# Pop, Crackle, Snap (and Pow):
Some Facets of Shards

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

Reading cut the hyperplanes in a real central arrangement $\mathcal{H}$ into
pieces called _shards_ , which reflect order-theoretic properties of the
arrangement. We show that shards have a natural interpretation as certain
generators of the fundamental group of the complement of the complexification
of $\mathcal{H}$. Taking only positive expressions in these generators yields
a new poset that we call the _pure shard monoid_.

When $\mathcal{H}$ is simplicial, its poset of regions is a lattice, so it
comes equipped with a pop-stack sorting operator $\mathsf{Pop}$. In this case,
we use $\mathsf{Pop}$ to define an embedding $\mathsf{Crackle}$ of Reading’s
shard intersection order into the pure shard monoid. When $\mathcal{H}$ is the
reflection arrangement of a finite Coxeter group, we also define a poset
embedding $\mathsf{Snap}$ of the shard intersection order into the positive
braid monoid; in this case, our three maps are related by
$\mathsf{Snap}=\mathsf{Crackle}\cdot\mathsf{Pop}$.

##  1\. Introduction

###  1.1. Introduction (Section 1)

Throughout this paper, we let $\mathcal{H}$ be a finite central irreducible
real hyperplane arrangement. Salvetti introduced a certain CW complex
associated to $\mathcal{H}$ and used it to provide a presentation of the
fundamental group of the complement of the complexification of $\mathcal{H}$.
We prove that Salvetti’s generating set is parameterized by shards, which
Reading introduced in order to define his shard intersection order in the case
when $\mathcal{H}$ is simplicial. We introduce the _pure shard monoid_ , which
is the monoid generated by Salvetti’s generators; it comes equipped with a
natural partial order that we believe deserves further attention. In the case
when $\mathcal{H}$ is an arrangement of rank $2$ with $m$ hyperplanes, we
prove that the interval from the identity element to the _full twist_ in the
pure shard monoid is a planar lattice with rank generating function
$1+\left(\sum_{k=1}^{m-1}\left(2\binom{m}{k}-2\right)q^{k}\right)+q^{m}$ and
with $m2^{m-2}$ maximal chains.

We then assume $\mathcal{H}$ is simplicial. In this case, there is a known
characterization of the shard intersection order involving the pop-stack
sorting operator $\mathsf{Pop}$ on the poset of regions of $\mathcal{H}$. We
introduce a new map $\mathsf{Crackle}$, and we prove that it is a poset
embedding of the shard intersection order into the pure shard monoid.

Next, we specialize further to the case when $\mathcal{H}$ is the reflection
arrangement of a finite Coxeter group $W$. We introduce another map
$\mathsf{Snap}$, and we prove that it is a poset embedding of the shard
intersection order on $W$ into the weak order on the positive braid monoid.
The restriction of $\mathsf{Snap}$ to the set $\mathrm{Sort}(W,c)$ of
$c$-sortable elements of $W$ originally arose in connection with Deodhar
decompositions of noncrossing Catalan varieties. In this setting, we obtain as
a corollary that $\mathsf{Snap}$ restricts to a poset embedding of the shard
intersection order on $\mathrm{Sort}(W,c)$—which is isomorphic to the
noncrossing partition lattice of $W$—into the weak order on the positive braid
monoid.

Finally, we turn back to arbitrary finite central irreducible real
arrangements and define a fourth map $\mathsf{Pow}$. We prove that
$\mathsf{Pow}$ is a poset embedding of the poset of regions of $\mathcal{H}$
into the pure shard monoid. In the final section of the paper, we provide a
conjectural characterization of the positive expressions in Salvetti’s
generators that represent the full twist, and we conjecture that the interval
from the identity element to the full twist in the pure shard monoid is self-
dual.

###  1.2. The Salvetti Complex (Section 2)

We write $\mathcal{R}$ for the set of regions (connected components of the
complement) of $\mathcal{H}$, and we fix a base region $B\in\mathcal{R}$ and a
point $x_{B}\in B$. Write $\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}}$
for the complexified hyperplane complement and
$\pi_{1}(\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}},x_{B})$ for its
fundamental group with base point $x_{B}$. Let
$\mathrm{Weak}(\mathcal{H},B):=(\mathcal{R},\leq)$ be the usual poset of
regions of $\mathcal{H}$ with respect to the base region $B$.

Following [Sal87, Del72], we construct a CW complex
$\mathrm{Sal}(\mathcal{H})$ by gluing together oriented dual zonotopes for
$\mathcal{H}$ along compatible faces—one zonotope for each choice of base
region $B$, oriented from $B$ to $-B$. The resulting CW complex
$\mathrm{Sal}(\mathcal{H})$ has the same fundamental group as the complexified
hyperplane complement:

$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)=\pi_{1}(\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}},x_{B}).$

The 1-skeleton of $\mathrm{Sal}(\mathcal{H})$ is given by orienting all edges
of $\mathrm{Weak}(\mathcal{H},B)$ away from $B$, and then for each edge $e$,
adding in a reversed edge $e^{*}$. An illustration is given in Figure 1.

$C_{2}$$\Sigma_{3}$$C_{3}$$\Sigma_{8}$$-B$$\Sigma_{1}$$C_{6}$$\Sigma_{6}$$C_{7}$$\Sigma_{7}$$C_{8}$$\Sigma_{8}$$B$$\Sigma_{1}$$C_{1}$$\Sigma_{2}$$e_{1}$$e_{1}^{*}$$e_{2}$$e_{2}^{*}$$e_{3}$$e_{3}^{*}$$e_{4}$$e_{4}^{*}$$e_{5}$$e_{5}^{*}$$e_{6}$$e_{6}^{*}$$e_{7}$$e_{7}^{*}$$e_{8}$$e_{8}^{*}$

Figure 1. Top: the hyperplane arrangement for the dihedral group $I_{2}(4)$
has eight regions, and its four hyperplanes are cut into six shards. Bottom:
the eight 2-cells of $\mathrm{Sal}(\mathcal{H})$, indicated in blue. One
2-cell is attached for each of the eight homotopies $e_{1}e_{2}e_{3}e_{4}\cong
e_{8}e_{7}e_{6}e_{5},e_{2}e_{3}e_{4}e_{5}^{*}\cong
e_{1}^{*}e_{8}e_{7}e_{6},\ldots,e_{8}^{*}e_{1}e_{2}e_{3}\cong
e_{7}e_{6}e_{5}e_{4}^{*}$.

Given regions $D,D^{\prime}\in\mathcal{R}$, we fix a positive minimal gallery
$\mathrm{gal}(D^{\prime},D)$ from $D^{\prime}$ to $D$ in
$\mathrm{Sal}(\mathcal{H})$; any teo such galleries from $D"$ to $D$ are
homotopic. If $e$ is the edge $C^{\prime}\xrightarrow{e}C$, we define the
corresponding loop $\ell_{e}\in\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ by

(1) $\ell_{e}:=\mathrm{gal}(B,C^{\prime})\cdot
ee^{*}\cdot\mathrm{gal}(B,C^{\prime})^{-1}\in\pi_{1}(\mathrm{Sal}(\mathcal{H}),B).$

Because of homotopies, this definition of the loop does not depend on the
choice of the gallery. Write
$\mathcal{L}_{\mathrm{edge}}=\mathcal{L}_{\mathrm{edge}}(\mathcal{H},B)$ for
the set of all such loops $\ell_{e}$. By definition, the group
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ is generated by
$\mathcal{L}_{\mathrm{edge}}$.

###  1.3. Shards (Section 3)

_Shards_ are certain closed polyhedral subsets of the hyperplanes in
$\mathcal{H}$ that were introduced by Reading in [Rea11]; we write
${\mbox{{Sh}}}(\mathcal{H},B)$ for the set of shards. Each cover relation
$C^{\prime}\lessdot C$ in $\mathrm{Weak}(\mathcal{H},B)$ can be labeled by a
shard $\Sigma(C^{\prime}\lessdot C)$, which is the unique shard separating the
region $C^{\prime}$ from the region $C$; in this case, we call
$\Sigma(C^{\prime}\lessdot C)$ a _lower shard_ of $C$. Let
$\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)$ be the set of lower shards of $C$.

Now assume $\mathcal{H}$ is simplicial. Then $\mathrm{Weak}(\mathcal{H},B)$ is
a semidistributive lattice [BEZ90], and the set of shards forms an elegant
geometric realization of the set of join-irreducible elements. Furthermore,
shard intersections encode the canonical join representations of
$\mathrm{Weak}(\mathcal{H},B)$. The map
$C\mapsto\bigcap\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)$ defines a bijection
from $\mathcal{R}$ to the set of arbitrary intersections of shards. In
[Rea11], Reading introduced another poset
$\mathrm{Shard}(\mathcal{H},B):=(\mathcal{R},\preceq)$ called the _shard
intersection order_ , which is defined by

$C\preceq D\quad\text{if and only
if}\quad\bigcap\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)\supseteq\bigcap\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(D).$

As with $\mathrm{Weak}(\mathcal{H},B)$, the poset
$\mathrm{Shard}(\mathcal{H},B)$ is a lattice—but while
$\mathrm{Weak}(\mathcal{H},B)$ is “tall and slender” (with height equal to the
number of hyperplanes and with the number of atoms equal to the dimension),
$\mathrm{Shard}(\mathcal{H},B)$ is “short and wide” (with height equal to
dimension and with the number of atoms equal to the number of shards). When
$\mathcal{H}$ is the reflection arrangement of a finite Coxeter group $W$, the
relationship between noncrossing partitions and sortable elements allowed
Reading to embed the $W$-noncrossing partition lattice into the shard
intersection order, thereby giving a uniform proof that the noncrossing
partition lattice is indeed a lattice. An example is illustrated in Figure 2.

$B$$R_{1}$$R_{2}$$R_{3}$$-B$$R_{6}$$R_{7}$$R_{8}$ Figure 2. The poset
$\mathrm{Shard}(\mathcal{H},B)$ for the arrangement of Figure 1, which is the
reflection arrangement of the dihedral group $I_{2}(4)$. Gray indicates
elements in the image of Reading’s embedding of the $I_{2}(4)$-noncrossing
partition lattice (with respect to a certain Coxeter element); see Section
7.5.

###  1.4. Shards and the Salvetti Complex (Section 4)

Whenever we have an edge $C^{\prime}\xrightarrow{e}C$ in
$\mathrm{Sal}(\mathcal{H})$, we will write $\Sigma(e)$ for the associated
shard $\Sigma(C^{\prime}\lessdot C)$. Although the generators in
$\mathcal{L}_{\mathrm{edge}}$ are a priori indexed by cover relations in
$\mathrm{Weak}(\mathcal{H},B)$, our first theorem says that they are really
indexed by the much smaller set of shards.

######  Theorem 1.1.

Let $\mathcal{H}$ be a central real hyperplane arrangement. Given edges
$C^{\prime}\xrightarrow{e}C$ and $D^{\prime}\xrightarrow{f}D$ in
$\mathrm{Sal}(\mathcal{H})$, we have $\ell_{e}\simeq\ell_{f}$ if and only if
$\Sigma(e)=\Sigma(f)$.

By Theorem 1.1, it makes sense write
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}=\mathcal{L}_{\mathrm{edge}}$, indexing the
loops in $\mathcal{L}_{\mathrm{edge}}$ by shards. Thus, for any shard
$\Sigma\in{\mbox{{Sh}}}(\mathcal{H},B)$, we define the _shard loop_
$\ell_{\Sigma}=\ell_{e}$, where $e$ is any edge such that $\Sigma(e)=\Sigma$.

We define the _pure shard monoid_ , denoted $\mathbf{P}^{+}(\mathcal{H},B)$,
to be the submonoid of $\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ generated by
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$. That is, an element of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ is in $\mathbf{P}^{+}(\mathcal{H},B)$
if and only if it can be represented by a word in the alphabet
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$. This allows us to endow
$\mathbf{P}^{+}(\mathcal{H},B)$ with a partial order $\leq$ by declaring that
$p\leq p^{\prime}$ if there is a word over
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$ representing $p^{\prime}$ that contains a
prefix representing $p$. Figure 3 shows the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ between the identity element and
the _full twist_ $\Delta^{2}$ (see Section 4.2) in the pure shard monoid of
the arrangement from Figure 1. This interval is “tall and wide,” but it is
_not_ a lattice in general (see Figure 7), preventing the use of Garside
theory to study $\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$.

In general, the combinatorics of the pure shard monoid can be quite involved.
However, for a rank-2 arrangement with $m$ hyperplanes, we will prove the
following precise theorem, which tells us that there are exactly $m2^{m-2}$
words over the alphabet $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$ representing
$\Delta^{2}$.

######  Theorem 1.2.

Let $\mathcal{H}$ be hyperplane arrangement of rank $2$ with $m$ hyperplanes.
The interval $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ is a planar lattice
with rank generating function
$1+\left(\sum_{k=1}^{m-1}\left(2\binom{m}{k}-2\right)q^{k}\right)+q^{m}$ and
with $m2^{m-2}$ maximal chains.

###  1.5. Pop (Section 5)

In the next three subsections, we discuss three incarnations of the shard
intersection order given by three maps $\mathsf{Pop}$, $\mathsf{Crackle}$, and
$\mathsf{Snap}$. The first one, defined using the map $\mathsf{Pop}$, is not
new, but it inspired our terminology for the other two. For these three
subsections, we assume that $\mathcal{H}$ is simplicial so that
$\mathrm{Weak}(\mathcal{H},B)$ is a semidistributive lattice.

Let $L$ be a locally finite meet-semilattice with meet operation denoted by
$\wedge$. Motivated by work on _pop-stacks_ from enumerative combinatorics and
theoretical computer science [Ung82, CG19, AV21], the first author defined the
_pop-stack sorting operator_ $\mathsf{Pop}\colon L\to L$ in [Def22a] (see also
[Def22b]) by

(2) $\mathsf{Pop}(x):=x\wedge\bigwedge\\{y\in L:y\lessdot x\\}.$

In the case when $L=\mathrm{Weak}(\mathcal{H},B)$, we can use $\mathsf{Pop}$
to characterize $\mathrm{Shard}(\mathcal{H},B)$. For $C\in\mathcal{R}$, let
$\Sigma([\mathsf{Pop}(C),C])$ be the set of shards that label the cover
relations in the interval $[\mathsf{Pop}(C),C]$; that is,

$\Sigma([\mathsf{Pop}(C),C]):=\left\\{\Sigma(D^{\prime}\lessdot
D):\mathsf{Pop}(C)\leq D^{\prime}\lessdot D\leq C\right\\}.$

By [Rea11, Proposition 5.7], we have

(3) $C^{\prime}\preceq C\quad\text{if and only
if}\quad\Sigma([\mathsf{Pop}(C^{\prime}),C^{\prime}])\subseteq\Sigma([\mathsf{Pop}(C),C]).$

###  1.6. Crackle (Section 6)

We define the _crackle map_
$\mathsf{Crackle}\colon\mathcal{R}\to\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ by

(4)
$\mathsf{Crackle}(C):=\mathrm{gal}(B,\mathsf{Pop}(C))\cdot\mathrm{gal}(\mathsf{Pop}(C),C)\cdot\mathrm{gal}(C,\mathsf{Pop}(C))\cdot\mathrm{gal}(B,\mathsf{Pop}(C))^{-1}.$

This map generalizes the shard loops $\ell_{\Sigma}$ of Equation 1 and Theorem
1.1: if $J$ is a join-irreducible region of $\mathrm{Weak}(\mathcal{H},B)$,
then $\mathsf{Pop}(J)$ is the unique region covered by $J$, so
$\mathsf{Crackle}(J)=\ell_{\Sigma(\mathsf{Pop}(J)\lessdot J)}$.

Just as Equation 3 characterized $\mathrm{Shard}(\mathcal{H},B)$ using
$\mathsf{Pop}$, we can characterize $\mathrm{Shard}(\mathcal{H},B)$ using
$\mathsf{Crackle}$. As above, we write
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ for the interval between the
identity element $\mathbbm{1}$ and the full twist $\Delta^{2}$ in the pure
shard monoid $\mathbf{P}^{+}(\mathcal{H},B)$. Recall that if $P$ and $Q$ are
posets, then a map $\psi\colon P\to Q$ is called a _poset embedding_ if it is
a poset isomorphism from $P$ to its image $\psi(P)\subseteq Q$.

######  Theorem 1.3.

The map $\mathsf{Crackle}$ is a poset embedding from
$\mathrm{Shard}(\mathcal{H},B)$ into the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$.

Theorem 1.3 is illustrated in Figures 3 and 7.

$\mathbbm{1}$$\ell_{\Sigma_{6}}$$\ell_{\Sigma_{7}}$$\ell_{\Sigma_{8}}$$\ell_{\Sigma_{1}}$$\ell_{\Sigma_{2}}$$\ell_{\Sigma_{3}}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\Delta^{2}$678123788321876112832116388761167238
Figure 3. The interval $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ in the pure
shard monoid $\mathbf{P}^{+}(\mathcal{H},B)$ between the identity element and
the full twist $\Delta^{2}$, where $(\mathcal{H},B)$ is as in Figure 1. An
edge $p\lessdot p^{\prime}$ is labeled $i$ when
$p^{\prime}=p\cdot\ell_{\Sigma_{i}}$. Circled elements are in the image of
$\mathsf{Crackle}$. An element is colored gray if it appears as a prefix of a
word for $\Delta^{2}$ using only generators corresponding to noncrossing
shards; see Section 7.5.

###  1.7. Snap (Section 7)

We now specialize to the case when $\mathcal{H}$ is the reflection arrangement
of a finite Coxeter group $W$. We identify the base region $B$ with the
identity element of $W$; the free transitive action of $W$ on $\mathcal{R}$
then allows us to identity regions of $\mathcal{H}$ with elements of $W$. We
write $\mathrm{Shard}(W):=\mathrm{Shard}(\mathcal{H},B)$,
$\mathrm{Weak}(W):=\mathrm{Weak}(\mathcal{H},B)$,
$\mathbf{P}^{+}(W)=\mathbf{P}^{+}(\mathcal{H},B)$, etc.

In this setting, the group
$\mathbf{P}(W):=\pi_{1}(\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}},x_{B})$
is called the _pure braid group_ of $W$, while the group
$\mathbf{B}(W):=\pi_{1}((\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}})/W,x_{B})$
is called the _braid group_ of $W$. The Coxeter group $W$ fits into the
following well-known exact sequence with its braid and pure braid groups:

$1\to\mathbf{P}(W)\to\mathbf{B}(W)\xrightarrow{\varphi}W\to 1.$

Let ${\bf S}$ be the set of simple generators of $\mathbf{B}(W)$ obtained by
lifting the set $S$ of simple reflections of $W$. The generators in $\bf S$
satisfy the same braid relations as the corresponding simple reflections of
$W$; the difference is that $W$ also includes the relations stating that the
simple reflections are involutions. Thus, the projection
$\varphi\colon\mathbf{B}(W)\to W$ is the quotient map that sends each
generator ${\bf s}\in{\bf S}$ to the corresponding $s\in S$ and imposes these
additional relations. The submonoid of $\mathbf{B}(W)$ generated by ${\bf S}$
is called the _positive braid monoid_ of $W$ and is denoted by
$\mathbf{B}^{+}(W)$. The weak order $(W,\leq)$ is defined by saying $u\leq v$
if and only if any reduced word for $u$ appears as a prefix of some reduced
word for $v$. Analogously, the weak order
$\mathrm{Weak}(\mathbf{B}^{+}(W)):=(\mathbf{B}^{+}(W),\leq)$ is defined by
saying ${\bf u}\leq\mathbf{v}$ if and only if any word over $\bf S$
representing $\bf u$ appears as a prefix of some word over $\bf S$
representing $\mathbf{v}$. In this setting, the full twist $\Delta^{2}$ is
equal to the lift $\mathbf{w}_{\circ}^{2}$ of the long element $w_{\circ}$ of
$W$; we will also write $\Delta=\mathbf{w}_{\circ}$.

We can rephrase Theorem 1.1 when $\mathcal{H}$ is the reflection arrangement
of $W$ as follows.

######  Corollary 1.4.

Suppose $u,v\in W$ and $s,t\in S$ are such that $u\lessdot us$ and $v\lessdot
vt$. Let ${\bf u},{\bf v},{\bf s},{\bf t}$ be the lifts of $u,v,s,t$,
respectively, to $\mathbf{B}^{+}(W)$. We have ${\bf u}\mathbf{s}{\bf
u}^{-1}=\mathbf{v}\mathbf{t}\mathbf{v}^{-1}$ if and only if $\Sigma(w\lessdot
ws)=\Sigma(u\lessdot ut)$.

For $u,v\in W$, we let $u^{v}=v^{-1}uv$. Write
$\mathbf{c}=(\mathbf{s}_{1},\ldots,\mathbf{s}_{n})$ for an ordering of the
elements of $\bf S$. For $\mathbf{s}\in{\bf S}$ and a positive braid
$\mathbf{w}=\mathbf{s}_{i_{1}}\cdots\mathbf{s}_{i_{k}}\in\mathbf{B}^{+}(W)$
with projection $w=\varphi(\mathbf{w})\in W$, let
$\mathbf{s}^{\mathbf{w}}=(t,j)$, where $t=s^{w}$ and $j$ counts the number of
times $t$ appears in the sequence
$s^{s_{i_{k}}},s^{s_{i_{k}}s_{i_{k-1}}},\ldots,s^{s_{i_{k}}s_{i_{k-1}}\cdots
s_{1}}$.

In [GLTW22], a new set of noncrossing $W$-Catalan objects was introduced as
the set of subwords of $\mathbf{c}^{h+1}$ that represent the full twist
$\Delta^{2}=\mathbf{w}_{\circ}^{2}$ and satisfy an additional _Deodhar
condition_. When interpreted in the positive braid monoid $\mathbf{B}^{+}(W)$,
this Deodhar condition is equivalent to restricting to those $c$-sortable
elements $\mathbf{w}$ in the interval
$[\mathbbm{1},\mathbf{w}_{\circ}^{2}]_{\mathbf{B}^{+}}$ with the property that
for each descent $\mathbf{s}$ of $\mathbf{w}$, we have
$\mathbf{s}^{\mathbf{w}}=(t,j)$ with $j$ even—this is a nonstandard _Deodhar
embedding_ of the $c$-sortable elements into the interval
$[\mathbbm{1},\mathbf{w}_{\circ}^{2}]_{\mathbf{B}^{+}}$ (different from the
usual lift of $W$ into $\mathbf{B}^{+}(W)$). The second author speculated that
restricting the $2$nd $c$-Fuss–Cambrian lattice to the image of this Deodhar
embedding would recover the noncrossing partition lattice $\mathrm{NC}(W,c)$.
As the $2$nd $c$-Fuss–Cambrian lattice is a subposet of
$\mathrm{Weak}(\mathbf{B}^{+}(W))$, it makes sense to generalize this Deodhar
embedding of $c$-sortable elements to all elements of $W$.

The _snap map_ $\mathsf{Snap}\colon W\to\mathbf{B}^{+}(W)$ is our
generalization of the Deodhar embedding. We write $\mathsf{Pop}$ for the pop-
stack sorting operators on the lattice $\mathrm{Weak}(W)$ and the meet-
semilattice $\mathrm{Weak}(\mathbf{B}^{+}(W))$, relying on the argument of the
operator to indicate the context. For $w\in W$, let $\mathrm{des}(w)$ denote
the right descent set of $w$, let $w_{\circ}(\mathrm{des}(w))$ be the longest
element of the parabolic subgroup of $W$ generated by $\mathrm{des}(w)$, and
write $\mathbf{w}$ and $\mathbf{w}_{\circ}(\mathrm{des}(w))$ for the usual
lifts of $w$ and $w_{\circ}(\mathrm{des}(w))$ to $\mathbf{B}^{+}(W)$. Define

(5)
$\mathsf{Snap}(w):=\mathsf{Pop}(\mathbf{w})\cdot(\mathbf{w}_{\circ}(\mathrm{des}(w)))^{2}.$

Since
$\mathsf{Crackle}(w)=\mathsf{Pop}(\mathbf{w})\cdot(\mathbf{w}_{\circ}(\mathrm{des}(w)))^{2}\cdot\mathsf{Pop}(\mathbf{w})^{-1}$
in $\mathbf{P}^{+}(W)\subseteq\mathbf{B}(W)$, it follows that

$\mathsf{Snap}(w)=\mathsf{Crackle}(w)\mathsf{Pop}(\mathbf{w}).$

$\mathbbm{1}$$\mathbf{s}$$\mathbf{s}\mathbf{s}$$\mathbf{s}\mathbf{s}\mathbf{t}$$\mathbf{s}\mathbf{s}\mathbf{t}\mathbf{s}$$\mathbf{s}\mathbf{t}$$\mathbf{s}\mathbf{t}\mathbf{t}$$\mathbf{s}\mathbf{t}\mathbf{t}\mathbf{s}$$\mathbf{s}\mathbf{t}\mathbf{t}\mathbf{s}\mathbf{t}$$\mathbf{s}\mathbf{t}\mathbf{s}$$\mathbf{s}\mathbf{t}\mathbf{s}\mathbf{s}$$\mathbf{s}\mathbf{t}\mathbf{s}\mathbf{s}\mathbf{t}$$\mathbf{s}\mathbf{t}\mathbf{s}\mathbf{s}\mathbf{t}$$\Delta$$\Delta\mathbf{s}$$\Delta\mathbf{s}\mathbf{t}$$\Delta\mathbf{s}\mathbf{t}\mathbf{s}$$\Delta^{2}$$\Delta\mathbf{t}$$\Delta\mathbf{t}\mathbf{s}$$\Delta\mathbf{t}\mathbf{s}\mathbf{t}$$\mathbf{t}$$\mathbf{t}\mathbf{t}$$\mathbf{t}\mathbf{t}\mathbf{s}$$\mathbf{t}\mathbf{t}\mathbf{s}\mathbf{t}$$\mathbf{t}\mathbf{s}$$\mathbf{t}\mathbf{s}\mathbf{s}$$\mathbf{t}\mathbf{s}\mathbf{s}\mathbf{t}$$\mathbf{t}\mathbf{s}\mathbf{s}\mathbf{t}\mathbf{s}$$\mathbf{t}\mathbf{s}\mathbf{t}$$\mathbf{t}\mathbf{s}\mathbf{t}\mathbf{t}$$\mathbf{t}\mathbf{s}\mathbf{t}\mathbf{t}\mathbf{s}$$\mathbf{t}\mathbf{s}\mathbf{t}\mathbf{t}\mathbf{s}\mathbf{t}$
Figure 4. The interval $[\mathbbm{1},\Delta^{2}]_{\mathbf{B}^{+}}$ in
$\mathrm{Weak}(\mathbf{B}^{+}(I_{2}(4)))$, where $I_{2}(4)$ is the dihedral
group of order $8$ with simple reflections $s$ and $t$. The reflection
arrangement of $I_{2}(4)$ is shown in Figure 1. Circled elements are in the
image of $\mathsf{Snap}$. Gray indicates that the element is the image of a
$c$-sortable element under $\mathsf{Snap}$, where $c=st$; see Section 7.5.

Just as the shard intersection order was characterized via $\mathsf{Pop}$ in
Equation 3 and via $\mathsf{Crackle}$ in Theorem 1.3, it is also characterized
via $\mathsf{Snap}$.

######  Theorem 1.5.

The map $\mathsf{Snap}$ is a poset embedding from $\mathrm{Shard}(W)$ into
$[\mathbbm{1},\Delta^{2}]_{\mathbf{B}^{+}}$.

Theorem 1.5 is illustrated in Figure 4.

###  1.8. Pow (Section 8)

In her history “The Untold Tale of Pow!, the Fourth Rice Krispies Elf: A look
into the era when the cereal mascots were more than just Snap!, Crackle! and
Pop!” [Smi14], Smith writes:

> “Lost in the shuffle, however, was a fourth Rice Krispies elf named Pow! His
> short life is a time-capsule of an era when everyone was dreaming big.”

Inspired by this fourth mascot, we introduce a map
$\mathsf{Pow}:\mathcal{R}\to\mathbf{P}^{+}(\mathcal{H},B)$ in the general
setting when $\mathcal{H}$ is a central irreducible real hyperplane
arrangement. Given a region $C\in\mathcal{R}$ and a positive minimal gallery

$B=C_{0}\xrightarrow{e_{1}}C_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}C_{k-1}\xrightarrow{e_{k}}C_{k}=C,$

we let

$\mathsf{Pow}(C):=\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}.$

(We prove in Proposition 8.1 that $\mathsf{Pow}$ is well-defined.) Just as
$\mathsf{Crackle}$ embeds the “short and wide” poset
$\mathrm{Shard}(\mathcal{H},B)$ into the “tall and wide” interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$, the map $\mathsf{Pow}$ embeds the
“tall and slender” poset $\mathrm{Weak}(\mathcal{H},B)$ into
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$.

######  Theorem 1.6.

The map $\mathsf{Pow}$ is a poset embedding from
$\mathrm{Weak}(\mathcal{H},B)$ into
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$.

Theorem 1.6 is illustrated in Figure 5.

###  1.9. Future Work (Section 9)

The final section of the paper proposes some ideas for further investigation.
In particular, Conjecture 9.1 gives a conjectural description of the words
over the alphabet $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$ that represent the full
twist (i.e., the maximal chains in the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$) when $\mathcal{H}$ is an
arbitrary finite central irreducible real hyperplane arrangement. This
conjecture extends the characterization of the
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-words for the full twist that we prove in
Proposition 4.7 when $\mathcal{H}$ has rank $2$. We also conjecture that the
interval $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ is self-dual (Conjecture
9.2).

##  2\. The Salvetti Complex

###  2.1. Real Hyperplane Arrangements

We recall some definitions regarding real hyperplane arrangements, referring
the reader to [Sta04] for additional details.

A real _hyperplane_ is an affine subspace of $\mathbb{R}^{n}$ of codimension
1, and a _hyperplane arrangement_ (or just _arrangement_ , for short) is a set
of distinct hyperplanes in $\mathbb{R}^{n}$. A hyperplane is _linear_ if it
contains the origin, and an arrangement is _central_ if all of its hyperplanes
are linear. A _subarrangement_ of an arrangement $\mathcal{H}$ is simply a
subset of $\mathcal{H}$. We say $\mathcal{H}$ is _irreducible_ if there do not
exist a linear automorphism $\theta$ of $\mathbb{R}^{n}$ and a partition
$\mathcal{H}=\mathcal{H}_{1}\sqcup\mathcal{H}_{2}$ with
$\mathcal{H}_{1},\mathcal{H}_{2}\neq\emptyset$ such that all of the normal
vectors of hyperplanes in $\theta(\mathcal{H}_{1})$ are orthogonal to all of
the normal vectors of hyperplanes in $\theta(\mathcal{H}_{2})$. Throughout
this paper, we always assume that $\mathcal{H}$ is a finite central
irreducible real hyperplane arrangement in $\mathbb{R}^{n}$; note that this
forces $\mathcal{H}$ to contain at least $3$ hyperplanes.

The _rank_ of an arrangement (or subarrangement) is the codimension of the
intersection of its hyperplanes. We call a subarrangement
$\mathcal{A}\subseteq\mathcal{H}$ _full_ if it contains all hyperplanes of
$\mathcal{H}$ containing a particular subset of $\mathbb{R}^{n}$.

A _region_ (or _chamber_) of $\mathcal{H}$ is a connected component of
$\mathbb{R}^{n}\setminus\bigcup\mathcal{H}$; we write
$\mathcal{R}=\mathcal{R}_{\mathcal{H}}$ for the set of regions of
$\mathcal{H}$. We will fix a choice of a _base region_ $B$. A _bounding
hyperplane_ of a region $C\in\mathcal{R}$ is a hyperplane in $\mathcal{H}$
whose intersection with the boundary of $C$ is $(n-1)$-dimensional. We say
$\mathcal{H}$ is _simplicial_ if every region has exactly $n$ bounding
hyperplanes.

###  2.2. The Poset of Regions

Let $C,C^{\prime}\in\mathcal{R}$ be two regions of $\mathcal{H}$, and choose
points $x\in C$ and $x^{\prime}\in C^{\prime}$. The hyperplanes intersecting
the line segment with endpoints $x$ and $x^{\prime}$ are said to _separate_
$C$ from $C^{\prime}$, and we write $\mathrm{inv}(C,C^{\prime})$ for the set
of all hyperplanes separating $C$ from $C^{\prime}$. We write $-C$ for the
unique region such that $\mathrm{inv}(C,-C)=\mathcal{H}$. Let
$\mathrm{inv}(C)=\mathrm{inv}(B,C)$, where $B$ is our fixed base region. The
_poset of regions_ of $\mathcal{H}$ is the poset
$\mathrm{Weak}(\mathcal{H},B):=(\mathcal{R},\leq)$, where we write $C\leq
C^{\prime}$ if $\mathrm{inv}(C)\subseteq\mathrm{inv}(C^{\prime})$.

######  Theorem 2.1 ([BEZ90, Theorem 3.4]).

If $\mathcal{H}$ is a simplicial hyperplane arrangement, then the poset of
regions $\mathrm{Weak}(\mathcal{H},B)$ is a lattice.

We write $C^{\prime}\xrightarrow{e}C$ when $e$ is the edge in the Hasse
diagram of $\mathrm{Weak}(\mathcal{H},B)$ corresponding to a cover relation
$C^{\prime}\lessdot C$. In this case, we write $H_{e}$ for the unique
hyperplane in $\mathrm{inv}(C)\setminus\mathrm{inv}(C^{\prime})$, and we call
$H_{e}$ a _lower cover_ of $C$. Let $\mathrm{cov}(C)$ be the set of lower
covers of $C$.

###  2.3. The Salvetti Complex

Let $\mathcal{H}=\\{H_{i}\\}_{i=1}^{N}$. The _Salvetti complex_ is a
combinatorial model for the complement in $\mathbb{C}^{n}$ of the complexified
hyperplane arrangement $\mathcal{H}_{\mathbb{C}}$ [Sal87]. Choose $\alpha_{i}$
to be a normal vector to $H_{i}$. We write $Z(\mathcal{H})$ for the _dual
zonotope_ of $\mathcal{H}$, defined as the Minkowski sum
$\alpha_{1}+\cdots+\alpha_{N}$. As a polytope, the zonotope $Z(\mathcal{H})$
comes equipped with the structure of a CW complex. For $C\in\mathcal{R}$,
write $Z(\mathcal{H},C)$ for an oriented copy of $Z(\mathcal{H})$, where each
face of dimension at least $1$ is given an orientation pointing from $C$ to
$-C$. The Salvetti complex is then the CW complex defined as the union of the
CW complexes $Z(\mathcal{H},C)$ over all $C\in\mathcal{R}$:

(6) $\mathrm{Sal}(\mathcal{H}):=\bigcup_{C\in\mathcal{R}}Z(\mathcal{H},C).$

This is not a disjoint union: for any $C,C^{\prime}\in\mathcal{R}$, we
identify faces in $Z(\mathcal{H},C)$ and $Z(\mathcal{H},C^{\prime})$ that have
equivalent orientations. Vertices of $\mathrm{Sal}(\mathcal{H})$ correspond to
regions of $\mathcal{H}$, so it makes sense to think of our fixed base region
$B$ as a point in $\mathrm{Sal}(\mathcal{H})$.

######  Theorem 2.2 ([Sal87]).

If $\mathcal{H}$ is a central real arrangement, then
$\mathrm{Sal}(\mathcal{H})$ embeds in
$\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}}$ and is a deformation retract
of $\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}}$. Furthermore,

$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)=\pi_{1}(\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}},x_{B}),$

where $B\in\mathcal{R}$ and $x_{B}\in B$.

Salvetti further described the abelianization
$H_{1}(\mathrm{Sal}(\mathcal{H}),B)$ of the group
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ (see also Theorem 2.5 below).

######  Theorem 2.3 ([Sal87]).

The abelianization $H_{1}(\mathrm{Sal}(\mathcal{H}),B)$ of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ is a free abelian group with generating
set $\\{\overline{\ell}_{H}\\}_{H\in\mathcal{H}}$ indexed by the hyperplanes
in $\mathcal{H}$. The natural homomorphism
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)\to H_{1}(\mathrm{Sal}(\mathcal{H}),B)$
sends $\ell_{e}$ to $\overline{\ell}_{H_{e}}$.

A direct description of the restriction of $\mathrm{Sal}(\mathcal{H})$ to its
1-skeleton can be built directly from the Hasse diagram of
$\mathrm{Weak}(\mathcal{H},B)$ (see [Sal87, Part Two]). The vertices are given
by the regions of $\mathcal{H}$. The edges are given by the cover relations of
the Hasse diagram of $\mathrm{Weak}(\mathcal{H},B)$ as follows. For each cover
relation $C^{\prime}\lessdot C$ in $\mathrm{Weak}(\mathcal{H},B)$, introduce
an oriented edge $e$ from $C^{\prime}$ to $C$ and an oriented edge $e^{*}$
from $C$ to $C^{\prime}$. We extend the notation $C^{\prime}\xrightarrow{e}C$
to accommodate these extra (oriented) edges using
$C\xrightarrow{e^{*}}C^{\prime}$. Note that $e^{*}$ is not the same as
$e^{-1}$. We will always use un-starred letters for edges oriented away from
$B$ and starred letter for edges oriented toward $B$.

Given arbitrary regions $C,C^{\prime}\in\mathcal{R}$, we define a _gallery_
from $C^{\prime}$ to $C$ to be a sequence of edges (of the form $e$, $e^{*}$,
$e^{-1}$, and $(e^{*})^{-1}$) that starts at $C^{\prime}$ and ends at $C$. A
gallery from $C^{\prime}$ to $C$ is _positive_ if it only uses edges of the
form $e$ and $e^{*}$ (not $e^{-1}$ or $(e^{*})^{-1}$). The gallery is
_minimal_ if its length is equal to $|\mathrm{inv}(C^{\prime},C)|$, and it is
called a _loop_ if $C=C^{\prime}$. If $\alpha$ and $\beta$ are galleries such
that the ending point of $\alpha$ is the starting point of $\beta$, then
$\alpha\cdot\beta$ is the gallery that we traverse by first traversing
$\alpha$ and then traversing $\beta$. We say two galleries $\alpha,\beta$ in
$\mathrm{Sal}(\mathcal{H})$ are _homotopic_ and write $\alpha\cong\beta$ if
$\alpha$ can be obtained from $\beta$ by repeated insertion or deletion of
boundary paths of 1-cells and 2-cells in $\mathrm{Sal}(\mathcal{H})$. A
gallery is minimal if and only if it has the shortest length among all
galleries in its homotopy class. Any two positive minimal galleries from a
region $C^{\prime}$ to a region $C$ are homotopic, and we write
$\mathrm{gal}(C^{\prime},C)$ for a choice of one such gallery. We abbreviate
the concatenation of a sequence of positive minimal galleries with the
notation
$\mathrm{gal}(C_{1},C_{2},C_{3}\ldots,C_{k-1},C_{k})=\mathrm{gal}(C_{1},C_{2})\cdot\mathrm{gal}(C_{2},C_{3})\cdot\cdots\cdot\mathrm{gal}(C_{k-1},C_{k})$.

###  2.4. Generators and Relations of the Fundamental Group

This subsection follows [Sal87, Part Two]. If $C^{\prime}\xrightarrow{e}C$, we
define the corresponding loop $\ell_{e}$ by

$\ell_{e}:=\mathrm{gal}(B,C^{\prime})\cdot
ee^{*}\cdot\mathrm{gal}(B,C^{\prime})^{-1}\in\pi_{1}(\mathrm{Sal}(\mathcal{H}),B).$

Note that $\ell_{e}$ does not depend on the choice of the minimal gallery
$\mathrm{gal}(B,C^{\prime})$. Write $\mathcal{L}_{\mathrm{edge}}$ for the set
of all such loops $\ell_{e}$; this set generates
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$, but it is redundant. It turns out to
be enough to reduce to a single generator from each hyperplane, as we now
explain.

Fix a positive minimal gallery

$\delta=\left(B=C_{0}\xrightarrow{e_{1}}C_{1}\xrightarrow{e_{2}}C_{2}\xrightarrow{e_{3}}\cdots\xrightarrow{e_{N}}C_{N}=-B\right)$

from $B$ to $-B$ in $\mathrm{Sal}(\mathcal{H})$. Writing
$\delta_{i}:=\ell_{e_{i}}$ and $H_{i}:=H_{e_{i}}$ specifies the subset
$\mathcal{L}_{\delta}=\\{\delta_{i}\\}_{i=1}^{N}\subseteq\mathcal{L}_{\mathrm{edge}}$
and totally orders the hyperplanes by

$H_{1}<_{\delta}H_{2}<_{\delta}\cdots<_{\delta}H_{N}.$

######  Example 2.4.

Continuing the example drawn in Figure 1, we fix

$\delta=\left(B\xrightarrow{e_{1}}C_{1}\xrightarrow{e_{2}}C_{2}\xrightarrow{e_{3}}C_{3}\xrightarrow{e_{4}}-B\right),$

which gives the loops in $\mathcal{L}_{\delta}$:

$\displaystyle\delta_{1}$ $\displaystyle=\ell_{e_{1}}=e_{1}e_{1}^{*},$
$\displaystyle\delta_{2}$
$\displaystyle=\ell_{e_{2}}=e_{1}e_{2}e_{2}^{*}e_{1}^{-1},$
$\displaystyle\delta_{3}$
$\displaystyle=\ell_{e_{3}}=e_{1}e_{2}e_{3}e_{3}^{*}e_{2}^{-1}e_{1}^{-1},$
$\displaystyle\delta_{4}$
$\displaystyle=\ell_{e_{4}}=e_{1}e_{2}e_{3}e_{4}e_{4}^{*}e_{3}^{-1}e_{2}^{-1}e_{1}^{-1}.$

The next theorem shows that any generator in $\mathcal{L}_{\mathrm{edge}}$ is
conjugate to an element of $\mathcal{L}_{\delta}$.

######  Theorem 2.5 ([Sal87, Lemma 12, Corollary 12]).

For any positive minimal gallery $\delta$ from $B$ to $-B$,
$\mathcal{L}_{\delta}$ is a generating set of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$. Specifically, for any edge
$D^{\prime}\xrightarrow{e}D$, there is a unique edge
$C_{k-1}\xrightarrow{e_{k}}C_{k}$ such that $H_{e_{k}}=H_{e}$, and we have

$\ell_{e}=\left(\delta_{i_{s}}\delta_{i_{s-1}}\cdots\delta_{i_{1}}\right)^{-1}\delta_{k}\left(\delta_{i_{s}}\delta_{i_{s-1}}\cdots\delta_{i_{1}}\right),$

where $i_{1}<\cdots<i_{s}$ are the elements $i$ of $\\{1,\ldots,k-1\\}$ such
that $H_{i}\not\in\mathrm{inv}(D)$.

######  Example 2.6.

Continuing Example 2.4, we have

$\displaystyle\ell_{e_{5}}$ $\displaystyle=\delta_{1},$
$\displaystyle\ell_{e_{6}}$
$\displaystyle=\delta_{1}^{-1}\delta_{2}\delta_{1},$
$\displaystyle\ell_{e_{7}}$
$\displaystyle=\delta_{1}^{-1}\delta_{2}^{-1}\delta_{3}\delta_{2}\delta_{1},$
$\displaystyle\ell_{e_{8}}$
$\displaystyle=\delta_{1}^{-1}\delta_{2}^{-1}\delta_{3}^{-1}\delta_{4}\delta_{3}\delta_{2}\delta_{1}=\delta_{4},$

where the identity
$\delta_{1}^{-1}\delta_{2}^{-1}\delta_{3}^{-1}\delta_{4}\delta_{3}\delta_{2}\delta_{1}=\delta_{4}$
follows from Theorem 2.7 below.

Having found a set $\mathcal{L}_{\delta}$ of generators for the group
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$, we now need relations. To present
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$, we need one family of relations for
each full rank-2 subarrangement $\mathcal{A}$ of $\mathcal{H}$.

######  Theorem 2.7 ([Sal87, Page 616]).

Fix a positive minimal gallery $\delta$ from $B$ to $-B$ in
$\mathrm{Sal}(\mathcal{H})$. For each full rank-2 subarrangement $\mathcal{A}$
of $\mathcal{H}$, choose one 2-cell in $\mathrm{Sal}(\mathcal{H})$ with edges
named $e_{1},e_{2},\ldots,e_{m}$ and $e_{m+1},\ldots,e_{2m}$ as in Figure 1 so
that $\\{H_{e_{i}}\\}_{i=1}^{m}=\mathcal{A}$ and
$H_{e_{1}}<_{\delta}\cdots<_{\delta}H_{e_{m}}$. Write
$[\mathcal{A}]_{\mathcal{L}_{\delta}}$ for the family of relations

$\ell_{e_{m}}\cdots\ell_{e_{2}}\ell_{e_{1}}=\ell_{e_{1}}\ell_{e_{m}}\cdots\ell_{e_{2}}=\cdots=\ell_{e_{m{-}1}}\cdots\ell_{e_{2}}\ell_{e_{1}}\ell_{e_{m}},$

where each $\ell_{e_{i}}$ has been rewritten using the elements of
$\mathcal{L}_{\delta}$ using Theorem 2.5. Then

$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)=\left\langle\mathcal{L}_{\delta}:[\mathcal{A}]_{\mathcal{L}_{\delta}}\right\rangle,$

where the relations range over all full rank-2 subarrangements
$\mathcal{A}\subseteq\mathcal{H}$.

######  Example 2.8.

Continuing Example 2.4, we have the relations
$\delta_{4}\delta_{3}\delta_{2}\delta_{1}=\delta_{1}\delta_{4}\delta_{3}\delta_{2}=\delta_{2}\delta_{1}\delta_{4}\delta_{3}=\delta_{3}\delta_{2}\delta_{1}\delta_{4}$;
by Theorem 2.5, these relations imply the further relations
$\ell_{e_{5}}\ell_{e_{6}}\ell_{e_{7}}\ell_{e_{8}}=\ell_{e_{8}}\ell_{e_{5}}\ell_{e_{6}}\ell_{e_{7}}=\ell_{e_{7}}\ell_{e_{8}}\ell_{e_{5}}\ell_{e_{6}}=\ell_{e_{6}}\ell_{e_{7}}\ell_{e_{8}}\ell_{e_{5}}$.

We end this section with a corollary of Theorem 2.5 that we will find useful
in subsequent sections.

######  Corollary 2.9.

Let $C\in\mathcal{R}$. Consider a positive minimal gallery

$B=C_{0}\xrightarrow{e_{1}}C_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}C_{k-1}\xrightarrow{e_{k}}C_{k}=C$

from $B$ to $C$, and let

$-C=-C_{k}\xrightarrow{f_{k}}-C_{k-1}\xrightarrow{f_{k-1}}\cdots\xrightarrow{f_{2}}-C_{1}\xrightarrow{f_{1}}-C_{0}=-B$

be the corresponding positive minimal gallery from $-C$ to $-B$. Then

$\ell_{e_{k}}\ell_{e_{k-1}}\cdots\ell_{e_{1}}=\ell_{f_{1}}\ell_{f_{2}}\cdots\ell_{f_{k}}.$

###### Proof.

We can extend the given positive minimal gallery from $B$ to $C$ to a positive
minimal gallery $\delta$ from $B$ to $-B$. In the notation of Theorem 2.5, we
have $\delta_{i}=\ell_{e_{i}}$ for all $1\leq i\leq k$. Therefore, that
theorem tells us that

$\ell_{f_{i}}=\big{(}\ell_{e_{1}}^{-1}\cdots\ell_{e_{i-1}}^{-1}\big{)}\ell_{e_{i}}\big{(}\ell_{e_{i-1}}\cdots\ell_{e_{1}}\big{)}$

for all $1\leq i\leq k$. The result follows by multiplying and canceling
terms. ∎

##  3\. Shards

We recall some constructions and results from [Rea03, Rea11]. As before, fix a
base region $B\in\mathcal{R}$. For a full rank-2 subarrangement $\mathcal{A}$
of $\mathcal{H}$, let $B_{\mathcal{A}}$ be the region of $\mathcal{A}$
containing $B$. We say a hyperplane $H\in\mathcal{A}$ is _basic_ if its
intersection with the boundary of $B_{\mathcal{A}}$ has dimension $n-1$. For
$H,H^{\prime}\in\mathcal{A}$, we say $H$ _cuts_ $H^{\prime}$ if $H$ is basic
in $\mathcal{A}$ but $H^{\prime}$ is not. Each hyperplane $H\in\mathcal{H}$ is
broken into a number of connected pieces if we remove all points in $H$
contained the the hyperplanes of $\mathcal{H}$ that cut $H$—a _shard_ of $H$
is then the closure of one of these connected pieces. We write $H_{\Sigma}$
for the unique hyperplane containing a shard $\Sigma$. Let
${\mbox{{Sh}}}(\mathcal{H},B)$ denote the set of all shards.

Extending the notation for hyperplanes, when $C^{\prime}\xrightarrow{e}C$, we
write $\Sigma(e)$ for the shard crossed by $e$ and call $\Sigma(e)$ a _lower
shard_ of $C$. Let $\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)$ be the set of
lower shards of $C$. The _shard intersection order_ is the partial order
$\preceq$ on $\mathcal{R}$ defined by saying

$C^{\prime}\preceq C\quad\text{if and only
if}\quad\bigcap\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C^{\prime})\supseteq\bigcap\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C).$

For each shard $\Sigma$, we fix a minimal element $J_{\Sigma}$ of
$\\{C\in\mathcal{R}:\Sigma\in\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)\\}$ in the
poset $\mathrm{Weak}(\mathcal{H},B)$. Recall that an element in a finite
lattice is called _join-irreducible_ if it covers exactly one element. When
$\mathcal{H}$ is simplicial, we know by Theorem 2.1 that
$\mathrm{Weak}(\mathcal{H},B)$ is a lattice, and the following results of
Reading show that shards give a geometric interpretation of this lattice
structure.

######  Theorem 3.1 ([Rea03, Proposition 2.2],[Rea11, Proposition 3.3]).

If $\mathcal{H}$ is simplicial, then there is a bijection between
${\mbox{{Sh}}}(\mathcal{H},B)$ and the set of join-irreducible regions in
$\mathrm{Weak}(\mathcal{H},B)$. The shard $\Sigma$ corresponds to the join-
irreducible region $J_{\Sigma}$, which is the unique minimal element of
$\\{C\in\mathcal{R}:\Sigma\in\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)\\}$. The
shard corresponding to a join-irreducible region $J$ is the unique element of
$\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(J)$.

######  Theorem 3.2 ([Rea11, Theorem 3.6]).

If $\mathcal{H}$ is simplicial and $C\in\mathcal{R}$, then

$C=\bigvee_{\Sigma\in\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)}J_{\Sigma}.$

(There is actually a stronger version of Theorem 3.2, which says that
$\\{J_{\Sigma}:\Sigma\in\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)\\}$ is the
_canonical join representation_ of $C$ in the semidistributive lattice
$\mathrm{Weak}(\mathcal{H},B)$—the lattice property is strongly tied to the
simpliciality of $\mathcal{H}$; see [Rea11].)

##  4\. Shards and the Salvetti Complex

Our goal in this section is to establish Theorem 1.1, which tells us that two
loops $\ell_{e}$ and $\ell_{f}$ in $\mathcal{L}_{\mathrm{edge}}$ are homotopic
if and only if they go around the same shard (that is, $\Sigma(e)=\Sigma(f)$).
Thus, while Theorems 2.5 and 2.3 tell us that the elements of
$\mathcal{L}_{\mathrm{edge}}$ up to conjugation are indexed by the hyperplanes
in $\mathcal{H}$, Theorem 1.1 tells us that the elements of
$\mathcal{L}_{\mathrm{edge}}$ (not up to conjugation) are indexed by the
shards of $\mathcal{H}$.

###  4.1. Proof of Theorem 1.1

We work toward understanding general arrangements by first understanding
rank-2 arrangements. We need the following general lemma, which will also be
useful elsewhere in the paper. Given a subarrangement $\mathcal{A}$ of
$\mathcal{H}$, we let

$\iota_{\mathcal{A}}\colon\mathcal{R}\to\mathcal{R}_{\mathcal{A}}$

be the map that sends a region $D\in\mathcal{R}$ to the unique region of
$\mathcal{A}$ containing $D$, and we let
$B_{\mathcal{A}}=\iota_{\mathcal{A}}(B)$.

######  Lemma 4.1.

Let $\mathcal{A}$ be a subarrangement of a central arrangement $\mathcal{H}$.
The map $\iota_{\mathcal{A}}\colon\mathcal{R}\to\mathcal{R}_{\mathcal{A}}$
induces a quotient map

$\overline{\iota}_{\mathcal{A}}\colon\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)\to\pi_{1}(\mathrm{Sal}(\mathcal{A}),B_{\mathcal{A}})$

that sends the generating set $\mathcal{L}_{\mathrm{edge}}(\mathcal{H},B)$ to
$\mathcal{L}_{\mathrm{edge}}(\mathcal{A},B_{\mathcal{A}})\cup\\{\mathbbm{1}\\}$,
where $\mathbbm{1}$ denotes the identity element of
$\pi_{1}(\mathrm{Sal}(\mathcal{A}),B_{\mathcal{A}})$.

###### Proof.

By identifying regions via $\iota_{\mathcal{A}}$, we collapse all edges
$C^{\prime}\xrightarrow{e}C$ and $C\xrightarrow{e^{*}}C^{\prime}$ with
$\iota_{\mathcal{A}}(C^{\prime})=\iota_{\mathcal{A}}(C)$ and identify all
edges $C^{\prime}\xrightarrow{e}C$ and $D^{\prime}\xrightarrow{f}D$ with
$\iota_{\mathcal{A}}(C^{\prime})=\iota_{A}(D^{\prime})$ and
$\iota_{A}(C)=\iota_{A}(D)$. The quotient therefore introduces some relations
of the form $\ell_{e}=\mathbbm{1}$ (when $H_{e}\not\in\mathcal{A}$) and some
relations of the form $\ell_{e}=\ell_{e^{\prime}}$. ∎

######  Lemma 4.2.

Suppose $\mathcal{H}=\\{H_{1},H_{2},H_{3}\\}$ is a rank-2 hyperplane
arrangement with three hyperplanes that has base region $B$ and edges named
$e_{1},e_{2},e_{3}$ and $e_{4},e_{5},e_{6}$ (in the same manner as in Figure
1). Then $\ell_{e_{2}}\neq\ell_{e_{5}}$.

###### Proof.

If follows from Theorem 2.5 that
$\ell_{e_{5}}=\ell_{e_{1}}^{-1}\ell_{e_{2}}\ell_{e_{1}}$, so we just need to
show that $\ell_{e_{1}}$ and $\ell_{e_{2}}$ do not commute in
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$. By Theorem 2.7, this group has
presentation
$\langle\ell_{e_{1}},\ell_{e_{2}},\ell_{e_{3}}:\ell_{e_{1}}\ell_{e_{2}}\ell_{e_{3}}=\ell_{e_{3}}\ell_{e_{1}}\ell_{e_{2}}=\ell_{e_{2}}\ell_{e_{3}}\ell_{e_{1}}\rangle$.
Consider the free group $F_{2}=\langle x,y\rangle$. The map
$\\{\ell_{e_{1}},\ell_{e_{2}},\ell_{e_{3}}\\}\to F_{2}$ given by
$\ell_{e_{1}}\mapsto y^{-1}x^{-1}$, $\ell_{e_{2}}\mapsto x$,
$\ell_{e_{3}}\mapsto y$ extends to a homomorphism from
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ to $F_{2}$; this implies that
$\ell_{e_{2}}$ and $\ell_{e_{3}}$ do not commute. ∎

######  Lemma 4.3.

Let $\mathcal{H}$ be a rank-2 hyperplane arrangement with base region $B$ and
with edges named $e_{1},e_{2},\ldots,e_{m}$ and $e_{m+1},\ldots,e_{2m}$ as in
Figure 1. For all distinct $i,j\in\\{1,\ldots,2m\\}$ such that $\\{i,j\\}$ is
not $\\{m,2m\\}$ or $\\{1,m+1\\}$, we have $\ell_{e_{i}}\neq\ell_{e_{j}}$.

###### Proof.

We may assume $i<j$. If $j\neq i+m$, then the hyperplanes $H_{e_{i}}$ and
$H_{e_{j}}$ are distinct. In this case, Theorem 2.3 tells us that the natural
homomorphism $\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)\to
H_{1}(\mathrm{Sal}(\mathcal{H}))$ sends $\ell_{e_{i}}$ and $\ell_{e_{j}}$ to
different elements, so $\ell_{e_{i}}\neq\ell_{e_{j}}$. Now suppose $j=m+1$ so
that $H_{e_{i}}=H_{e_{j}}$. By hypothesis, we have $1<i<m$. Consider the
subarrangement $\mathcal{A}=\\{H_{e_{1}},H_{e_{i}},H_{e_{m}}\\}$. It follows
from Lemma 4.1 (and its proof) and Lemma 4.2 that the quotient map
$\overline{\iota}_{\mathcal{A}}$ sends $\ell_{e_{i}}$ and $\ell_{e_{j}}$ to
distinct elements of
$\mathcal{L}_{\mathrm{edge}}(\mathrm{Sal}(\mathcal{A}),B_{\mathcal{A}})$.
Hence, $\ell_{e_{i}}\neq\ell_{e_{j}}$. ∎

######  Proposition 4.4.

Let $\mathcal{H}$ be a central arrangement. If $C^{\prime}\xrightarrow{e}C$
and $D^{\prime}\xrightarrow{f}D$ are edges in $\mathrm{Sal}(\mathcal{H})$ such
that $\Sigma(e)=\Sigma(f)$, then $\ell_{e}=\ell_{f}$.

###### Proof.

Write $\Sigma:=\Sigma(e)=\Sigma(f)$. Choose a positive gallery $g$ from $C$ to
$D$, and let $C=E_{1},E_{2},\ldots,E_{r}=D$ be the regions through which $g$
passes. Let $1=k_{1}<k_{2}<\cdots<k_{s}=r$ be the indices such that
$\Sigma\in\mathrm{cov}(E_{k_{i}})$. We may assume $g$ is chosen so that for
each $1\leq i\leq s-1$, the hyperplanes that $g$ crosses when passing from
$E_{k_{i}}$ to $E_{k_{i+1}}$ all belong to a single full rank-2 subarrangement
$\mathcal{A}_{i}$ with $H_{\Sigma}\in\mathcal{A}_{i}$. For $1\leq i\leq s-1$,
there is an edge $E_{k_{i}}^{\prime}\xrightarrow{d_{i}}E_{k_{i}}$ with
$\Sigma(d_{i})=\Sigma$. It suffices to prove that
$\ell_{d_{i}}=\ell_{d_{i+1}}$ for each such $i$.

Fix $1\leq i\leq s-1$, and let
$B_{\mathcal{A}_{i}}=\iota_{\mathcal{A}_{i}}(B)$. Let $m=|\mathcal{A}_{i}|$.
The hyperplane $H_{\Sigma}$ must be basic in $\mathcal{A}_{i}$ (otherwise, the
basic hyperplanes in $\mathcal{A}_{i}$ would cut $\Sigma$), so either
$E_{k_{i}}$ or $E_{k_{i+1}}$ is contained in $-B_{\mathcal{A}_{i}}$. For
simplicity, let us assume $E_{k_{i+1}}\subseteq-B_{\mathcal{A}_{i}}$; the
other case is virtually the same. Then $E_{k_{i}}^{\prime}\subseteq
B_{\mathcal{A}_{i}}$. We can find a positive minimal gallery
$E_{k_{i}}^{\prime}\xrightarrow{e_{1}}\cdots\xrightarrow{e_{m-1}}E_{k_{i+1}}^{\prime}$
that only passes through hyperplanes in $\mathcal{A}_{i}$. Let
$E_{k_{i}}\xrightarrow{e_{2m-1}}\cdots\xrightarrow{e_{m+1}}E_{k_{i+1}}$ be the
part of the gallery $g$ from $E_{k_{i}}$ to $E_{k_{i+1}}$, and note that this
is a positive minimal gallery from $E_{k_{i}}$ to $E_{k_{i+1}}$. Let
$e_{2m}=d_{i}$ and $e_{m}=d_{i+1}$. The collection of edges
$e_{1},\ldots,e_{m},e_{m+1},\ldots,e_{2m}$ forms a $2$-cell in
$\mathrm{Sal}(\mathcal{H})$, so we have the homotopy $e_{1}e_{2}\cdots
e_{m}\cong e_{2m}e_{2m-1}\cdots e_{m+1}$. Similarly, the edges
$e_{1},\ldots,e_{m-1},e_{m}^{*},e_{m+1},\ldots,e_{2m-1},e_{2m}^{*}$ form a
$2$-cell in $\mathrm{Sal}(\mathcal{H})$, so we have the homotopy
$e_{2m-1}\cdots e_{m+1}e_{m}^{*}\cong e_{2m}^{*}e_{1}\cdots e_{m-1}$. Hence,

$\displaystyle\ell_{d_{i+1}}$
$\displaystyle\cong\mathrm{gal}(B,E_{k_{i}}^{\prime})(e_{1}e_{2}\cdots
e_{m})e_{m}^{*}e_{m-1}^{-1}\cdots
e_{1}^{-1}\mathrm{gal}(B,E_{k_{i}}^{\prime})^{-1}$
$\displaystyle\cong\mathrm{gal}(B,E_{k_{i}}^{\prime})e_{2m}(e_{2m-1}\cdots
e_{m+1}e_{m}^{*})e_{m-1}^{-1}\cdots
e_{1}^{-1}\mathrm{gal}(B,E_{k_{i}}^{\prime})^{-1}$
$\displaystyle\cong\mathrm{gal}(B,E_{k_{i}}^{\prime})e_{2m}(e_{2m}^{*}e_{1}\cdots
e_{m-1})e_{m-1}^{-1}\cdots e_{1}^{-1}\mathrm{gal}(B,E_{k_{i}}^{\prime})^{-1}$
$\displaystyle\cong\mathrm{gal}(B,E_{k_{i}}^{\prime})e_{2m}e_{2m}^{*}\mathrm{gal}(B,E_{k_{i}}^{\prime})^{-1}$
$\displaystyle\cong\ell_{d_{i}}.$

Thus, the generators $\ell_{d_{i}}$ and $\ell_{d_{i+1}}$ are the same. ∎

######  Proposition 4.5.

Let $\mathcal{H}$ be a central arrangement. If $C^{\prime}\xrightarrow{e}C$
and $D^{\prime}\xrightarrow{f}D$ are edges in $\mathrm{Sal}(\mathcal{H})$ such
that $\Sigma(e)\neq\Sigma(f)$, then $\ell_{e}\not\cong\ell_{f}$.

###### Proof.

Let $\Sigma=\Sigma(e)$ and $\Sigma^{\prime}=\Sigma(f)$. If the hyperplanes
$H_{\Sigma}$ and $H_{\Sigma^{\prime}}$ are different, then the result follows
from Theorem 2.3 by passing to the abelianization
$H_{1}(\mathrm{Sal}(\mathcal{H}),B)$.

Now suppose $\Sigma$ and $\Sigma^{\prime}$ are shards for the same hyperplane
$H$. Let $H^{\prime}$ be a hyperplane that cuts $H$ so that $\Sigma$ and
$\Sigma^{\prime}$ are on opposite sides of $H^{\prime}$, and let $\mathcal{A}$
be the full rank-2 subarrangement of $\mathcal{H}$ containing $H$ and
$H^{\prime}$. Note that $H$ is not basic in $\mathcal{A}$. It follows from
Lemma 4.3 and the proof of Lemma 4.1 that the quotient map
$\overline{\iota}_{\mathcal{A}}$ sends $\ell_{e}$ and $\ell_{f}$ to different
elements of $\mathcal{L}_{\mathrm{edge}}(\mathcal{A},B_{\mathcal{A}})$, so
$\ell_{e}\neq\ell_{f}$. ∎

Put together, Propositions 4.4 and 4.5 imply Theorem 1.1.

###  4.2. The Pure Shard Monoid

We now know that the generators in $\mathcal{L}_{\mathrm{edge}}$ are indexed
by shards, so we may write $\ell_{\Sigma}$ to refer to the _shard loop_
indexed by the shard $\Sigma$ and write
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}=\mathcal{L}_{\mathrm{edge}}$.

######  Definition 4.6.

The _pure shard monoid_ $\mathbf{P}^{+}(\mathcal{H},B)$ is the submonoid of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ generated by
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$.

Our name for $\mathbf{P}^{+}(\mathcal{H},B)$ comes from the fact that the
generating set $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$ is indexed by shards and
the fact that if $\mathcal{H}$ is the reflection arrangement of a finite
Coxeter group $W$, then $\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ is isomorphic
to the pure braid group of $W$.

An _$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$ -word_ is a word over the alphabet
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$. By definition,
$\mathbf{P}^{+}(\mathcal{H},B)$ is the set of elements of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ represented by
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-words. We view
$\mathbf{P}^{+}(\mathcal{H},B)$ as a poset, where the partial order $\leq$ is
defined by saying $p\leq p^{\prime}$ if there is an
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word representing $p^{\prime}$ that
contains an $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word representing $p$ as a
prefix.

The _full twist_ is the element $\Delta^{2}$ of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ defined by

$\Delta^{2}:=\mathrm{gal}(B,-B)\cdot\mathrm{gal}(-B,B).$

For $\mathcal{H}$ a finite irreducible simplicial arrangement, it is known
that the center of $\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ is an infinite
cyclic group generated by $\Delta^{2}$ [Cor94]. We are especially interested
in $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$, the interval between the
identity element $\mathbbm{1}$ and the full twist $\Delta^{2}$ in
$\mathbf{P}^{+}(\mathcal{H},B)$. This interval is “tall and wide,” but it is
_not_ a lattice in general, preventing the use of Garside theory to study
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ (see Figure 10).

###  4.3. Proof of Theorem 1.2

To initiate the study of the pure shard monoid, we investigate the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ when $\mathcal{H}$ is an
arrangement of rank $2$ with $m$ hyperplanes. As in Figure 1, index the shards
of $\mathcal{H}$ as $\Sigma_{1},\Sigma_{2},\ldots,\Sigma_{2m}$ so that
$\Sigma_{1}=\Sigma_{m+1}$ and $\Sigma_{m}=\Sigma_{2m}$ are basic hyperplanes
and so that $H_{\Sigma_{i}}=H_{\Sigma_{m+i}}$ for all $1\leq i\leq m$. For
simplicity, let us write $H_{i}=H_{\Sigma_{i}}$. Index the generators of
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$ as $\mathcal{r}_{i}=\ell_{\Sigma_{i}}$
and $\mathcal{l}_{i}=\ell_{\Sigma_{m+i}}$, and note that
$\mathcal{r}_{1}=\mathcal{l}_{1}$ and $\mathcal{r}_{m}=\mathcal{l}_{m}$. Let
$\binom{[m]}{k}$ be the collection of $k$-element subsets of $[m]$. For each
$I=\\{i_{1}<\cdots<i_{k}\\}\in\binom{[m]}{k}$, let

$\mathcal{r}_{I}=\mathcal{r}_{i_{k}}\mathcal{r}_{i_{k-1}}\cdots\mathcal{r}_{i_{1}}\quad\text{and}\quad\mathcal{l}_{I}=\mathcal{l}_{i_{1}}\mathcal{l}_{i_{2}}\cdots\mathcal{l}_{i_{k}}.$

Figures 5 and 6 depict the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ when $m=4$ and $m=5$,
respectively.

A finite poset is called _planar_ if its Hasse diagram can be drawn in the
$xy$-plane so that each cover relation is represented by a curve with a
strictly increasing $y$-coordinate and so that no two curves representing
cover relations intersect except possibly at their endpoint. Our goal in this
subsection is to prove Theorem 1.2, which states that
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ is a planar lattice with rank
generating function
$1+\left(\sum_{k=1}^{m-1}\left(2\binom{m}{k}-2\right)q^{k}\right)+q^{m}$ and
with $m2^{m-2}$ maximal chains. We break the proof into Propositions 4.7, 4.8
and 4.10 below.

$\mathbbm{1}$$\mathcal{l}_{2}$$\mathcal{l}_{3}$$\mathcal{l}_{4}=\mathcal{r}_{4}$$\mathcal{l}_{1}=\mathcal{r}_{1}$$\mathcal{r}_{2}$$\mathcal{r}_{3}$$\mathcal{l}_{23}$$\mathcal{l}_{24}$$\mathcal{l}_{34}=\mathcal{r}_{43}$$\mathcal{r}_{42}$$\mathcal{r}_{41}$$\mathcal{l}_{14}$$\mathcal{l}_{13}$$\mathcal{l}_{12}=\mathcal{r}_{21}$$\mathcal{r}_{31}$$\mathcal{r}_{32}$$\mathcal{l}_{234}=\mathcal{r}_{432}$$\mathcal{r}_{431}$$\mathcal{r}_{421}$$\mathcal{l}_{134}$$\mathcal{l}_{124}$$\mathcal{l}_{123}=\mathcal{r}_{321}$$\Delta^{2}$
Figure 5. A redrawing of the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ in the pure shard monoid
$\mathbf{P}^{+}(\mathcal{H},B)$ from Figure 3, where elements are labeled
using the conventions of Section 4.3. Boxed elements are in the image of
$\mathsf{Pow}$ (see Section 8).

$\mathbbm{1}$$5$$\mathcal{l}_{4}$$\mathcal{l}_{3}$$\mathcal{l}_{2}$$\mathcal{r}_{4}$$\mathcal{r}_{3}$$\mathcal{r}_{2}$$1$$51$$52$$53$$45$$35$$34$$25$$24$$23$$15$$14$$13$$21$$31$$32$$41$$42$$43$$521$$531$$532$$541$$542$$345$$245$$235$$234$$145$$135$$134$$125$$124$$321$$421$$431$$432$$5321$$5421$$5431$$2345$$4321$$1235$$1245$$1345$$\Delta^{2}$

Figure 6. The interval $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ in the pure
shard monoid $\mathbf{P}^{+}(\mathcal{H},B)$, where $\mathcal{H}$ is a rank-2
arrangement with $m=5$ hyperplanes. Elements are labeled using the conventions
of Section 4.3. With the exception of singleton sets $I=\\{i\\}$ for $1<i<m$,
elements $\mathcal{r}_{I}$ are abbreviated as decreasing sequences, elements
$\mathcal{l}_{I}$ are abbreviated as increasing sequences, and boxed elements
are in the image of $\mathsf{Pow}$ (see Section 8) and may be viewed as
increasing or decreasing. Figure 7. The interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ in $\mathbf{P}^{+}(\mathcal{H},B)$
for the braid arrangement of type $A_{3}$. It has 152 elements and 588 maximal
chains. Large circles represent elements in the image of $\mathsf{Crackle}$.
This interval is not a lattice, as witnessed by the 4 elements that have been
drawn as stars.

Maximal chains in $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ correspond to
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-words for $\Delta^{2}$; we
will first characterize these words, showing that there are $m2^{m-2}$ of
them. Let us say a $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word is _unimodal_ if
it is of the form $\mathcal{l}_{I}\mathcal{r}_{[m]\setminus I}$, where
$I\subseteq[m]$ is such that $1\in I$ and $m\not\in I$. The name comes from
the fact that the indices in this word increase from $1$ up to $\max(I)$ and
then decrease from $m$ down to $\min([m]\setminus I)$. Note that there are
$2^{m-2}$ unimodal $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-words.

######  Proposition 4.7.

Let $\mathcal{H}$ be a rank-2 arrangement with $m$ hyperplanes. An
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word represents $\Delta^{2}$ if and only
if it is a cyclic rotation of a unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word. Hence, the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ has exactly $m2^{m-2}$ maximal
chains.

###### Proof.

We will write $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$ instead of
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$ since we will need to pass to
subarrangements. For each subarrangement we consider, we will employ the
quotient map from Lemma 4.1. One can check directly that the desired result
holds when $m=3$, so we may assume $m\geq 4$ and proceed by induction on $m$.
By direct computation, we find that $\Delta^{2}$ is represented by the
unimodal $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word
$\mathcal{l}_{[m-1]}\mathcal{r}_{m}=\mathcal{l}_{1}\mathcal{l}_{2}\cdots\mathcal{l}_{m-1}\mathcal{r}_{m}$.
Therefore, we can pass to the abelianization of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ and invoke Theorem 2.3 to see that for
each $1\leq i\leq m$, an
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word for $\Delta^{2}$ must
use exactly one of $\mathcal{r}_{i}$ or $\mathcal{l}_{i}$; in particular, such
a word has no repeated letters. Since $\Delta^{2}$ is in the center of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$, the set of
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-words for $\Delta^{2}$ is
closed under taking cyclic rotations. Thus, it suffices to prove that the
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-words for $\Delta^{2}$
that begin with $\mathcal{l}_{1}$ are precisely the unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-words. As illustrated in
Figure 8, each unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word can be obtained from
the word
$\mathcal{l}_{1}\mathcal{l}_{2}\cdots\mathcal{l}_{m-1}\mathcal{r}_{m}$ by
repeatedly using relations of the form
$\mathcal{l}_{i}\mathcal{l}_{i+1}\cdots\mathcal{l}_{m-1}\mathcal{r}_{m}=\mathcal{r}_{m}\mathcal{r}_{m-1}\cdots\mathcal{r}_{i}$,
which follow from Corollary 2.9. Hence, all unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-words represent
$\Delta^{2}$.

Now suppose we have an arbitrary
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word
$\mathcal{w}_{1}\cdots\mathcal{w}_{m}$ for $\Delta^{2}$ such that
$\mathcal{w}_{1}=\mathcal{l}_{1}$; we will show that it is unimodal. We
consider two cases depending on whether $\mathcal{r}_{2}$ or $\mathcal{l}_{2}$
appears in $\mathcal{w}_{1}\cdots\mathcal{w}_{m}$.

Case 1. Assume $\mathcal{w}_{p}=\mathcal{r}_{2}$ for some $p\geq 2$. We claim
that $p=m$. Suppose instead that $p<m$, and let $i$ be such that
$\mathcal{w}_{m}\in\\{\mathcal{l}_{i},\mathcal{r}_{i}\\}$. Consider the
subarrangement $\mathcal{A}=\\{H_{1},H_{2},H_{i}\\}$. Let
$\overline{\iota}_{\mathcal{A}}$ be the quotient map from Lemma 4.1. The basic
hyperplanes of $\mathcal{A}$ are $H_{1}$ and $H_{i}$, so
$\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{i})=\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{i})$.
It follows that
$\overline{\iota}_{\mathcal{A}}(\mathcal{w}_{1}\cdots\mathcal{w}_{m})=\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{1})\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{2})\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{i})$,
so
$\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{1})\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{2})\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{i})$
is an $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{A},B_{\mathcal{A}})$-word
representing $\overline{\iota}_{\mathcal{A}}(\Delta^{2})$, which is the full
twist in $\pi_{1}(\mathrm{Sal}(\mathcal{A}),B_{\mathcal{A}})$. However, this
word is not a cyclic shift of a unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{A},B_{\mathcal{A}})$-word, so
this contradicts our induction hypothesis. Hence, $p=m$.

We now know that our $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word
is of the form
$\mathcal{l}_{1}\mathcal{w}_{2}\cdots\mathcal{w}_{m-1}\mathcal{r}_{2}$. Let
$\mathcal{H}^{\prime}$ be the subarrangement $\mathcal{H}\setminus\\{H_{1}\\}$
of $\mathcal{H}$. The basic hyperplanes of $\mathcal{H}^{\prime}$ are $H_{2}$
and $H_{m}$, so
$\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{l}_{2})=\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{r}_{2})$.
Now,
$\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{2})\cdots\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{m-1})\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{r}_{2})$
is an
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}^{\prime},B_{\mathcal{H}^{\prime}})$-word
representing the full twist in
$\pi_{1}(\mathrm{Sal}(\mathcal{H}^{\prime}),B_{\mathcal{H}^{\prime}})$, so by
induction, it is a cyclic rotation of a unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}^{\prime},B_{\mathcal{H}^{\prime}})$-word.
This shows that the
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}^{\prime},B_{\mathcal{H}^{\prime}})$-word
$\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{r}_{2})\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{2})\cdots\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{m-1})$
is unimodal, so
$\mathcal{l}_{1}\mathcal{w}_{2}\cdots\mathcal{w}_{m-1}\mathcal{r}_{2}$ is a
unimodal $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word.

Case 2. Assume $\mathcal{w}_{p}=\mathcal{l}_{2}$ for some $p\geq 2$. We claim
that $p=2$. Suppose instead that $p>2$, and let $i$ be such that
$\mathcal{w}_{2}\in\\{\mathcal{l}_{i},\mathcal{r}_{i}\\}$. Consider the
subarrangement $\mathcal{A}=\\{H_{1},H_{2},H_{i}\\}$. The basic hyperplanes of
$\mathcal{A}$ are $H_{1}$ and $H_{i}$, so
$\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{i})=\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{i})$.
It follows that
$\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{1})\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{i})\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{2})$
is an $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{A},B_{\mathcal{A}})$-word
representing $\overline{\iota}_{\mathcal{A}}(\Delta^{2})$, which is the full
twist in $\pi_{1}(\mathrm{Sal}(\mathcal{A}),B_{\mathcal{A}})$. This word is
not a cyclic rotation of a unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{A},B_{\mathcal{A}})$-word, so
this contradicts our induction hypothesis. Hence, $p=2$.

We now know that our $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word
is of the form
$\mathcal{l}_{1}\mathcal{l}_{2}\mathcal{w}_{3}\cdots\mathcal{w}_{m}$. Let
$\mathcal{H}^{\prime}$ be the subarrangement $\mathcal{H}\setminus\\{H_{1}\\}$
of $\mathcal{H}$. The basic hyperplanes of $\mathcal{H}^{\prime}$ are $H_{2}$
and $H_{m}$, so
$\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{l}_{2})=\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{r}_{2})$.
Now,
$\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{l}_{2})\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{3})\cdots\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{m})$
is an
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}^{\prime},B_{\mathcal{H}^{\prime}})$-word
representing the full twist in
$\pi_{1}(\mathrm{Sal}(\mathcal{H}^{\prime}),B_{\mathcal{H}^{\prime}})$. By
induction, we know that
$\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{l}_{2})\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{3})\cdots\overline{\iota}_{\mathcal{H}^{\prime}}(\mathcal{w}_{m})$
is a unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}^{\prime},B_{\mathcal{H}^{\prime}})$-word.
It follows that
$\mathcal{l}_{1}\mathcal{l}_{2}\mathcal{w}_{3}\cdots\mathcal{w}_{m}$ is a
unimodal $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$-word. ∎

$\mathcal{l}_{1}\mathcal{l}_{2}\mathcal{l}_{3}\mathcal{l}_{4}\mathcal{r}_{5}$$\mathcal{l}_{1}\mathcal{l}_{2}\mathcal{l}_{3}\mathcal{r}_{5}\mathcal{r}_{4}$$\mathcal{l}_{1}\mathcal{l}_{2}\mathcal{r}_{5}\mathcal{r}_{4}\mathcal{r}_{3}$$\mathcal{l}_{1}\mathcal{r}_{5}\mathcal{r}_{4}\mathcal{r}_{3}\mathcal{r}_{2}$$\mathcal{l}_{1}\mathcal{l}_{2}\mathcal{l}_{4}\mathcal{r}_{5}\mathcal{r}_{3}$$\mathcal{l}_{1}\mathcal{l}_{3}\mathcal{l}_{4}\mathcal{r}_{5}\mathcal{r}_{2}$$\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{r}_{5}\mathcal{r}_{3}\mathcal{r}_{2}$$\mathcal{l}_{1}\mathcal{l}_{3}\mathcal{r}_{5}\mathcal{r}_{4}\mathcal{r}_{2}$
Figure 8. The $2^{m-2}=8$ different unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-words when $\mathcal{H}$ is a rank-2
hyperplane arrangement with $m=5$ hyperplanes. All other words representing
$\Delta^{2}$ are cyclic rotations of these.

######  Proposition 4.8.

Let $\mathcal{H}$ be a rank-2 arrangement with $m$ hyperplanes. The elements
of $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ are precisely the elements of
the form $\mathcal{l}_{I}$ and $\mathcal{r}_{I}$ for $I\subseteq[m]$, and all
of these elements are distinct except that
$\mathcal{l}_{\\{1,\ldots,k\\}}=\mathcal{r}_{\\{1,\ldots,k\\}}$ and
$\mathcal{l}_{\\{m-k+1,\ldots,m\\}}=\mathcal{r}_{\\{m-k+1,\ldots,m\\}}$ for
all $0\leq k\leq m$. In particular, the rank generating function of
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ is
$1+\left(\sum_{k=1}^{m-1}\left(2\binom{m}{k}-2\right)q^{k}\right)+q^{m}$.

###### Proof.

It is clear that there is $1$ element of rank $0$ (the identity $\mathbbm{1}$)
and that there is $1$ element of rank $m$ (the full twist $\Delta^{2}$). Now
fix $1\leq k\leq m-1$. For each $I\in\binom{[m]}{k}$, we can use the fact that
$\mathcal{r}_{1}=\mathcal{l}_{1}$ and $\mathcal{r}_{m}=\mathcal{l}_{m}$ to see
that $\mathcal{l}_{I}\mathcal{r}_{[m]\setminus I}$ and
$\mathcal{r}_{I}\mathcal{l}_{[m]\setminus I}$ (viewed as words) are cyclic
rotations of unimodal $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-words. Therefore,
it follows from Proposition 4.7 that $\mathcal{l}_{I}\leq\Delta^{2}$ and
$\mathcal{r}_{I}\leq\Delta^{2}$. If $I,I^{\prime}\in\binom{[m]}{k}$ as
distinct, then we can pass to the abelianiztion of
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$ and invoke Theorem 2.3 to see that
$\mathcal{l}_{I}\neq\mathcal{l}_{I^{\prime}}$,
$\mathcal{r}_{I}\neq\mathcal{r}_{I^{\prime}}$, and
$\mathcal{l}_{I}\neq\mathcal{r}_{I^{\prime}}$. It follows from Theorems 2.5
and 2.7 that $\mathcal{l}_{\\{1,\ldots,k\\}}=\mathcal{r}_{\\{1,\ldots,k\\}}$
and $\mathcal{l}_{\\{m-k+1,\ldots,m\\}}=\mathcal{r}_{\\{m-k+1,\ldots,m\\}}$.
We claim that if $I$ is not $\\{1,\ldots,k\\}$ or $\\{m-k+1,\ldots,m\\}$, then
$\mathcal{l}_{I}\neq\mathcal{r}_{I}$. It will then follow that we have found
$2\binom{m}{k}-2$ distinct elements of rank $k$ in
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$.

We can check the claim directly when $m=3$, so we may assume $m\geq 4$ and
proceed by induction on $m$. Suppose $I$ is not $\\{1,\ldots,k\\}$ or
$\\{m-k+1,\ldots,m\\}$. Let $x=\min(I)$. If there exist $y\in[m]\setminus I$
and $z\in I$ with $x<y<z$, then we can let
$\mathcal{A}=\\{H_{x},H_{y},H_{z}\\}$ and use the quotient map
$\overline{\iota}_{\mathcal{A}}$ from Lemma 4.1. In this case, it follows by
induction that
$\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{I})=\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{x})\overline{\iota}_{\mathcal{A}}(\mathcal{l}_{z})\neq\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{z})\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{x})=\overline{\iota}_{\mathcal{A}}(\mathcal{r}_{I})$,
so $\mathcal{l}_{I}\neq\mathcal{r}_{I}$. Now suppose no such $y$ and $z$
exist. This prohibits $1$ and $m$ from being in $I$. Let
$\mathcal{A}^{\prime}=\\{H_{1},H_{x},H_{m}\\}$, and consider the quotient map
$\overline{\iota}_{\mathcal{A}^{\prime}}$. In this case, it follows by
induction (or, more directly, from Lemma 4.2) that
$\overline{\iota}_{\mathcal{A}^{\prime}}(\mathcal{l}_{I})=\overline{\iota}_{\mathcal{A}^{\prime}}(\mathcal{l}_{x})\neq\overline{\iota}_{\mathcal{A}^{\prime}}(\mathcal{r}_{x})=\overline{\iota}_{\mathcal{A}^{\prime}}(\mathcal{r}_{I})$,
so $\mathcal{l}_{I}\neq\mathcal{r}_{I}$. This establishes the claim.

To complete the proof, we need to show that every element of rank $k$ in
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ is equal to $\mathcal{l}_{I}$ or
$\mathcal{r}_{I}$ for some $I\in\binom{[m]}{k}$. Thus, let us choose an
arbitrary element $u\in[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ of rank $k$.
Let $\mathcal{w}_{1}\cdots\mathcal{w}_{k}$ be an
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word representing $u$. Let
$I=\\{i_{1}<\cdots<i_{k}\\}\in\binom{[m]}{k}$ be such that
$\\{H_{i_{1}},\ldots,H_{i_{k}}\\}$ is the set of hyperplanes containing the
shards that index the shard loops $\mathcal{w}_{1},\ldots,\mathcal{w}_{k}$. It
is possible to extend the word $\mathcal{w}_{1}\cdots\mathcal{w}_{k}$ to a
word
$\mathcal{w}_{1}\cdots\mathcal{w}_{k}\mathcal{w}_{k+1}\cdots\mathcal{w}_{m}$
representing $\Delta^{2}$, and we know by Proposition 4.7 that
$\mathcal{w}_{1}\cdots\mathcal{w}_{k}\mathcal{w}_{k+1}\cdots\mathcal{w}_{m}$
is a cyclic rotation of a unimodal $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word.
If $1$ and $m$ are not in $I$, then this immediately implies that
$\mathcal{w}_{1}\cdots\mathcal{w}_{k}$ is either $\mathcal{l}_{I}$ or
$\mathcal{r}_{I}$. Let us now assume $1$ and $m$ are both in $I$; we omit the
proof in the case when exactly one of these indices is in $I$ because it is
very similar. Let us also assume $\mathcal{l}_{1}$ appears before
$\mathcal{r}_{m}$ in the word $\mathcal{w}_{1}\cdots\mathcal{w}_{k}$; the
other case is similar. Then $\mathcal{w}_{1}\cdots\mathcal{w}_{k}$ is of the
form
$\mathcal{r}_{K_{1}}\mathcal{l}_{1}\mathcal{l}_{K_{2}}\mathcal{r}_{m}\mathcal{r}_{K_{3}}$,
where $K_{1}\sqcup K_{2}\sqcup K_{3}\sqcup\\{1,m\\}=I$ and $x<y<z$ for all
$x\in K_{1}$, $y\in[m]\setminus I$, and $z\in K_{3}$. By repeatedly applying
relations of the form
$\mathcal{l}_{i}\mathcal{l}_{i+1}\cdots\mathcal{l}_{m-1}\mathcal{r}_{m}=\mathcal{r}_{m}\mathcal{r}_{m-1}\cdots\mathcal{r}_{i}$
(which follow from Corollary 2.9), we can transform
$\mathcal{r}_{K_{1}}\mathcal{l}_{1}\mathcal{l}_{K_{2}}\mathcal{r}_{m}\mathcal{r}_{K_{3}}$
into the word $\mathcal{r}_{K_{1}}\mathcal{l}_{1}\mathcal{l}_{K_{2}\cup
K_{3}}\mathcal{r}_{m}$ (see Example 4.9). Then, by repeatedly applying
relations of the form
$\mathcal{l}_{1}\mathcal{l}_{2}\cdots\mathcal{l}_{i}=\mathcal{r}_{i}\mathcal{r}_{i-1}\cdots\mathcal{r}_{2}\mathcal{l}_{1}$
(which follow from Corollary 2.9), we can transform
$\mathcal{r}_{K_{1}}\mathcal{l}_{1}\mathcal{l}_{K_{2}\cup
K_{3}}\mathcal{r}_{m}$ into $\mathcal{l}_{1}\mathcal{l}_{K_{1}\cup K_{2}\cup
K_{3}}\mathcal{r}_{m}$, which is the same as $\mathcal{l}_{I}$ because
$\mathcal{r}_{m}=\mathcal{l}_{m}$. Hence, $u$ is represented by
$\mathcal{l}_{I}$. ∎

######  Example 4.9.

Suppose $m=10$. The word
$\mathcal{r}_{2}\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{l}_{7}\mathcal{l}_{8}\mathcal{r}_{10}\mathcal{r}_{9}\mathcal{r}_{6}$
represents an element $u$ of rank $8$ in
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ because it is a prefix of
$\mathcal{r}_{2}\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{l}_{7}\mathcal{l}_{8}\mathcal{r}_{10}\mathcal{r}_{9}\mathcal{r}_{6}\mathcal{r}_{5}\mathcal{r}_{3}$,
which is a cyclic rotation of a unimodal
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word. In the notation of the proof of
Proposition 4.8, we have $I=\\{1,2,4,6,7,8,9,10\\}$, $K_{1}=\\{2\\}$,
$K_{2}=\\{4,7,8\\}$, and $K_{3}=\\{6,9\\}$. We can apply relations of the form
$\mathcal{l}_{i}\mathcal{l}_{i+1}\cdots\mathcal{l}_{9}\mathcal{r}_{10}=\mathcal{r}_{10}\mathcal{r}_{9}\cdots\mathcal{r}_{i}$
to find that

$\mathcal{r}_{2}\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{l}_{7}\mathcal{l}_{8}\mathcal{r}_{10}\mathcal{r}_{9}\mathcal{r}_{6}=\mathcal{r}_{2}\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{l}_{7}\mathcal{l}_{8}\mathcal{l}_{9}\mathcal{r}_{10}\mathcal{r}_{6}=\mathcal{r}_{2}\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{r}_{10}\mathcal{r}_{9}\mathcal{r}_{8}\mathcal{r}_{7}\mathcal{r}_{6}=\mathcal{r}_{2}\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{l}_{6}\mathcal{l}_{7}\mathcal{l}_{8}\mathcal{l}_{9}\mathcal{r}_{10};$

this last expression is
$\mathcal{r}_{K_{2}}\mathcal{l}_{1}\mathcal{l}_{K_{2}\cup
K_{3}}\mathcal{r}_{m}$. Applying the relation
$\mathcal{l}_{1}\mathcal{l}_{2}\cdots\mathcal{l}_{i}=\mathcal{r}_{i}\mathcal{r}_{i-1}\cdots\mathcal{r}_{2}\mathcal{l}_{1}$
when $i=2$, we find that

$\mathcal{r}_{2}\mathcal{l}_{1}\mathcal{l}_{4}\mathcal{l}_{6}\mathcal{l}_{7}\mathcal{l}_{8}\mathcal{l}_{9}\mathcal{r}_{10}=\mathcal{l}_{1}\mathcal{l}_{2}\mathcal{l}_{4}\mathcal{l}_{6}\mathcal{l}_{7}\mathcal{l}_{8}\mathcal{l}_{9}\mathcal{r}_{10};$

this last expression is the same as $\mathcal{l}_{I}$ because
$\mathcal{r}_{10}=\mathcal{l}_{10}$.

######  Proposition 4.10.

If $\mathcal{H}$ is a rank-2 arrangement, then the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ is a planar lattice.

###### Proof.

According to [Bir40, Ex. 7, p. 20], a finite planar poset with a unique
minimal element and a unique maximal element is automatically a lattice.
Hence, we just need to prove planarity. We are going to describe an explicit
drawing of the Hasse diagram of $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$,
and we will give a full description of the edges in this Hasse diagram. Using
the ideas discussed earlier in this subsection, one can verify that these are
in fact all of the edges.

Let $m$ be the number of hyperplanes in $\mathcal{H}$. Suppose $1\leq k\leq
m-1$. Proposition 4.8 gives us an explicit description of the elements of rank
$k$ in $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$; we will construct a
sequence that lists these elements. The sequence consists of four
subsequences, where the first two subsequences overlap in a single element and
the last two subsequences also overlap in a single element. The first
subsequence lists the elements $\mathcal{r}_{I}$ with $m\in I$ so that the
indexing subsets appear in lexicographically-increasing order; thus, this
subsequence starts with $\mathcal{r}_{\\{1,\ldots,k-1,m\\}}$ and ends with
$\mathcal{r}_{\\{m-k+1,\ldots,m\\}}$. The second subsequence lists the
elements $\mathcal{l}_{I}$ with $1\not\in I$ so that the indexing subsets
appear in lexicographically-decreasing order; this subsequence starts with
$\mathcal{l}_{\\{m-k+1,\ldots,m\\}}$ (which is the same as the last element in
the first subsequence) and ends with $\mathcal{l}_{\\{2,\ldots,k+1\\}}$. The
third subsequence lists the elements $\mathcal{r}_{I}$ with $m\not\in I$ so
that the indexing subsets appear in lexicographically-decreasing order; this
subsequence starts with $\mathcal{r}_{\\{m-k,\ldots,m-1\\}}$ and ends with
$\mathcal{r}_{\\{1,\ldots,k\\}}$. The fourth subsequence lists the elements
$\mathcal{l}_{I}$ with $1\in I$ so that the indexing subsets appear in
lexicographically-increasing order; this subsequence starts with
$\mathcal{l}_{\\{1,\ldots,k\\}}$ (which is the same as the last element in the
third subsequence) and ends with $\mathcal{l}_{\\{1,m-k+2,\ldots,m\\}}$.

Let us draw the Hasse diagram of $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$
so that elements of rank $k$ are drawn at height $k$ and so that all edges are
line segments. For each $1\leq k\leq m-1$, we can draw the elements of rank
$k$ so that they appear from left to right in the order specified by the
sequence in the preceding paragraph; see Figures 5 and 6. We claim that this
drawing is planar. To see this, we note that we can break this drawing of the
Hasse diagram into four pieces, where some pieces overlap on their boundaries.
We have a _northwest_ piece consisting of elements of the form
$\mathcal{r}_{I}$ with $m\in I$, a _southwest_ piece consisting of elements
$\mathcal{l}_{I}$ with $1\not\in I$, a _southeast_ piece consisting of
elements $\mathcal{r}_{I}$ with $m\not\in I$, and a _northeast_ piece
consisting of elements $\mathcal{l}_{I}$ with $1\in I$. If we can show that
each of these pieces is planar, then it will follow that the entire drawing of
the Hasse diagram is planar. We will prove that the southwest piece is planar;
the other pieces are similar.

For $2\leq i\leq m$, let $Q_{i}$ be the interval
$[\mathcal{l}_{i},\mathcal{l}_{\\{i,i+1,\ldots,m\\}}]_{\mathbf{P}^{+}}$. The
set of elements in the southwest piece is the disjoint union
$\\{\mathbbm{1}\\}\sqcup Q_{2}\sqcup\cdots\sqcup Q_{m}$. For $2\leq
i<i^{\prime}\leq m$, there are no cover relations between elements of $Q_{i}$
and elements of $Q_{i^{\prime}}$ unless $i^{\prime}=i+1$, in which case the
only such cover relation is
$\mathcal{l}_{\\{i+1,\ldots,m\\}}\lessdot\mathcal{l}_{\\{i,\ldots,m\\}}$.
Thus, it suffices to show that our drawing of each interval $Q_{i}$ is planar.
We do so by reverse induction on $i$, noting first that our drawing of $Q_{m}$
is certainly planar because $Q_{m}$ is the single element $\mathcal{l}_{m}$.
Now suppose $2\leq i\leq m-1$. The interval $Q_{i}$ is the disjoint union of
the two subintervals
$Q_{i}^{\swarrow}=[\mathcal{l}_{i},\mathcal{l}_{\\{i,i+2,i+3,\ldots,m\\}}]_{\mathbf{P}^{+}}$
and
$Q_{i}^{\nearrow}=[\mathcal{l}_{\\{i,i+1\\}},\mathcal{l}_{\\{i,\ldots,m\\}}]_{\mathbf{P}^{+}}$.
The subinterval $Q_{i}^{\swarrow}$ consists of all elements
$\mathcal{l}_{I}\in Q_{i}$ with $i+1\not\in I$, while the subinterval
$Q_{I}^{\nearrow}$ consists of all elements $\mathcal{l}_{I}\in Q_{i}$ with
$i+1\in I$. Both $Q_{i}^{\swarrow}$ and $Q_{i}^{\nearrow}$ are isomorphic to
$Q_{i+1}$, so by induction, they are planar (and our drawing represents them
in a planar manner). Finally, the only cover relations between elements of
$Q_{i}^{\swarrow}$ and elements of $Q_{i}^{\nearrow}$ are
$\mathcal{l}_{i}\lessdot\mathcal{l}_{\\{i,i+1\\}}$ and
$\mathcal{l}_{\\{i,i+2,i+3,\ldots,m\\}}\lessdot\mathcal{l}_{\\{i,\ldots,m\\}}$.
Thus, our drawing of $Q_{i}$ is planar. ∎

##  5\. Pop

As in Section 1.5, we consider the _pop-stack sorting operator_
$\mathsf{Pop}\colon L\to L$, where $L$ is a locally finite meet-semilattice,
defined by

(7) $\mathsf{Pop}(x):=x\wedge\bigwedge\\{y\in L:y\lessdot x\\}$

for all $x\in L$. For a simplicial arrangement $\mathcal{H}$ with
$C\in\mathcal{R}$, let

$\Sigma([\mathsf{Pop}(C),C]):=\left\\{\Sigma(D^{\prime}\lessdot
D):\mathsf{Pop}(C)\leq D^{\prime}\lessdot D\leq C\right\\}.$

In [Rea11], Reading provided the following characterization of the shard
intersection order using $\mathsf{Pop}$.

######  Proposition 5.1 ([Rea11, Proposition 5.7]).

Suppose $\mathcal{H}$ is a simplicial arrangement. For
$C,C^{\prime}\in\mathcal{R}$, we have

$C^{\prime}\preceq C\quad\text{if and only
if}\quad\Sigma([\mathsf{Pop}(C^{\prime}),C^{\prime}])\subseteq\Sigma([\mathsf{Pop}(C),C]),$

where $\mathsf{Pop}$ denotes the pop-stack sorting operator on the lattice
$\mathrm{Weak}(\mathcal{H},B)$.

Generalizing [STW22, Theorem 2.10.5], Reading gave a second characterization
of the shard intersection order.

######  Theorem 5.2 ([Rea16, Theorem 9-7.24],[STW22, Theorem 2.10.5]).

Suppose $\mathcal{H}$ is a simplicial arrangement. For
$C,C^{\prime}\in\mathcal{R}$, we have

$C^{\prime}\preceq C\text{ if and only if }C^{\prime}\leq C\text{ and
}\bigcap\mathrm{cov}(C^{\prime})\supseteq\bigcap\mathrm{cov}(C).$

##  6\. Crackle

We assume throughout this section that $\mathcal{H}$ is a simplicial real
arrangement with base region $B$.

###  6.1. Full Twists, Shard Generators, and Crackle

We have $[\mathsf{Pop}(-B),-B]=\mathrm{Weak}(\mathcal{H},B)$ since
$\mathsf{Pop}(-B)=B$ by [Ede84, Corollary 1.6]. Given a region
$C\in\mathcal{R}$, we can think of the interval $[\mathsf{Pop}(C),C]$ in
$\mathrm{Weak}(\mathcal{H},B)$ as a “nonstandard parabolic subarrangement” of
$\mathcal{H}$. This motivates us to define $\mathsf{Crackle}(C)$ to be the
“parabolic full twist” of this subarrangement.

######  Definition 6.1.

Let $C\in\mathcal{R}$. We define

$\mathsf{Crackle}(C):=\mathrm{gal}(B,\mathsf{Pop}(C))\cdot\mathrm{gal}(\mathsf{Pop}(C),C,\mathsf{Pop}(C))\cdot\mathrm{gal}(B,\mathsf{Pop}(C))^{-1}\in\pi_{1}(\mathrm{Sal}(\mathcal{H}),B).$

Definition 6.1 generalizes both the full twist and the shard loops of Section
4: on the one hand, we have $\mathsf{Crackle}(-B)=\Delta^{2}$ by construction,
so $\mathsf{Crackle}$ generalizes the full twist; on the other hand, if $J$ is
a join-irreducible region of $\mathrm{Weak}(\mathcal{H},B)$, then
$\mathsf{Pop}(J)$ is the unique region covered by $J$, so
$\mathsf{Crackle}(J)=\ell_{\Sigma(\mathsf{Pop}(J)\lessdot J)}$.

Our primary goal in this section is to prove Theorem 1.3, which uses
$\mathsf{Crackle}$ to characterize the shard intersection order. Recall that
this theorem states that $\mathsf{Crackle}$ is a poset embedding from
$\mathrm{Shard}(\mathcal{H},B)$ into
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ and is illustrated in Figures 3
and 7.

###  6.2. Crackle as a Product of Shard Generators

Our first task is to prove Proposition 6.3, which tells us that
$\mathsf{Crackle}$ maps $\mathcal{R}$ into the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ in the pure shard monoid.

######  Lemma 6.2.

Consider $E^{\prime},E\in\mathcal{R}$ with $E^{\prime}\leq E$. If

$E^{\prime}=E_{0}\xrightarrow{e_{1}}E_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}E_{k-1}\xrightarrow{e_{k}}E_{k}=E$

is a positive minimal gallery in $\mathrm{Sal}(\mathcal{H})$ from $E^{\prime}$
to $E$, then

$\mathrm{gal}(B,E^{\prime})\cdot\mathrm{gal}(E^{\prime},E,E^{\prime})\cdot\mathrm{gal}(B,E^{\prime})^{-1}=\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}.$

###### Proof.

We have

$\displaystyle\mathrm{gal}(B,E^{\prime})\cdot\mathrm{gal}(E^{\prime},E,E^{\prime})\cdot\mathrm{gal}(B,E^{\prime})^{-1}$
$\displaystyle=\mathrm{gal}(B,E^{\prime})\cdot\left(e_{1}\cdots
e_{k}e_{k}^{*}\cdots e_{1}^{*}\right)\cdot\mathrm{gal}(B,E^{\prime})^{-1}$
$\displaystyle\cong\prod_{i=k}^{1}\left(\mathrm{gal}(B,E^{\prime})\cdot\left(e_{1}\cdots
e_{i-1}e_{i}e_{i}^{*}e_{i-1}^{-1}\cdots
e_{k}^{-1}\right)\cdot\mathrm{gal}(B,E^{\prime})^{-1}\right)$
$\displaystyle=\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}.\qed$

######  Proposition 6.3.

Let $C\in\mathcal{R}$. If

$\mathsf{Pop}(C)=E_{0}\xrightarrow{e_{1}}E_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}E_{k-1}\xrightarrow{e_{k}}E_{k}=C$

is a positive minimal gallery in $\mathrm{Sal}(\mathcal{H})$ from
$\mathsf{Pop}(C)$ to $C$, then

$\mathsf{Crackle}(C)=\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}.$

Moreover, $\mathsf{Crackle}(C)\leq\Delta^{2}$ in
$\mathbf{P}^{+}(\mathcal{H},B)$.

###### Proof.

The first statement is immediate from Lemma 6.2 (with $E=C$ and
$E^{\prime}=\mathsf{Pop}(C)$). Now let
$B\xrightarrow{f_{1}}\cdots\xrightarrow{f_{r}}\mathsf{Pop}(C)$ be a positive
minimal gallery from $B$ to $\mathsf{Pop}(C)$, and let
$C\xrightarrow{f_{r+1}}\cdots\xrightarrow{f_{m}}-B$ be a positive minimal
gallery from $C$ to $-B$. Applying Lemma 6.2 (with $E^{\prime}=B$ and $E=-B$),
we find that

$\displaystyle\Delta^{2}=\mathsf{Crackle}(-B)$
$\displaystyle=\ell_{\Sigma(f_{m})}\cdots\ell_{\Sigma(f_{r+1})}\ell_{\Sigma(e_{k})}\cdots\ell_{\Sigma(e_{1})}\ell_{\Sigma(f_{r})}\cdots\ell_{\Sigma(f_{1})}$
$\displaystyle=\ell_{\Sigma(e_{k})}\cdots\ell_{\Sigma(e_{1})}\ell_{\Sigma(f_{r})}\cdots\ell_{\Sigma(f_{1})}\ell_{\Sigma(f_{m})}\cdots\ell_{\Sigma(f_{r+1})},$

where the last equality follows from the fact that $\Delta^{2}$ is central in
$\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$. This proves the second statement since
$\ell_{\Sigma(e_{k})}\cdots\ell_{\Sigma(e_{1})}=\mathsf{Crackle}(C)$. ∎

###  6.3. Intervals, Subarrangements, and Crackle

Suppose $\mathcal{A}$ is a subarrangement of $\mathcal{H}$. Let
$\iota_{\mathcal{A}}\colon\mathcal{R}\to\mathcal{R}_{\mathcal{A}}$ denote the
map that sends a region $D$ of $\mathcal{H}$ to the unique region of
$\mathcal{A}$ containing $D$, and let $\overline{\iota}_{\mathcal{A}}$ be the
quotient map from Lemma 4.1. For $A\in\mathcal{R}$, let $\mathcal{H}_{A}$ be
the subarrangement of $\mathcal{H}$ consisting of all hyperplanes that contain
the intersection of the lower covers of $A$; in this case, we write
$\mathcal{R}_{A}$, $B_{A}$, $\iota_{A}$, and $\overline{\iota}_{A}$ instead of
$\mathcal{R}_{\mathcal{A}}$, $B_{\mathcal{A}}$, $\iota_{\mathcal{A}}$, and
$\overline{\iota}_{\mathcal{A}}$. We will often intentionally confuse
$\mathcal{H}_{A}$ with its essentialization, which is simplicial; thus, it
also makes sense to write $\mathsf{Pop}_{A}$ and $\mathsf{Crackle}_{A}$ for
the pop-stack sorting operator and the crackle map on
$\mathrm{Weak}(\mathcal{H}_{A},B_{A})$.

######  Lemma 6.4.

For all $A,C\in\mathcal{R}$, we have

$\overline{\iota}_{A}(\mathsf{Crackle}(C))=\mathsf{Crackle}_{A}(\iota_{A}(C)).$

###### Proof.

The quotient $\overline{\iota}_{A}$ introduces the relation
$\ell_{\Sigma}=\mathbbm{1}$ when $H_{\Sigma}\not\in\mathcal{H}_{A}$ and the
relation $\ell_{\Sigma}=\ell_{\Sigma^{\prime}}$ when $\Sigma$ and
$\Sigma^{\prime}$ are contained in the same shard of $\mathcal{H}_{A}$, so the
desired identity follows from Proposition 6.3. ∎

For $C\in\mathcal{R}$, it follows from Reading’s work in [Rea11, Propositions
5.7 and 5.8] that there are two natural ways to identify some of the regions
of $\mathcal{H}$ with the regions of $\mathcal{H}_{C}$: the map $\iota_{C}$
restricts to a bijection $\iota_{C}^{\preceq}\colon\\{D\in\mathcal{R}:D\preceq
C\\}\to\mathcal{R}_{C}$ and also restricts to a bijection
$\iota_{C}^{\mathsf{Pop}}\colon[\mathsf{Pop}(C),C]\to\mathcal{R}_{C}$. The
next result, which is illustrated in Figure 9, also follows from Reading’s
work.

######  Theorem 6.5 ([Rea11, Propositions 5.7 and 5.8]).

Consider regions $C,D\in\mathcal{R}$ with $D\preceq C$. There is a poset
isomorphism

$\omega_{C}:[\mathsf{Pop}(D),D]\to[(\iota^{\mathsf{Pop}}_{C})^{-1}(\mathsf{Pop}_{C}(\iota_{C}(D))),(\iota^{\mathsf{Pop}}_{C})^{-1}(\iota_{C}(D))],$

where both intervals are taken in $\mathrm{Weak}(\mathcal{H},B)$. Suppose
$E^{\prime}\xrightarrow{e}E$ is an edge in $\mathrm{Sal}(\mathcal{H})$ such
that $\mathsf{Pop}(D)\leq E^{\prime}\lessdot E\leq D$, and let $\omega_{C}(e)$
denote the corresponding edge
$\omega_{C}(E^{\prime})\xrightarrow{\omega_{C}(e)}\omega_{C}(E)$. Then

$\Sigma(e)=\Sigma(\omega_{C}(e)).$

$\bullet$$\bullet$$\mathsf{Pop}(D)$$D$$\bullet$$-B$$\bullet$$C$$\bullet$$\mathsf{Pop}(C)$$\bullet$$B$$\bullet$$\bullet$$\omega_{C}(\mathsf{Pop}(D))$$\omega_{C}(D)$
Figure 9. An illustration of Theorem 6.5.
$C$$\mathsf{Pop}(C)$$D$$\mathsf{Pop}(D)$$\omega_{C}(D)$$B$ Figure 10. The
braid arrangement $\mathcal{H}$ of type $A_{3}$, drawn by intersecting
hyperplanes with a sphere and applying a stereographic projection.
Illustrating Theorem 6.5, we have indicated a region $C$, the region
$\mathsf{Pop}(C)$, a region $D\preceq C$, and the region $\omega_{C}(D)$. The
black hyperplanes are in $\mathcal{H}_{C}$, while the gray dashed hyperplanes
are not. The gray shading indicates the region
$\iota_{C}(D)\in\mathcal{R}_{C}$.

######  Lemma 6.6.

Let $C,D\in\mathcal{R}$ be such that $D\preceq C$. For every region
$E\in[\mathsf{Pop}(D),D]$, we have $E\leq\omega_{C}(E)$.

###### Proof.

It follows from Theorem 6.5 that
$\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(D)\subseteq\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(\omega_{C}(D))$,
so

$D=\bigvee_{\Sigma\in\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(D)}J_{\Sigma}\leq\bigvee_{\Sigma\in\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(\omega_{C}(D))}J_{\Sigma}=\omega_{C}(D)$

by Theorem 3.2. Now let $E\xrightarrow{e_{1}}\cdots\xrightarrow{e_{k}}D$ be a
positive minimal gallery from $E$ to $D$. Then
$\omega_{C}(E)\xrightarrow{\omega_{C}(e_{1})}\cdots\xrightarrow{\omega_{C}(e_{k})}\omega_{C}(D)$
is a positive minimal gallery from $\omega_{C}(E)$ to $\omega_{C}(D)$. For
each $1\leq i\leq k$, we have $H_{e_{i}}=H_{\omega_{C}(e_{i})}$ because
$\Sigma(e_{i})=\Sigma(\omega_{C}(e_{i}))$ by Theorem 6.5. It follows that

$\mathrm{inv}(E)=\mathrm{inv}(D)\setminus\\{H_{e_{1}},\ldots,H_{e_{k}}\\}\subseteq\mathrm{inv}(\omega_{C}(D))\setminus\\{H_{e_{1}},\ldots,H_{e_{k}}\\}=\mathrm{inv}(\omega_{C}(E)).$

so $E\leq\omega_{C}(E)$. ∎

The map
$(\iota_{C}^{\mathsf{Pop}})^{-1}\colon\mathcal{R}_{C}\to[\mathsf{Pop}(C),C]$
is defined on regions in $\mathcal{H}_{C}$; since a positive gallery $g$ in
$\mathrm{Sal}(\mathcal{H}_{C})$ can be viewed as a sequence of regions, it
makes sense to apply $(\iota_{C}^{\mathsf{Pop}})^{-1}$ to all of $g$, thereby
yielding a sequence of regions in the interval $[\mathsf{Pop}(C),C]$.

######  Lemma 6.7.

For each $C\in\mathcal{R}$, there is an injective group homomorphism

$\displaystyle\Phi_{C}:\pi_{1}(\mathrm{Sal}(\mathcal{H}_{C}),B_{C})$
$\displaystyle\hookrightarrow\pi_{1}(\mathrm{Sal}(\mathcal{H}),B)$
$\displaystyle g$
$\displaystyle\mapsto\mathrm{gal}(B,\mathsf{Pop}(C))\cdot(\iota_{C}^{\mathsf{Pop}})^{-1}(g)\cdot\mathrm{gal}(B,\mathsf{Pop}(C))^{-1}.$

If $D\preceq C$, then

$\Phi_{C}(\mathsf{Crackle}_{C}(\iota_{C}(D)))=\mathsf{Crackle}(D).$

###### Proof.

The first statement is immediate. Now suppose $D\preceq C$. Let

$\mathsf{Pop}(D)=E_{0}\xrightarrow{e_{1}}E_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}E_{k-1}\xrightarrow{e_{k}}E_{k}=D$

be a positive minimal gallery in $\mathrm{Sal}(\mathcal{H})$ from
$\mathsf{Pop}(D)$ to $D$, and let

$\omega_{C}(\mathsf{Pop}(D))=\omega_{C}(E_{0})\xrightarrow{\omega_{C}(e_{1})}\omega_{C}(E_{1})\xrightarrow{\omega_{C}(e_{2})}\cdots\xrightarrow{\omega_{C}(e_{k-1})}\omega_{C}(E_{k-1})\xrightarrow{\omega_{C}(e_{k})}\omega_{C}(E_{k})=\omega_{C}(D)$

be the corresponding positive minimal gallery obtained by applying the
isomorphism $\omega_{C}$ from Theorem 6.5. We have

$(\iota_{C}^{\mathsf{Pop}})^{-1}(B_{C})=\mathsf{Pop}(C),\quad(\iota_{C}^{\mathsf{Pop}})^{-1}(\mathsf{Pop}_{C}(\iota_{C}(D)))=\omega_{C}(\mathsf{Pop}(D)),\quad\text{and}\quad(\iota_{C}^{\mathsf{Pop}})^{-1}(\iota_{C}(D))=\omega_{C}(D).$

Consequently, since $\mathsf{Crackle}_{C}(\iota_{C}(D))$ is equal to

$\mathrm{gal}(B_{C},\mathsf{Pop}_{C}(\iota_{C}(D)))\cdot\mathrm{gal}(\mathsf{Pop}_{C}(\iota_{C}(D)),\iota_{C}(D),\mathsf{Pop}_{C}(\iota_{C}(D)))\cdot\mathrm{gal}(B_{C},\mathsf{Pop}_{C}(\iota_{C}(D)))^{-1}$

by definition, we can write
$(\iota_{C}^{\mathsf{Pop}})^{-1}(\mathsf{Crackle}_{C}(\iota_{C}(D)))$ as

$\mathrm{gal}(B,\omega_{C}(\mathsf{Pop}(D)))\cdot\mathrm{gal}(\omega_{C}(\mathsf{Pop}(D)),\omega_{C}(D),\omega_{C}(\mathsf{Pop}(D)))\cdot\mathrm{gal}(B,\omega_{C}(\mathsf{Pop}(D)))^{-1}.$

Setting $E^{\prime}=\omega_{C}(\mathsf{Pop}(D))$ and $E=\omega_{C}(D)$ in
Lemma 6.2 allows us to rewrite this gallery as

$\ell_{\Sigma(\omega_{C}(e_{k}))}\ell_{\Sigma(\omega_{C}(e_{k-1}))}\cdots\ell_{\Sigma(\omega_{C}(e_{1}))},$

and we know by Theorem 6.5 that this is equal to

$\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}.$

This completes the proof since Proposition 6.3 tells us that
$\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}=\mathsf{Crackle}(D)$.
∎

###  6.4. Proof of Theorem 1.3

Recall that we use Theorem 3.1 to write $J_{\Sigma}$ for the join-irreducible
region in $\mathrm{Weak}(\mathcal{H},B)$ corresponding to the shard $\Sigma$.

######  Proposition 6.8.

Let $\Sigma\in{\mbox{{Sh}}}(\mathcal{H},B)$ be a shard, and let
$C\in\mathcal{R}$ be a region. Then $J_{\Sigma}\preceq C$ if and only if there
exists an $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word representing
$\mathsf{Crackle}(C)$ that uses the shard loop $\ell_{\Sigma}$.

###### Proof.

First suppose $J_{\Sigma}\preceq C$. By Proposition 5.1, there is an edge
$D^{\prime}\xrightarrow{e}D$ with $\mathsf{Pop}(C)\leq D^{\prime}\lessdot
D\leq C$ such that $\Sigma(e)=\Sigma$. We can find a positive minimal gallery
from $\mathsf{Pop}(C)$ to $C$ that uses $e$, so it follows from Proposition
6.3 that there is an $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word representing
$\mathsf{Crackle}(C)$ that uses $\ell_{\Sigma}$.

To prove the converse, suppose we can write
$\mathsf{Crackle}(C)=\ell_{\Sigma_{1}}\cdots\ell_{\Sigma_{m}}$, where
$\Sigma_{i}=\Sigma$ for some $1\leq i\leq m$. Let
$\mathsf{Pop}(C)=E_{0}\xrightarrow{e_{1}}E_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}E_{k-1}\xrightarrow{e_{k}}E_{k}=C$
be a positive minimal gallery in $\mathrm{Sal}(\mathcal{H})$ from
$\mathsf{Pop}(C)$ to $C$; by Proposition 6.3, we have
$\mathsf{Crackle}(C)=\ell_{\Sigma(e_{k})}\cdots\ell_{\Sigma(e_{1})}$. When we
pass to the abelianization $H_{1}(\mathrm{Sal}(\mathcal{H}),B)$, we obtain the
equality
$\overline{\ell}_{H_{\Sigma_{1}}}\cdots\overline{\ell}_{H_{\Sigma_{m}}}=\overline{\ell}_{H_{\Sigma(e_{1})}}\cdots\overline{\ell}_{H_{\Sigma(e_{m})}}$.
Theorem 2.3 tells us that $H_{1}(\mathrm{Sal}(\mathcal{H}),B)$ is a free
abelian group with free generating set
$\\{\overline{\ell}_{H}\\}_{H\in\mathcal{H}}$, so $H_{\Sigma_{i}}$ must be one
of the hyperplanes $H_{\Sigma(e_{1})},\ldots,H_{\Sigma(e_{k})}$. In other
words, there exists $1\leq j\leq k$ such that the shards $\Sigma=\Sigma_{i}$
and $\Sigma(e_{j})$ belong to the same hyperplane $H_{\Sigma}$. It follows
from Proposition 5.1 that $J_{\Sigma(e_{j})}\preceq C$.

Suppose by way of contradiction that $J_{\Sigma}\not\preceq C$. Since
$\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(J_{\Sigma})=\\{\Sigma\\}$, it follows
from the definition of the shard intersection order that there exists a point
$x\in\bigcap\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)\setminus\Sigma$. On the
other hand, since $J_{\Sigma(e_{j})}\preceq C$, we have $x\in\Sigma(e_{j})$.
The shard $\Sigma$ is a polyhedral cone defined as the intersection of
$H_{\Sigma}$ with some closed half-spaces, where each closed half-space is
determined by a hyperplane that cuts $H_{\Sigma}$. Because $x\not\in\Sigma$,
at least one of these closed half-spaces does not contain $x$. In other words,
there exists a hyperplane $H\in\mathcal{H}$ that cuts $H_{\Sigma}$ such that
$\Sigma$ and $x$ are on opposite sides of $H$ and $x\not\in H$. There exists a
region $A\in\mathcal{R}$ such that $\mathcal{H}_{A}$ is the full rank-$2$
subarrangement of $\mathcal{H}$ containing $H_{\Sigma}$ and $H$. Then
$H_{\Sigma}$ is a basic hyperplane of $\mathcal{H}_{A}$; let $H^{\prime}$ be
the other basic hyperplane of $\mathcal{H}_{A}$. Then $H^{\prime}$ also cuts
$H_{\Sigma}$, and $\Sigma$ and $x$ are on opposite sides of $H^{\prime}$ (with
$x\not\in H^{\prime}$). Since $x\in\Sigma(e_{j})$, we find that $\Sigma$ and
$\Sigma(e_{j})$ are on opposite sides of $H$ and on opposite sides of
$H^{\prime}$. Either $H$ or $H^{\prime}$ is in $\mathrm{inv}(J_{\Sigma})$;
without loss of generality, say $H\in\mathrm{inv}(J_{\Sigma})$. Then $\Sigma$
and $B$ are on opposite sides of $H$, so $x$ and $B$ must be on the same side
of $H$. Because $x\in\bigcap\mathrm{cov}_{\text{{\mbox{{Sh}}}}}(C)$, this
means that $C$ and $B$ are on the same side of $H$; that is,
$H\not\in\mathrm{inv}(C)$.

Because $H_{\Sigma}\in\mathcal{H}_{A}$ and $\Sigma$ and $\Sigma(e_{j})$ are
contained in different shards of $\mathcal{H}_{A}$, it follows from Theorem
1.1 and Lemma 6.4 that $\overline{\iota}_{A}(\ell_{\Sigma})$ and
$\overline{\iota}_{A}(\ell_{\Sigma(e_{j})})$ are generators in
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}_{A},B_{A})$ and that

(8)
$\overline{\iota}_{A}(\ell_{\Sigma})\neq\overline{\iota}_{A}(\ell_{\Sigma(e_{j})}).$

Since $J_{\Sigma(e_{j})}\preceq C$, we have $H_{\Sigma}\in\mathrm{inv}(C)$ and
$H\not\in\mathrm{inv}(C)$, so $\iota_{A}(C)$ is join-irreducible in
$\mathrm{Weak}(\mathcal{H}_{A},B_{A})$. This means that
$\mathsf{Crackle}_{A}(\iota_{A}(C))$ is a single generator in
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}_{A},B_{A})$. According to
Lemma 6.4, $\overline{\iota}_{A}(\mathsf{Crackle}(C))$ is a single generator
in $\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}_{A},B_{A})$. We have
assumed that there is an expression
$\mathsf{Crackle}(C)=\ell_{\Sigma_{1}}\cdots\ell_{\Sigma_{m}}$ with
$\Sigma=\Sigma_{i}$; if we apply the quotient map $\overline{\iota}_{A}$ to
this expression, then, by Lemma 4.1, we obtain an expression for
$\overline{\iota}_{A}(\mathsf{Crackle}(C))$ as a product of elements of
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}_{A},B_{A})\cup\\{\mathbbm{1}\\}$
such that one of the elements is $\overline{\iota}_{A}(\ell_{\Sigma})$. Hence,
$\overline{\iota}_{A}(\ell_{\Sigma})=\overline{\iota}_{A}(\mathsf{Crackle}(C))$.
On the other hand, $J_{\Sigma(e_{j})}\preceq C$, so we know by the first
paragraph of the proof that there is an
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word for $\mathsf{Crackle}(C)$ using
$\ell_{\Sigma(e_{j})}$. Applying $\overline{\iota}_{A}$ to this expression
yields an expression for $\overline{\iota}_{A}(\mathsf{Crackle}(C))$ as a
product of elements of
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}_{A},B_{A})\cup\\{\mathbbm{1}\\}$
such that one of the elements is $\overline{\iota}_{A}(\ell_{\Sigma(e_{j})})$.
Hence,
$\overline{\iota}_{A}(\ell_{\Sigma(e_{j})})=\overline{\iota}_{A}(\mathsf{Crackle}(C))$.
This shows that
$\overline{\iota}_{A}(\ell_{\Sigma})=\overline{\iota}_{A}(\ell_{\Sigma(e_{j})})$,
which contradicts Equation 8. ∎

We can now complete the proof of Theorem 1.3. We saw in Proposition 6.3 that
$\mathsf{Crackle}$ maps $\mathrm{Weak}(\mathcal{H},B)$ into the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$; we are left to show that it is a
poset embedding.

Suppose first that $C,D\in\mathcal{R}$ are such that $D\preceq C$. We can
restrict to the subarrangement $\mathcal{H}_{C}$ and note that
$\iota_{C}(C)=-B_{C}$. By Proposition 6.3,
$\mathsf{Crackle}_{C}(\iota_{C}(D))\leq\mathsf{Crackle}_{C}(\iota_{C}(C))$ in
the pure shard monoid $\mathbf{P}^{+}(\mathcal{H}_{C},B_{C})$. Thus, there
exist $\ell_{e_{1}},\ldots,\ell_{e_{k}},\ell_{e_{k+1}},\ldots,\ell_{e_{m}}$ in
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H}_{C},B_{C})$ such that

$\ell_{e_{1}}\cdots\ell_{e_{k}}=\mathsf{Crackle}_{C}(\iota_{C}(D))\quad\text{and}\quad\ell_{e_{1}}\cdots\ell_{e_{k}}\ell_{e_{k+1}}\cdots\ell_{e_{m}}=\mathsf{Crackle}_{C}(\iota_{C}(C)).$

For $1\leq i\leq m$, note that
$\Phi_{C}(\ell_{e_{i}})=\mathrm{gal}(B,\mathsf{Pop}(C))\cdot(\iota_{C}^{\mathsf{Pop}})^{-1}(\ell_{e_{i}})\cdot\mathrm{gal}(B,\mathsf{Pop}(C))^{-1}$
is one of the shard loops in
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}(\mathcal{H},B)$. Because $D\preceq C$,
Lemma 6.7 tells us that

$\mathsf{Crackle}(D)=\Phi_{C}(\mathsf{Crackle}_{C}(\iota_{C}(D)))=\Phi_{C}(\ell_{e_{1}})\cdots\Phi_{C}(\ell_{e_{k}})$

and

$\mathsf{Crackle}(C)=\Phi_{C}(\mathsf{Crackle}_{C}(\iota_{C}(C)))=\Phi_{C}(\ell_{e_{1}})\cdots\Phi_{C}(\ell_{e_{k}})\Phi_{C}(\ell_{e_{k+1}})\cdots\Phi_{C}(\ell_{e_{m}}).$

This proves that $\mathsf{Crackle}(D)\leq\mathsf{Crackle}(C)$ in
$\mathbf{P}^{+}(\mathcal{H},B)$.

To prove the converse, assume $\mathsf{Crackle}(D)\leq\mathsf{Crackle}(C)$ in
$\mathbf{P}^{+}(\mathcal{H},B)$. Then every
$\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word representing $\mathsf{Crackle}(D)$
can be extended to an $\mathcal{L}_{\text{{\mbox{{Sh}}}}}$-word representing
$\mathsf{Crackle}(C)$. According to Proposition 6.8, we have

$\\{\Sigma\in{\mbox{{Sh}}}:J_{\Sigma}\preceq
D\\}\subseteq\\{\Sigma\in{\mbox{{Sh}}}:J_{\Sigma}\preceq C\\}.$

It follows from Proposition 5.1 that

$\\{\Sigma\in{\mbox{{Sh}}}:J_{\Sigma}\preceq
D\\}=\Sigma([\mathsf{Pop}(D),D])\quad\text{and}\quad\\{\Sigma\in{\mbox{{Sh}}}:J_{\Sigma}\preceq
C\\}=\Sigma([\mathsf{Pop}(C),C]).$

Therefore, Proposition 5.1 tells us that $D\preceq C$, as desired.

##  7\. Snap

We now specialize to the setting of reflection arrangements of finite Coxeter
groups and prove Corollary 1.4 and Theorem 1.5. We will also collect
corollaries specializing these results to sortable elements and noncrossing
partitions.

###  7.1. Coxeter Groups and Braid Groups

Let $\mathcal{H}$ be the reflection arrangement of a finite Coxeter group $W$.
Then $\mathcal{H}$ is simplicial, and $W$ can be seen as the group of
orthogonal transformations of $\mathbb{R}^{n}$ generated by the reflections
through the hyperplanes in $\mathcal{H}$. There is bijection between $W$ and
$\mathcal{R}$ that maps an element $w\in W$ to the region $w(B)$, where $B$ is
the fixed base region of $\mathcal{H}$. We use this bijection to identify
regions of $\mathcal{H}$ with elements of $W$.

Let $S$ be the set of simple reflections of $W$. Then $W$ has a presentation
of the form $\langle S:(ss^{\prime})^{m(s,s^{\prime})}=\mathbbm{1}\rangle$,
where $\mathbbm{1}$ is the identity element of $W$, $m(s,s)=1$ for all $s\in
S$, and $m(s,s^{\prime})=m(s^{\prime},s)\in\\{2,3,\ldots\\}\cup\\{\infty\\}$
for all distinct $s,s^{\prime}\in S$. Given symbols $\alpha,\beta$ and a
nonnegative integer $r$, we write $[\alpha|\beta]_{r}$ for the word
$\alpha\beta\alpha\cdots$ of length $r$ that starts with $\alpha$ and
alternates between $\alpha$ and $\beta$. Thus, the braid relations of $W$ can
be written as
$[s|s^{\prime}]_{m(s,s^{\prime})}=[s^{\prime}|s]_{m(s,s^{\prime})}$. For each
$s\in S$. let ${\bf s}$ be a formal copy of $s$. The braid group
$\mathbf{B}(W)$ has a generating set ${\bf S}=\\{{\bf s}:s\in S\\}$ and
presentation $\langle{\bf S}:(ss^{\prime})^{m(s,s^{\prime})}=\mathbbm{1}\text{
for }s\neq s^{\prime}\rangle$. Thus, the generators in $\bf S$ have infinite
order in $\mathbf{B}(W)$, while the elements of $S$ are involutions in $W$.
The _positive braid monoid_ of $W$ is the monoid $\mathbf{B}^{+}(W)$ generated
by ${\bf S}$. There is a natural homomorphic quotient map
$\varphi\colon\mathbf{B}(W)\to W$ defined by $\varphi({\bf s})=s$ for all
$s\in S$. The _pure braid group_ of $W$, denoted $\mathbf{P}(W)$, is the
kernel of this map: $\mathbf{P}(W)=\ker(\varphi)$.

A _reduced word_ for an element $w\in W$ is a word in the alphabet $S$ that
represents $w$ and has minimum length among all such words. The length of a
reduced word for $w$ is called the _length_ of $w$ and is denoted by
$\mathrm{len}(w)$. The (right) _weak order_ on $W$ is defined by saying $u\leq
v$ if there is a reduced word for $v$ that contains a reduced word for $u$ as
a prefix. This defines a poset $\mathrm{Weak}(W)$, which coincides with
$\mathrm{Weak}(\mathcal{H},B)$ under the identification of $W$ with
$\mathcal{R}$. For each $w\in W$, there is a natural _lift_ ${\bf
w}\in\mathbf{B}^{+}(W)$ obtained by taking a reduced word for $w$ and
replacing each simple reflection $s$ with the corresponding generator ${\bf
s}$. We often denote the lift of an element of $W$ using bold, but we will
also sometimes write $\mathrm{lift}(w)$ for the lift of $w$ when we are wary
about the possibility of confusion. An _${\bf S}$ -word_ for an element
$\mathbf{w}\in\mathbf{B}^{+}(W)$ is a word in the alphabet ${\bf S}$ that
represents $\mathbf{w}$. We can also define the _weak order_ on
$\mathbf{B}^{+}(W)$ by saying ${\bf u}\leq{\bf v}$ if there is an ${\bf
S}$-word for ${\bf v}$ that contains an ${\bf S}$-word for ${\bf u}$ as a
prefix. We write $\mathrm{Weak}(\mathbf{B}^{+}(W))$ for the poset
$(\mathbf{B}^{+}(W),\leq)$. For ${\bf u},{\bf v}\in\mathbf{B}^{+}(W)$ with
${\bf u}\leq{\bf v}$, we write $[{\bf u},{\bf v}]_{\mathbf{B}^{+}}$ for the
interval between $\bf u$ and $\bf v$ in $\mathrm{Weak}(\mathbf{B}^{+}(W))$.

A _reflection_ in $W$ is an element of the form $usu^{-1}$, where $u\in W$ and
$s\in S$. Reflections are precisely the elements that, when viewed as
orthogonal transformations of $\mathbb{R}^{n}$, are reflections through
hyperplanes in $\mathcal{H}$. An _inversion_ of an element $w\in W$ is a
reflection $t$ in $W$ such that $\mathrm{len}(tw)<\mathrm{len}(w)$; such an
inversion is called a _cover reflection_ of $w$ if $tw$ is covered by $w$ in
the weak order. Let $\mathrm{inv}(w)$ and $\mathrm{cov}(w)$ be the set of
inversions and the set of cover reflections of $w$, respectively. Write
$\langle\mathrm{cov}(w)\rangle$ for the parabolic subgroup of $W$ generated by
$\mathrm{cov}(w)$.

For $u,v\in W$, we have $u\leq v$ in the weak order if and only if
$\mathrm{inv}(u)\subseteq\mathrm{inv}(v)$. Furthermore, a reformulation of
Theorem 5.2 in this context states that

(9) $u\preceq v\quad\text{if and only
if}\quad\mathrm{inv}(u)\subseteq\mathrm{inv}(v)\text{ and
}\langle\mathrm{cov}(u)\rangle\subseteq\langle\mathrm{cov}(v)\rangle.$

###  7.2. Pop and Crackle for Coxeter Groups

A _descent_ of an element $w\in W$ is a simple reflection $s$ such that there
is a reduced word for $w$ that ends with $s$. Similarly, a descent of an
element ${\bf w}\in\mathbf{B}^{+}(W)$ is a generator ${\bf s}\in{\bf S}$ such
that there is an ${\bf S}$-word for ${\bf w}$ that ends with ${\bf s}$. We
write $\mathrm{des}(w)$ and $\mathrm{des}({\bf w})$ for the set of descents of
$w$ and the set of descents of ${\bf w}$, respectively. Given a subset
$J\subseteq S$, we write $w_{\circ}(J)$ for the longest element of the
parabolic subgroup of $W$ generated by $J$. Since $\mathrm{Weak}(W)$ and
$\mathrm{Weak}(\mathbf{B}^{+}(W))$ are locally finite meet-semilattices, they
come equipped with pop-stack sorting operators; we denote both of these
operators by $\mathsf{Pop}$. If $w\in W$ and ${\bf w}=\mathrm{lift}(w)$, then
$\mathrm{des}({\bf w})$ is the set of lifts of descents of $w$. We have
$\mathsf{Pop}(w)=ww_{\circ}(\mathrm{des}(w))$, so it follows that

$\mathsf{Pop}({\bf w})={\bf w}\cdot({\bf
w}_{\circ}(\mathrm{des}(w)))^{-1}=\mathrm{lift}(\mathsf{Pop}(w)),$

where ${\bf
w}_{\circ}(\mathrm{des}(w))=\mathrm{lift}(w_{\circ}(\mathrm{des}(w)))$.

There is a natural action of $W$ on
$\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}}$, and the braid group
$\mathbf{B}(W)$ is isomorphic to the fundamental group
$\pi_{1}((\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}})/W,x_{B})$. The pure
braid group $\mathbf{P}(W)$ is isomorphic to
$\pi_{1}(\mathbb{C}^{n}\setminus\mathcal{H}_{\mathbb{C}},x_{B})$. The
identification of $W$ with $\mathcal{R}$ allows us to naturally label the
edges of $\mathrm{Sal}(\mathcal{H})$ by the generators in $\mathbf{S}$. More
precisely, if $w\in W$ and $s\in S\setminus\mathrm{des}(w)$, then we label the
edges $w\xrightarrow{e}ws$ and $ws\xrightarrow{e^{*}}w$ with the generator
${\bf s}\in{\bf S}$. This labeling allows us to rephrase notions and results
about crackle maps from Section 6 in the language of braid groups. For
example, if $w\in W$ and $\mathbf{w}=\mathrm{lift}(w)$, then

$\displaystyle\mathsf{Crackle}(w)$
$\displaystyle=\mathsf{Pop}(\mathbf{w})\cdot\left(\mathbf{w}_{\circ}(\mathrm{des}(w))\right)^{2}\cdot\mathsf{Pop}(\mathbf{w})^{-1}\in\mathbf{P}^{+}(W)\subseteq\mathbf{B}(W),$

where
$\mathbf{w}_{\circ}(\mathrm{des}(w))=\mathrm{lift}(w_{\circ}(\mathrm{des}(w)))$.

######  Proposition 7.1.

The pure shard monoid $\mathbf{P}^{+}(W)$ is the submonoid of $\mathbf{P}(W)$
generated by

$\\{\mathsf{Crackle}(j):j\in W,|\mathrm{des}(j)|=1\\}.$

###### Proof.

An element $j\in W$ has exactly 1 descent if and only if it is join-
irreducible in $\mathrm{Weak}(W)$. Theorem 3.1 tells us that there is
bijection $\Sigma\mapsto J_{\Sigma}$ from ${\mbox{{Sh}}}(\mathcal{H},B)$ to
the set of join-irreducible elements of $W$ (which we identify with regions of
$\mathcal{H}$). Moreover, $\mathsf{Crackle}(J_{\Sigma})=\ell_{\Sigma}$. Hence,
the result follows from the original definition of the pure shard monoid. ∎

By Theorem 1.3, $\mathsf{Crackle}$ is a poset embedding from
$\mathrm{Shard}(W)$ into $\mathbf{P}^{+}(W)$.

###  7.3. Snap

Let $w_{\circ}$ be the long element of $W$. The standard lift map
$\mathrm{lift}\colon W\to\mathbf{B}^{+}(W)$ is a poset isomorphism from
$\mathrm{Weak}(W)$ to the interval
$[\mathbbm{1},\mathbf{w}_{\circ}]_{\mathbf{B}^{+}}$ in
$\mathrm{Weak}(\mathbf{B}^{+}(W))$, where
$\mathbf{w}_{\circ}=\mathrm{lift}(w_{\circ})$. We now define a nonstandard
lift from $W$ to $\mathbf{B}^{+}(W)$.

######  Definition 7.2.

The _snap map_ $\mathsf{Snap}\colon W\to\mathbf{B}^{+}(W)$ is defined by

$\displaystyle\mathsf{Snap}(w)$
$\displaystyle:=\mathsf{Pop}(\mathbf{w})\cdot(\mathbf{w}_{\circ}(\mathrm{des}(w)))^{2},$

where $\mathbf{w}=\mathrm{lift}(w)$ and
$\mathbf{w}_{\circ}(\mathrm{des}(w))=\mathrm{lift}(w_{\circ}(\mathrm{des}(w)))$.

Observe that
$\mathsf{Snap}(w)=\mathbf{w}\cdot\mathbf{w}_{\circ}(\mathrm{des}(w))$ and that
$\varphi(\mathsf{Snap}(w))=\mathsf{Pop}(w)$. Our aim in this section is to
prove Theorem 1.5, which states that $\mathsf{Snap}$ is a poset embedding from
$\mathrm{Shard}(W)$ into $[\mathbbm{1},\Delta^{2}]_{\mathbf{B}^{+}}$. This is
illustrated in Figure 4, which shows
$[\mathbbm{1},\Delta^{2}]_{\mathbf{B}^{+}}$ when $W=I_{2}(4)$ is the dihedral
group of order $8$.

The first step in this endeavor is the following lemma, which will allow us to
consider inversion multisets of elements of $\mathbf{B}^{+}(W)$.

######  Lemma 7.3.

Let ${\bf w}\in\mathbf{B}^{+}(W)$, and let ${\bf s}_{1}\cdots{\bf s}_{r}$ and
${\bf s}_{1}^{\prime}\cdots{\bf s}_{r}^{\prime}$ be two ${\bf S}$-words for
$\mathbf{w}$. The multisets

$\\{s_{1}\cdots s_{k-1}s_{k}s_{k-1}\cdots s_{1}:1\leq k\leq
r\\}\quad\text{and}\quad\\{s_{1}^{\prime}\cdots
s_{k-1}^{\prime}s_{k}^{\prime}s_{k-1}^{\prime}\cdots s_{1}^{\prime}:1\leq
k\leq r\\}$

are equal.

###### Proof.

It suffices to prove the result when the two ${\bf S}$-words differ by a braid
move. In fact, it suffices to prove the result when the braid move changes the
entire first word into the entire second word. Thus, we can assume the first
word is $[{\bf s}|{\bf s}^{\prime}]_{m(s,s^{\prime})}$ and the second word is
$[{\bf s}^{\prime}|{\bf s}]_{m(s,s^{\prime})}$. In this case, the result
follows from the observation that
$[s|s^{\prime}]_{k}([s|s^{\prime}]_{k-1})^{-1}=[s^{\prime}|s]_{m(s,s^{\prime})+1-k}([s^{\prime}|s]_{m(s,s^{\prime})-k})^{-1}$
for all $1\leq k\leq m(s,s^{\prime})$. ∎

For each $\mathbf{w}\in\mathbf{B}^{+}(W)$, Lemma 7.3 allows us to define the
multiset

$\mathrm{Inv}(\mathbf{w})=\\{s_{1}\cdots s_{k-1}s_{k}s_{k-1}\cdots s_{1}:1\leq
k\leq r\\},$

where ${\bf s}_{1}\cdots{\bf s}_{r}$ is an ${\bf S}$-word for $\mathbf{w}$. If
$\mathbf{w}=\mathrm{lift}(w)$ for some $w\in W$, then $s_{1}\cdots s_{r}$ is a
reduced word for $w$; in this case, the multiset $\mathrm{Inv}(\mathbf{w})$ is
actually a set, and it is equal to the inversion set $\mathrm{inv}(w)$.

######  Lemma 7.4.

Let $w\in W$. Let $s_{1}\cdots s_{m}$ be a reduced word for $\mathsf{Pop}(w)$,
and let $s_{m+1}\cdots s_{r}$ be a reduced word for
$w_{\circ}(\mathrm{des}(w))$ so that $s_{1}\cdots s_{r}$ is a reduced word for
$w$. Let $t_{k}=s_{1}\cdots s_{k-1}s_{k}s_{k-1}\cdots s_{1}$. Then
$\mathrm{Inv}(\mathsf{Snap}(w))=\\{t_{1},\ldots,t_{m},t_{m+1},\ldots,t_{r},t_{r},\ldots,t_{m+1}\\}$.
Furthermore, $\mathrm{inv}(w)=\\{t_{1},\ldots,t_{r}\\}$, and
$t_{m+1},\ldots,t_{r}$ are precisely the reflections appearing in the
parabolic subgroup $\langle\mathrm{cov}(w)\rangle$.

###### Proof.

Since $w_{\circ}(\mathrm{des}(w))$ is an involution, it also has $s_{r}\cdots
s_{m+1}$ as a reduced word, so

$\mathbf{s}_{1}\cdots\mathbf{s}_{m}\mathbf{s}_{m+1}\cdots\mathbf{s}_{r}\mathbf{s}_{r}\cdots\mathbf{s}_{m+1}$

is an ${\bf S}$-word for $\mathsf{Snap}(w)$. Therefore, to prove the first
statement, we just need to show that for each $m+1\leq k\leq r$, the
reflection $t_{k}$ is equal to $s_{1}\cdots s_{r}s_{r}\cdots
s_{k+1}s_{k}s_{k+1}\cdots s_{r}s_{r}\cdots s_{1}$. We can group the terms in
this expression as

$s_{1}\cdots s_{k-1}(s_{k}\cdots s_{r})(s_{r}\cdots s_{k})(s_{k+1}\cdots
s_{r})(s_{r}\cdots s_{k+1})s_{k}\cdots s_{1}$

to make it clear that it is indeed equal to $t_{k}$.

It is well known that $\mathrm{inv}(w)=\\{t_{1},\ldots,t_{r}\\}$. For the last
statement, note that, in the language of hyperplanes, the reflections
$t_{m+1},\ldots,t_{r}$ correspond to the hyperplanes that separate the region
$\mathsf{Pop}(w)$ from the region $w$; these are the reflections in
$\langle\mathrm{cov}(w)\rangle$. ∎

######  Example 7.5.

Suppose $W$ is the symmetric group $\mathfrak{S}_{4}$ so that
$S=\\{s_{1},s_{2},s_{3}\\}$, where $s_{i}$ is the simple transposition
$(i\,\,i+1)$. We have $m(s_{1},s_{2})=m(s_{2},s_{3})=3$ and
$m(s_{1},s_{3})=2$. Let $w=s_{1}s_{2}s_{3}s_{2}$. Then
$\mathrm{des}(w)=\\{s_{2},s_{3}\\}$,
$w_{\circ}(\mathrm{des}(w))=s_{2}s_{3}s_{2}$, and $\mathsf{Pop}(w)=s_{1}$.
Now,
$\mathsf{Snap}(w)=\mathbf{w}\,\mathbf{w}_{\circ}(\mathrm{des}(w))=\mathbf{s}_{1}\mathbf{s}_{2}\mathbf{s}_{3}\mathbf{s}_{2}\mathbf{s}_{2}\mathbf{s}_{3}\mathbf{s}_{2}$,
so

$\displaystyle\mathrm{Inv}(\mathsf{Snap}(w))$
$\displaystyle=\\{s_{1},s_{1}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{3}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{2}s_{2}s_{3}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{2}s_{3}s_{2}s_{2}s_{3}s_{2}s_{1},$
$\displaystyle\hskip
17.92537pts_{1}s_{2}s_{3}s_{2}s_{2}s_{3}s_{2}s_{3}s_{2}s_{2}s_{3}s_{2}s_{1}\\}$
$\displaystyle=\\{s_{1},s_{1}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{3}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{3}s_{2}s_{1},s_{1}s_{2}s_{3}s_{2}s_{1},s_{1}s_{2}s_{1}\\}$
$\displaystyle=\\{(1\,\,2),(1\,\,3),(1\,\,4),(3\,\,4),(3\,\,4),(1\,\,4),(1\,\,3)\\}.$

###  7.4. Proof of Theorem 1.5

We are now going to prove that $\mathsf{Snap}$ is a poset embedding of
$\mathrm{Weak}(W)$ into $\mathrm{Weak}(\mathbf{B}^{+}(W))$. We know that
$\mathsf{Snap}(w_{\circ})=\mathbf{w}_{\circ}^{2}=\Delta^{2}$, where
$w_{\circ}$ is the long element of $W$, $\mathbf{w}_{\circ}$ is the lift of
$w_{\circ}$, and $\Delta^{2}$ is the full twist. Therefore, it will follow
immediately that $\mathsf{Snap}(W)$ is contained in the interval
$[\mathbbm{1},\Delta^{2}]_{\mathbf{B}^{+}}$.

Suppose first that $u,v\in W$ are such that
$\mathsf{Snap}(u)\leq\mathsf{Snap}(v)$. Then there is an ${\bf S}$-word for
$\mathsf{Snap}(v)$ that contains an ${\bf S}$-word for $\mathsf{Snap}(u)$ as a
prefix. This implies that the multiset $\mathrm{Inv}(\mathsf{Snap}(u))$ is
contained in the multiset $\mathrm{Inv}(\mathsf{Snap}(v))$, so it follows from
the last sentence in Lemma 7.4 that $\mathrm{inv}(u)\subseteq\mathrm{inv}(v)$
and $\langle\mathrm{cov}(u)\rangle\subseteq\langle\mathrm{cov}(v)\rangle$.
According to Equation 9, we have $u\preceq v$.

To prove the converse, let us assume that $u,v\in W$ are such that $u\preceq
v$. Let $\mathsf{Pop}(u)=u_{0}\lessdot u_{1}\lessdot\cdots\lessdot u_{k}=u$ be
a saturated chain in $\mathrm{Weak}(W)$ from $\mathsf{Pop}(u)$ to $u$. Let
$t_{i}=u_{i}u_{i-1}^{-1}$, and let $p=t_{k}\cdots
t_{1}=u\,\mathsf{Pop}(u)^{-1}$. Each $t_{i}$ is the reflection through the
hyperplane $H_{\Sigma(u_{i-1}\lessdot u_{i})}$. Identifying $v$ with a region
of $\mathcal{H}$, we can consider the poset isomorphism $\omega_{v}$ from
Theorem 6.5. That theorem tells us that $H_{\Sigma(u_{i-1}\lessdot
u_{i})}=H_{\Sigma(\omega_{v}(u_{i-1})\lessdot\omega_{v}(u_{i}))}$, so
$t_{i}=\omega_{v}(u_{i})\omega_{v}(u_{i-1})^{-1}$. This shows that
$\omega_{v}(u)=p\,\omega_{v}(\mathsf{Pop}(u))$, so

(10)
$u\,\mathsf{Pop}(u)^{-1}=\omega_{v}(u)(\omega_{v}(\mathsf{Pop}(u)))^{-1}.$

Since $\omega_{v}$ is a poset isomorphism by Theorem 6.5, we have
$\omega_{v}(\mathsf{Pop}(u))\leq\omega_{v}(u)$. This means there exists $x\in
W$ such that $\omega_{v}(u)=\omega_{v}(\mathsf{Pop}(u))\,x$ and

(11)
$\mathrm{lift}(\omega_{v}(u))=\mathrm{lift}(\omega_{v}(\mathsf{Pop}(u)))\,{\bf
x}.$

Note that
$\mathrm{len}(x)=\mathrm{len}(\omega_{v}(u))-\mathrm{len}(\omega_{v}(\mathsf{Pop}(u)))=\mathrm{len}(u)-\mathrm{len}(\mathsf{Pop}(u))=\mathrm{len}(w_{\circ}(\mathrm{des}(u)))$.
Lemma 6.6 tells us that $\mathsf{Pop}(u)\leq\omega_{v}(\mathsf{Pop}(u))$, so
there exists $y\in W$ such that
$\omega_{v}(\mathsf{Pop}(u))=\mathsf{Pop}(u)\,y$ and

(12)
$\mathrm{lift}(\omega_{v}(\mathsf{Pop}(u)))=\mathrm{lift}(\mathsf{Pop}(u))\,{\bf
y}.$

Then

$\displaystyle\mathsf{Pop}(u)yxy^{-1}\mathsf{Pop}(u)^{-1}$
$\displaystyle=\omega_{v}(\mathsf{Pop}(u))x(\omega_{v}(\mathsf{Pop}(u)))^{-1}$
$\displaystyle=\omega_{v}(u)(\omega_{v}(\mathsf{Pop}(u)))^{-1}$
$\displaystyle=u\,\mathsf{Pop}(u)^{-1}$
$\displaystyle=\mathsf{Pop}(u)w_{\circ}(\mathrm{des}(u))\mathsf{Pop}(u)^{-1},$

where we have used Equation 10. Rearranging this equation yields
$yx=w_{\circ}(\mathrm{des}(u))y$. Because
$\mathrm{len}(\mathsf{Pop}(u)yx)=\mathrm{len}(\omega_{v}(u))=\mathrm{len}(\omega_{v}(\mathsf{Pop}(u)))+\mathrm{len}(x)=\mathrm{len}(\mathsf{Pop}(u))+\mathrm{len}(y)+\mathrm{len}(x)$,
we have
$\mathrm{len}(yx)=\mathrm{len}(y)+\mathrm{len}(x)=\mathrm{len}(y)+\mathrm{len}(w_{\circ}(\mathrm{des}(u)))$.
It follows that

${\bf yx}=\mathrm{lift}(yx)=\mathrm{lift}(w_{\circ}(\mathrm{des}(u))y)={\bf
w}_{\circ}(\mathrm{des}(u)){\bf y},$

where
$\mathbf{w}_{\circ}(\mathrm{des}(u))=\mathrm{lift}(w_{\circ}(\mathrm{des}(u)))$.
Consequently,

$\displaystyle\mathsf{Snap}(u)$ $\displaystyle={\bf u}\,{\bf
w}_{\circ}(\mathrm{des}(u))$
$\displaystyle=\mathrm{lift}(\mathsf{Pop}(u))\mathbf{w}_{\circ}(\mathrm{des}(u))^{2}$
$\displaystyle\leq\mathrm{lift}(\mathsf{Pop}(u))\mathbf{w}_{\circ}(\mathrm{des}(u))^{2}\,{\bf
y}$
$\displaystyle=\mathrm{lift}(\mathsf{Pop}(u))\mathbf{w}_{\circ}(\mathrm{des}(u))\,{\bf
yx}$ $\displaystyle=\mathrm{lift}(\mathsf{Pop}(u))\,{\bf yx}^{2}.$

Invoking Equations 11 and 12, we find that
$\mathrm{lift}(\mathsf{Pop}(u))\,{\bf
yx}^{2}=\mathrm{lift}(\omega_{v}(\mathsf{Pop}(u)))\,{\bf
x}^{2}=\mathrm{lift}(\omega_{v}(u))\,{\bf x}$. Therefore, to prove that
$\mathsf{Snap}(u)\leq\mathsf{Snap}(v)$, we just need to show that
$\mathrm{lift}(\omega_{v}(u))\,{\bf x}\leq\mathsf{Snap}(v)$.

We defined $x$ via the equation
$\omega_{v}(u)=\omega_{v}(\mathsf{Pop}(u))\,x$; since
$\omega_{v}(\mathsf{Pop}(u))$ and $\omega_{v}(u)$ are both in the interval
$[\mathsf{Pop}(v),v]$ (by the definition of $\omega_{v}$ in Theorem 6.5), this
implies that $x$ is in the parabolic subgroup $\langle\mathrm{des}(v)\rangle$.
It follows that ${\bf x}\leq{\bf w}_{\circ}(\mathrm{des}(v))$, so
$\mathrm{lift}(\omega_{v}(u))\,{\bf
x}\leq\mathrm{lift}(\omega_{v}(u))\,\mathbf{w}_{\circ}(\mathrm{des}(v))$.
Therefore, the proof will be complete if we can demonstrate that
$\mathrm{lift}(\omega_{v}(u))\,\mathbf{w}_{\circ}(\mathrm{des}(v))\leq\mathsf{Snap}(v)$.

Because $\omega_{v}(u)\leq v$, there exists
$z\in\langle\mathrm{des}(v)\rangle$ such that $v=\omega_{v}(u)\,z$ and ${\bf
v}=\mathrm{lift}(\omega_{v}(u))\,{\bf z}$. It is well known that for every
$s\in\mathrm{des}(v)$, there exists $s^{\prime}\in\mathrm{des}(v)$ such that
$\mathbf{s}\,\mathbf{w}_{\circ}(\mathrm{des}(v))=\mathbf{w}_{\circ}(\mathrm{des}(v))\,\mathbf{s}^{\prime}$.
It follows that there exists ${\bf z}^{\prime}\in\mathbf{B}^{+}(W)$ such that
${\bf
z}\,\mathbf{w}_{\circ}(\mathrm{des}(v))=\mathbf{w}_{\circ}(\mathrm{des}(v))\,{\bf
z}^{\prime}$. Finally,

$\displaystyle\mathrm{lift}(\omega_{v}(u))\,\mathbf{w}_{\circ}(\mathrm{des}(v))$
$\displaystyle\leq\mathrm{lift}(\omega_{v}(u))\,\mathbf{w}_{\circ}(\mathrm{des}(v))\,{\bf
z}^{\prime}$ $\displaystyle=\mathrm{lift}(\omega_{v}(u))\,{\bf
z}\,\mathbf{w}_{\circ}(\mathrm{des}(v))$ $\displaystyle={\bf
v}\,\mathbf{w}_{\circ}(\mathrm{des}(v))$ $\displaystyle=\mathsf{Snap}(v),$

as desired.

###  7.5. Catalan Combinatorics

A _standard Coxeter element_ $c$ of $W$ is a product of the simple reflections
in some order; fix a reduced expression for $c$ as $s_{1}s_{2}\cdots s_{n}$.
The _$c$ -sorting word_ $\mathbf{w}(c)$ of $\mathbf{w}\in\mathbf{B}^{+}(W)$ is
the lexicographically minimal subword of the $\bf S$-word
$(\mathbf{s}_{1}\mathbf{s}_{2}\cdots\mathbf{s}_{n})^{\infty}$ that represents
$\mathbf{w}$ [Rea07b, STW22]. Write
$\mathbf{w}(c)=\mathbf{w}_{1}\mathbf{w}_{2}\cdots\mathbf{w}_{k}$, with each
$\mathbf{w}_{i}$ a subword of
$\mathbf{s}_{1}\mathbf{s}_{2}\cdots\mathbf{s}_{n}$. We say $\mathbf{w}$ is
_$c$ -sortable_ if the set of letters appearing in $\mathbf{w}_{i}$ contains
the set of letters appearing in $\mathbf{w}_{i+1}$ for all $1\leq i\leq k-1$.
Let $m$ be a positive integer. Following [Rea07b, STW22], we write
$\mathrm{Sort}^{m}(W,c)$ for the set of all $c$-sortable elements of
$\mathbf{B}^{+}(W)$ in the interval
$[\mathbbm{1},\mathbf{w}_{\circ}^{m}]_{\mathbf{B}^{+}}$. Then
$(\mathrm{Sort}^{m}(W,c),\leq)$ is the $m$-th _$c$ -Fuss–Cambrian lattice_ of
type $W$—in particular [GLTW22],

$\left|\mathrm{Sort}^{m}(W,c)\right|=\mathrm{Cat}^{m}(W):=\prod_{i=1}^{n}\frac{mh+d_{i}}{d_{i}},$

where $h$ is the _Coxeter number_ of $W$ and $d_{1},\ldots,d_{n}$ are the
_degrees_ of $W$. We write $\mathrm{Sort}(W,c)=\mathrm{Sort}^{1}(W,c)$; under
the usual identification of the interval
$[\mathbbm{1},\mathbf{w}_{\circ}]_{\mathbf{B}^{+}}$ in
$\mathrm{Weak}(\mathbf{B}^{+}(W))$ with $\mathrm{Weak}(W)$, the set
$\mathrm{Sort}(W,c)$ forms a sublattice of $\mathrm{Weak}(W)$.

Continuing to identify $[\mathbbm{1},\mathbf{w}_{\circ}]_{\mathbf{B}^{+}}$
with $\mathrm{Weak}(W)$, we find that the $c$-corting word
$\mathbf{w}_{\circ}(c)$ naturally defines a positive minimal gallery from
$\mathbbm{1}$ (identified with $B$) to $\mathbf{w}_{\circ}$ (identified with
$-B$); we say that a shard is _$c$ -noncrossing_ if it labels one of the edges
in this gallery. The _$c$ -noncrossing partition lattice_ $\mathrm{NC}(W,c)$
is the interval from $\mathbbm{1}$ to $c$ in the absolute order on $W$. In
[Rea07a], Reading gave a beautiful bijection
$\mathrm{nc}_{c}:\mathrm{Sort}(W,c)\to\mathrm{NC}(W,c)$, which is a poset
isomorphism from $(\mathrm{Sort}(W,c),\preceq)$ to $\mathrm{NC}(W,c)$ [Rea11,
Theorem 8.5].

We obtain the following corollaries of Theorems 1.3 and 1.5, establishing
similar relationships between the noncrossing partition lattice and the images
of the set of sortable elements under the crackle and snap maps.

######  Corollary 7.6.

The map $\mathsf{Crackle}$ restricts to a poset embedding of
$(\mathrm{Sort}(W,c),\preceq)$ into
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$. In particular, the subposet
$(\mathsf{Crackle}(\mathrm{Sort}(W,c)),\leq)$ of
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$ is isomorphic to
$\mathrm{NC}(W,c)$.

Corollary 7.6 is illustrated in Figure 2, where $s$ is the reflection through
$H_{8}$, $t$ is the reflection through $H_{1}$, and $c=st$. In this example,
the noncrossing shards are $H_{1},\Sigma_{6},\Sigma_{7}$ and $H_{8}$, with
corresponding $c$-sortable regions $B$, $R_{1}$, $R_{6}$, $R_{7}$, $R_{8}$,
and $-B$.

######  Corollary 7.7.

The map $\mathsf{Snap}$ restricts to a poset embedding of
$(\mathrm{Sort}(W,c),\preceq)$ into
$[\mathbbm{1},\Delta^{2}]_{\mathbf{B}^{+}}$. In particular, the subposet
$(\mathsf{Snap}(\mathrm{Sort}(W,c)),\leq)$ of
$[\mathbbm{1},\Delta^{2}]_{\mathbf{B}^{+}}$ is isomorphic to
$\mathrm{NC}(W,c)$.

Corollary 7.7 is illustrated in Figure 4. We now link this result to the
discussion in Section 1.7. Recall that for $\mathbf{s}\in{\bf S}$ and a
positive braid
$\mathbf{w}=\mathbf{s}_{i_{1}}\cdots\mathbf{s}_{i_{k}}\in\mathbf{B}^{+}(W)$
with projection $w=\varphi(\mathbf{w})\in W$, we let
$\mathbf{s}^{\mathbf{w}}=(t,j)$, where $t=s^{w}=w^{-1}sw$ and $j$ counts the
number of times $t$ appears in the sequence
$s^{s_{i_{k}}},s^{s_{i_{k}}s_{i_{k-1}}},\ldots,s^{s_{i_{k}}s_{i_{k-1}}\cdots
s_{1}}$.

In [GLTW22], a new set of noncrossing $W$-Catalan objects was introduced as
the set of subwords of $\mathbf{c}^{h+1}$ that represent the full twist
$\Delta^{2}=\mathbf{w}_{\circ}^{2}$ and satisfy an additional _Deodhar
condition_. When interpreted in the positive braid monoid $\mathbf{B}^{+}(W)$,
this Deodhar condition is equivalent to restricting to the subset of
$\mathrm{Sort}^{2}(W,c)$ consisting of the $c$-sortable elements $\mathbf{w}$
with the property that for each $\mathbf{s}\in\mathrm{des}(\mathbf{w})$, we
have $\mathbf{s}^{\mathbf{w}}=(t,j)$ with $j$ even. Thus, Corollary 7.7 is
equivalent to the observation in Section 1.7 that the $c$-noncrossing
partition lattice appears as the subposet of the $m$-th $c$-Fuss–Cambrian
lattice for $m=2$ consisting of the $c$-sortable elements whose corresponding
subword complex facets satisfy the Deodhar conditions.

##  8\. Pow

In this section, $\mathcal{H}$ is an arbitrary finite central irreducible real
hyperplane arrangement. We will introduce a map $\mathsf{Pow}$ from the set
$\mathcal{R}$ of regions to the pure shard monoid
$\mathbf{P}^{+}(\mathcal{H},B)$. Just as $\mathsf{Crackle}$ embeds the “short
and wide” poset $\mathrm{Shard}(\mathcal{H},B)$ into the “tall and wide”
interval $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$, Theorem 1.6 states that
$\mathsf{Pow}$ embeds the “tall and slender” poset
$\mathrm{Weak}(\mathcal{H},B)$ inside
$[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$; this theorem is illustrated in
Figure 5. Thus, the interval $[\mathbbm{1},\Delta^{2}]_{\mathbf{P}^{+}}$
simultaneously contains $\mathrm{Shard}(\mathcal{H},B)$ and
$\mathrm{Weak}(\mathcal{H},B)$.

We require a short argument to show that $\mathsf{Pow}$ will be well-defined.

######  Proposition 8.1.

Let

$B=C_{0}\xrightarrow{e_{1}}C_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}C_{k-1}\xrightarrow{e_{k}}C_{k}=C\quad\text{and}\quad
B=C_{0}^{\prime}\xrightarrow{e_{1}^{\prime}}C_{1}^{\prime}\xrightarrow{e_{2}^{\prime}}\cdots\xrightarrow{e_{k-1}^{\prime}}C_{k-1}^{\prime}\xrightarrow{e_{k}^{\prime}}C_{k}^{\prime}=C$

be two positive minimal galleries from the base region $B$ to a region
$C\in\mathcal{R}$. Then

$\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}=\ell_{\Sigma(e^{\prime}_{k})}\ell_{\Sigma(e^{\prime}_{k-1})}\cdots\ell_{\Sigma(e^{\prime}_{1})}.$

###### Proof.

Any two minimal galleries from $B$ to $C$ are homotopic by a succession of
homotopies across $2$-cells of $\mathrm{Sal}(\mathcal{H})$ [Sal87, Lemma 11].
Hence, it suffices to prove the proposition when $\mathcal{H}$ has rank $2$
and $C=-B$. In this case, the result is immediate from Corollary 2.9. ∎

######  Definition 8.2.

Let $C\in\mathcal{R}$, and let

$B=C_{0}\xrightarrow{e_{1}}C_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}C_{k-1}\xrightarrow{e_{k}}C_{k}=C$

be a positive minimal gallery from $B$ to $C$. Define

$\mathsf{Pow}(C):=\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}.$

###  8.1. Proof of Theorem 1.6

Suppose $C,D\in\mathcal{R}$ are such that $D\leq C$ in
$\mathrm{Weak}(\mathcal{H},B)$. Let

$B=C_{0}\xrightarrow{e_{1}}C_{1}\xrightarrow{e_{2}}\cdots\xrightarrow{e_{k-1}}C_{k-1}\xrightarrow{e_{k}}C_{k}=C$

be a positive minimal gallery from $B$ to $C$, where $D=C_{m}$. Let

$-C=-C_{k}\xrightarrow{f_{k}}-C_{k-1}\xrightarrow{f_{k-1}}\cdots\xrightarrow{f_{2}}-C_{1}\xrightarrow{f_{1}}-C_{0}=-B$

be the corresponding positive minimal gallery from $-C$ to $-B$ (with
$-D=-C_{m}$). Using Corollary 2.9, we find that

$\mathsf{Pow}(C)=\ell_{\Sigma(e_{k})}\ell_{\Sigma(e_{k-1})}\cdots\ell_{\Sigma(e_{1})}=\ell_{\Sigma(f_{1})}\ell_{\Sigma(f_{2})}\cdots\ell_{\Sigma(f_{k})}$

and

$\mathsf{Pow}(D)=\ell_{\Sigma(e_{m})}\ell_{\Sigma(e_{m-1})}\cdots\ell_{\Sigma(e_{1})}=\ell_{\Sigma(f_{1})}\ell_{\Sigma(f_{2})}\cdots\ell_{\Sigma(f_{m})}.$

This demonstrates that $\mathsf{Pow}(D)\leq\mathsf{Pow}(C)$ in