, Algorithms, and Programming – version 3 release 4 patchlevel 4_ , Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1997.
  * [Spr74] T. A. Springer, _Regular elements of finite reflection groups_ , Invent. Math. 25 (1974), 159–198.
  * [The21] The Sage Developers, _Sagemath, the Sage Mathematics Software System (Version 9.2)_ , 2021, https://www.sagemath.org.
  * [Ung82] P. Ungar, _$2{N}$ noncollinear points determine at least $2{N}$ directions_, J. Combin. Theory Ser. A 33 (1982), 343–347.