# Automorphisms of the generalized cluster complex

Matthieu Josuat-Vergès

###### Abstract

It is proved that the generalized cluster complex defined by Fomin and Reading
has a dihedral symmetry. Together with diagram symmetries, they generate its
automorphism group. A consequence is a simple explicit formula for the order
of this automorphism group.

##  1 Introduction

Fomin and Zelevinsky [10] defined the cluster complex associated to a finite-
type cluster algebra: it is a simplicial complex having cluster variables as
vertices and clusters as facets. Furthermore, the construction of Fomin and
Zelevinsky involves a pair of maps $\tau_{+}$ and $\tau_{-}$ that act as
automorphisms of the cluster complex, generating a dihedral group of symmetry.
In type $A_{n+1}$, the cluster complex can be described combinatorially as the
complex of dissections of a regular $(n+2)$-gon (in particular, its facets are
triangulations of this polygon), and the dihedral group of symmetries
coincides with the apparent geometric symmetry of the $(n+2)$-gon. The
automorphism group of the cluster complex (which is essentially the dihedral
group generated by $\tau_{+}$ and $\tau_{-}$), and related automorphisms of
finite-type cluster algebras, have been investigated in [1, 2, 5, 11].

The generalized cluster complex $\Gamma^{(m)}$ was introduced by Fomin and
Reading [9], in the framework of finite-type Coxeter combinatorics. It depends
on an integer parameter $m\geq 1$, in such a way that $m=1$ (in
crystallographic types) corresponds to the original definition of Fomin and
Zelevinsky. There is no underlying cluster algebra, but there is a
representation theoretic interpretation that lead to many developments such as
higher cluster categories. We refer to [4, 12, 13, 15]. In particular, Stump,
Thomas, and Williams [12] give a thorough combinatorial treatment in [12]. In
type $A_{n+1}$, the generalized cluster complex can be described via
dissections of a $(mn+2)$-gon (and thus naturally embeds as a subcomplex of
the cluster complex of type $A_{mn+1}$).

The original definition of Fomin and Reading makes clear that the complex
$\Gamma^{(m)}$ has a cyclic symmetry, via the action of an automorphism
$\mathcal{R}$ (see Section 2). In the case $m=1$, we have
$\mathcal{R}=\tau_{+}\tau_{-}$. Here, we extend the dihedral symmetry by
introducing involutive automorphisms $\SS$ and $\mathcal{T}$ (see Section 5)
that specialize in $\tau_{+}$ and $\tau_{-}$ when $m=1$ and satisfy
$\mathcal{R}^{m}=\SS\mathcal{T}$. This dihedral symmetry group is almost the
full automorphism group of $\Gamma^{(m)}$, in the sense that we only need to
add extra automorphisms coming from symmetry of the Coxeter graph. The precise
statement is as follows.

######  Theorem 1.1.

Let $W$ be a finite and irreducible Coxeter group, and let $\Gamma^{(m)}$ be
the associated generalized cluster complex. Define two subgroups:

  * •

$\operatorname{Dih}\subset\operatorname{Aut}(\Gamma^{(m)})$, called the
_dihedral subgroup_ , is generated by $\mathcal{R}$ and $\SS$ (or
$\mathcal{R}$ and $\mathcal{T}$).

  * •

$\operatorname{Diag}\subset\operatorname{Aut}(\Gamma^{(m)})$ is the subgroup
of _diagram automorphisms_ (See Section 4).

Then we have a semidirect product

$\operatorname{Aut}(\Gamma^{(m)})=\operatorname{Dih}\rtimes(\operatorname{Diag}/\langle\mathcal{C}\rangle),$

where $\mathcal{C}$ is the _c_ anonical diagram automorphism (see Section
4.3), and

$\big{|}{\operatorname{Aut}(\Gamma^{(m)})}\big{|}=(mh+2)\omega$ (1)

where $h$ is the Coxeter number of $W$, and $\omega$ is the number of
automorphisms of its Coxeter graph (also, $\omega=|{\operatorname{Diag}}|$).

This article is organized as follows:

  * •

In Section 2, we review the definition and properties of $\Gamma^{(m)}$.

  * •

Sections 3 and 4 contain elementary facts about the reducible case and diagram
automorphisms, respectively. The more technical Section 5 introduces the
involutive automorphisms $\SS$ and $\mathcal{T}$.

  * •

The proof of Theorem 1.1 is contained in Sections 6, 7, 8. By a global
induction hypothesis, we will assume that the statements in these three
sections hold in rank $<n$ where $n$ is the rank of $W$.

There are a few final remarks in Section 9.

##  2 Review of the generalized cluster complex

Let $\Phi$ be a finite-type root system, associated to the finite Coxeter
group $W$. Let $\Phi=\Phi_{+}\mathrel{\uplus}\Phi_{-}$ be a decomposition in
positive and negative roots. Let $\Delta\subset\Phi_{+}$ be the set of simple
roots, $S$ the set of simple reflections, $T$ the set of reflections. For each
root $\alpha\in\Phi$, the associated reflection is denoted $t_{\alpha}\in T$.
For $w\in W$, let $\operatorname{Supp}(w)\subset S$ denote the support of $w$,
i.e., the set of simple reflections that appear in reduced words for $w$. The
required background on reflection groups is rather small, as our work is
mostly based on the combinatorial properties of $\Gamma^{(m)}$ that we review
here.

######  Definition 2.1.

A colored root is a pair $(\alpha,i)\in\Phi\times\llbracket 1,m\rrbracket$. It
is denoted $\alpha^{i}$ and $i$ is called its color. It is almost-positive if
$\alpha\in\Phi_{+}$, or $-\alpha\in\Delta$ and $i=1$. The set of almost-
positive colored roots is denoted $\Phi^{(m)}_{\geq-1}$.

This set $\Phi^{(m)}_{\geq-1}$ is the vertex set of the generalized cluster
complex $\Gamma^{(m)}$. This is a simplicial complex with many interesting
geometric and enumerative features, see [7, 9, 12, 14]. Besides the original
definition by Fomin and Reading [9], a review which is sufficient for our
purpose has been given in our previous work [7, Section 6]. We give an outline
and refer to loc. cit. for any further information.

###  2.1 The compatibility relation

Fomin and Reading originally defined $\Gamma^{(m)}$ as the flag complex
associated with a (symmetric) binary relation on $\Phi^{(m)}_{\geq-1}$, called
compatibility of almost-positive colored roots. This is relevant in the
present context, since the automorphism group of a flag complex is clearly the
automorphism group of its 1-skeleton.

There exists a black/white decomposition
$\Delta=\Delta_{\bullet}\uplus\Delta_{\circ}$ where $\Delta_{\bullet}$,
respectively $\Delta_{\circ}$, contains pairwise orthogonal roots. The
bipartite Coxeter element $c$ is defined by:

$c_{\bullet}=\prod_{\alpha\in\Delta_{\bullet}}t_{\alpha},\quad
c_{\circ}=\prod_{\alpha\in\Delta_{\circ}}t_{\alpha},\quad\text{ and }\quad
c=c_{\bullet}c_{\circ}.$

It is also possible to consider an arbitrary standard Coxeter element (see
[12]), but the complexes obtained this way are all isomorphic. So, we stick to
the bipartite Coxeter element as in [9].

######  Definition 2.2 ([9]).

The self-bijection $\mathcal{R}:\Phi^{(m)}_{\geq-1}\to\Phi^{(m)}_{\geq-1}$ is
defined by

$\mathcal{R}(\alpha^{i}):=\begin{cases}\alpha^{i+1}&\text{ if
}\alpha\in\Phi_{+}\text{ and }i<m,\\\ (-\alpha)^{1}&\text{ if
}\alpha\in\Delta_{\circ}\text{ and }i=m,\text{ or
}-\alpha\in\Delta_{\bullet}\text{ and }i=1,\\\ c(\alpha)^{1}&\text{ if
}\alpha\in\Phi_{+}\backslash\Delta_{\circ}\text{ and }i=m,\text{ or
}-\alpha\in\Delta_{\circ}\text{ and }i=1.\end{cases}$

######  Lemma 2.3 ([9]).

Let $X\subset\Phi^{(m)}_{\geq-1}$ be an $\mathcal{R}$-orbit. We have either:

  * •

$\\#X=\frac{mh+2}{2}$, and $X$ contains exactly one element of the form
$-\rho^{1}$ (with $\rho\in\Delta$).

  * •

$\\#X=mh+2$, and $X$ contains exactly two elements of the form $-\rho^{1}$
(with $\rho\in\Delta$). Moreover, these two elements have the form $-\rho^{1}$
and $w_{0}(\rho)^{1}$.

The order of $\mathcal{R}$ is given by

$\frac{|{\mathcal{R}}|}{|{w_{0}}|}=\frac{mh+2}{2}.$ (2)

######  Remark 2.4.

The long element $w_{0}$ sends $\Delta$ to $-\Delta$, more precisely
$\rho\mapsto-w_{0}(\rho)$ is an automorphism of the Coxeter diagram of $W$
(see also Section 4.3).

######  Definition 2.5 ([9]).

The compatibility relation is a symmetric binary relation on
$\Phi^{(m)}_{\geq-1}$, uniquely defined by the two conditions:

  * •

$\mathrel{\|}$ is preserved by $\mathcal{R}$ (i.e., we have
$\alpha\mathrel{\|}\beta$ if and only if
$\mathcal{R}(\alpha)\mathrel{\|}\mathcal{R}(\beta)$),

  * •

if $\alpha\in\Delta$, we have $-\alpha^{1}\mathrel{\|}\beta^{j}$ if and only
if $t_{\alpha}\notin\operatorname{Supp}(t_{\beta})$.

See [9] for details on the existence and unicity.

######  Definition 2.6 ([9]).

The generalized cluster complex $\Gamma^{(m)}$ is defined as the flag complex
associated with the binary relation $\mathrel{\|}$. This means that a subset
of $\Phi^{(m)}_{\geq-1}$ is a face of $\Gamma^{(m)}$ iff its elements are
pairwise compatible.

We will use the following rules, which make the compatibility relation more
explicit. Let $\alpha,\beta\in\Delta$, and $1\leq i<j\leq m$. We have:

$\displaystyle-\alpha^{1}\mathrel{\|}\beta^{i}$
$\displaystyle\Longleftrightarrow
t_{\alpha}\notin\operatorname{Supp}(t_{\beta}),$ (3)
$\displaystyle\alpha^{i}\mathrel{\|}\beta^{i}$
$\displaystyle\Longleftrightarrow\langle\alpha|\beta\rangle\geq 0,\text{ and
}t_{\alpha}t_{\beta}\leq c\text{ or }t_{\beta}t_{\alpha}\leq c,$ (4)
$\displaystyle\alpha^{i}\mathrel{\|}\beta^{j}$
$\displaystyle\Longleftrightarrow t_{\alpha}t_{\beta}\leq c.$ (5)

One way to see this is to use the total order on $\Phi^{(m)}_{\geq-1}$ in the
next section, see also [7, Section 6].

###  2.2 Reflection ordering

An alternative characterization of faces in $\Gamma^{(m)}$ has been given by
Tzanaki [14]. It makes a connection with factorization of the Coxeter element.
The idea is to use a total order on $\Phi^{(m)}_{\geq-1}$, akin to Steinberg’s
indexing of $\Phi$ used by Brady and Watt [3]. Recall that the Steinberg
ordering of $\Phi$ is given by indexing roots $(\alpha_{i})_{1\leq i\leq nh}$
with the conditions:

  * •

$-\Delta_{\circ}=\\{\alpha_{i}\;:\;1\leq i\leq r\\}$ (where
$r=\\#\Delta_{\circ}$)

  * •

$\Delta_{\bullet}=\\{\alpha_{i}\;:\;r+1\leq i\leq n\\}$

  * •

$c(\alpha_{i})=\alpha_{i+n}$ (where indices are taken modulo $nh$).

Almost-positive roots are the roots $(\alpha_{i})_{1\leq
i\leq\frac{nh}{2}+n}$, and the indexing gives a total order on
$\Phi_{\geq-1}$. There is a decomposition:

$\displaystyle\Phi_{\geq-1}=-\Delta_{\circ}\uplus\Delta_{\bullet}\uplus
c_{\bullet}(\Delta_{\circ})\uplus c(\Delta_{\bullet})\uplus\dots\uplus
c_{\circ}(\Delta_{\bullet})\uplus\Delta_{\circ}\uplus-\Delta_{\bullet},$ (6)

which coarsens the total order in the sense that each block contain
consecutive elements and the blocks are written increasingly. Almost-positive
roots in $\Phi_{\geq-1}$ are the roots $(\alpha_{i})_{1\leq
i\leq\frac{nh}{2}+n}$, and Brady and Watt showed that facets of the cluster
complex correspond to decreasing factorizations of the Coxeter element [3,
Section 8].

Tzanaki’s generalization is a total order $\prec$ on $\Phi^{(m)}_{\geq-1}$
given as follows. Rather than a total order, it is helpful to think as
$\Phi^{(m)}_{\geq-1}$ as a union of $mh+2$ blocks, generalizing (6). Denote
$X^{i}$ for $\\{\alpha^{i}\;:\;\alpha\in X\\}\subset\Phi^{(m)}_{\geq-1}$ if
$X\subset\Phi$. We have:

$\displaystyle\Phi^{(m)}_{\geq-1}=$
$\displaystyle-\Delta_{\circ}^{1}\uplus\Delta_{\bullet}^{m}\uplus\dots\uplus\Delta_{\circ}^{m}\uplus\Delta_{\bullet}^{m-1}\uplus\dots\uplus\Delta_{\circ}^{m-1}\uplus\quad\cdots\quad\uplus$
(7)
$\displaystyle\Delta_{\bullet}^{2}\uplus\dots\uplus\Delta_{\circ}^{2}\uplus\Delta_{\bullet}^{1}\uplus\dots\uplus\Delta_{\circ}^{1}\uplus-\Delta_{\bullet}^{1}.$
(8)

The total order is such that the decomposition in (7) is a coarsening (as
above), and each block is ordered via Steinberg’s ordering.

######  Proposition 2.7 ([14]).

Let $f$ be a tuple of $n$ elements of $\Phi^{(m)}_{\geq-1}$, indexed
$f=(\gamma_{i}^{k_{i}})_{1\leq i\leq n}$ such that
$\gamma_{1}^{k_{1}}\succ\dots\succ\gamma_{n}^{k_{n}}$. Then $f$ is a facet of
$\Gamma^{(m)}$ iff $c=t_{\gamma_{1}}\cdots t_{\gamma_{n}}$.

######  Remark 2.8.

We will see that the decomposition in (7) is well-behaved with respect to the
automorphisms of $\Gamma^{(m)}$. For example, $\mathcal{R}$ sends a block of
index $i$ to the block of index $i-m$ (where indices are taken modulo $mh+2$).
We will see that $\SS$ fixes $-\Delta_{\circ}^{1}$ and reverses the order on
the other blocks. Similarly, $\mathcal{T}$ fixes $-\Delta_{\bullet}^{1}$ and
reverses the order on the other blocks. Note that there might exist nontrivial
automorphisms that acts trivially on the decomposition in (7) (these are the
even diagram automorphisms, see Section 4).

A direct consequence of the previous proposition is the following:

######  Lemma 2.9.

If $\alpha^{k}\mathrel{\|}\beta^{\ell}$, we have $t_{\alpha}\neq t_{\beta}$,
and $t_{\alpha}t_{\beta}\leq c$ or $t_{\beta}t_{\alpha}\leq c$. (Here $\leq$
is the _absolute order_ on $W$.) More precisely,
$\alpha^{k}\mathrel{\|}\beta^{\ell}$ and $\alpha^{k}\prec\beta^{\ell}$ imply
$t_{\beta}t_{\alpha}\leq c$.

See [3] for more on the absolute order.

###  2.3 The other bipartite Coxeter element

Note that the definition of $\Gamma^{(m)}$ depends on the choice of
$\Delta_{\bullet}$ and $\Delta_{\circ}$. The exchange of $\bullet$ and $\circ$
gives an isomorphic complex and we make this explicit here.

Denote by $\check{\Gamma}^{(m)}$ the complex defined similar to $\Gamma^{(m)}$
but exchanging the roles of $\Delta_{\bullet}$ and $\Delta_{\circ}$.
Similarly, we denote $\check{\|}$ and $\check{\mathcal{R}}$ the analog of $\|$
and $\mathcal{R}$.

######  Proposition 2.10.

The involutive self-map $\iota:\Phi^{(m)}_{\geq-1}\to\Phi^{(m)}_{\geq-1}$
defined by

$\iota(\alpha^{i})=\begin{cases}\alpha^{i}&\text{ if }\alpha\in-\Delta,\\\
\alpha^{m+1-i}&\text{ if }\alpha\in\Phi_{+}\end{cases}$

induces an isomorphism $\iota:\Gamma^{(m)}\to\check{\Gamma}^{(m)}$. Moreover,
we have: $\iota\mathcal{R}\iota=\check{\mathcal{R}}$.

The proof is straightforward and omitted.

###  2.4 Links

A useful consequence of the definition is the following. Let
$\alpha\in\Delta$, and $W_{\alpha}$ the maximal standard parabolic subgroup of
$W$ obtained by removing $\alpha$ from the Coxeter graph of $W$. The link of a
face $f\in\Gamma^{(m)}$ is by definition

$\operatorname{Link}(f)=\Big{\\{}f^{\prime}\;:\;f\cap
f^{\prime}=\varnothing\text{ and }f\cup f^{\prime}\in\Gamma^{(m)}\Big{\\}}.$

Note that $\operatorname{Link}(f)$ is itself a simplicial complex.

######  Proposition 2.11.

Let $\rho\in\Delta$. We have
$\operatorname{Link}(\\{-\rho^{1}\\})\simeq\Gamma^{(m)}(W_{\rho})$.

It also follows that the link of any face in $f\in\Gamma^{(m)}$ is isomorphic
to $\Gamma^{(m)}(P)$ where $P$ is a standard parabolic subgroup of $W$. A
precise way to give the parabolic subgroup $P$ in terms of $f$ is given in [7,
Proposition 2.20] (this is not needed in the present work).

A concrete consequence of this property of links is the following:

######  Theorem 2.12.

Let $W$ and $W^{\prime}$ be finite Coxeter groups. If $\Gamma^{(m)}(W)$ and
$\Gamma^{(m^{\prime})}(W^{\prime})$ are isomorphic, then $m=m^{\prime}$ and
$W$ and $W^{\prime}$ are Coxeter-isomorphic.

###### Proof.

We show that $m$ and the Coxeter graph of $W$ can be recovered from
$\Gamma^{(m)}(W)$. The link of a 1-codimensional face in $\Gamma^{(m)}(W)$ is
$\Gamma^{(m)}(A_{1})$ (since $A_{1}$ is the unique Coxeter group of rank 1),
which consists in $m+1$ isolated vertices. So $m$ can be recovered from
$\Gamma^{(m)}(W)$. Note also that the rank of $W$ is $1+\dim(\Gamma^{(m)})$.

We then proceed by induction on $n$, the rank of $W$. The case $n=1$ is
trivial, and the case $n=2$ is settled by noting that the number of facets of
$\Gamma^{(m)}(I_{2}(k))$ strictly increases with $k$ (this number is known as
the Fuß-Catalan number, see [9] for the exact formula).

We need the following: if $n\geq 3$, the Coxeter graph of $W$ is uniquely
characterized by the collection of Coxeter graphs of $W_{\alpha}$
($\alpha\in\Delta$). It is straightforward to check this. As the links of
vertices of $\Gamma^{(m)}$ provide this collection (by induction hypothesis),
this completes the proof. ∎

##  3 The reducible case

We examine the situation where $W$ can be decomposed as a product

$W=W_{1}\times\dots\times W_{r}.$

According to [9], we have

$\Gamma^{(m)}(W)=\Gamma^{(m)}(W_{1})\star\dots\star\Gamma^{(m)}(W_{r})$ (9)

where $\star$ is the join operation on simplicial complexes. Concretely, this
means that each face $f\in\Gamma^{(m)}(W)$ can be written

$f=(f_{1},\dots,f_{r})$ (10)

where $f_{i}$ is a face of $\Gamma^{(m)}(W_{i})$, moreover
$\dim(f)=\sum_{i=1}^{r}\dim(f_{i})$. In particular, at the level of vertex
sets we have

$\Phi^{(m)}_{\geq-1}(W)=\biguplus_{i=1}^{r}\Phi^{(m)}_{\geq-1}(W_{i}).$ (11)

######  Definition 3.1.

An automorphism $\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ is monomial
if the partition of the vertex set in (11) is stabilized: for any $i$, there
exists $j$ such that we have
$\mathcal{F}\big{(}\Phi^{(m)}_{\geq-1}(W_{i})\big{)}=\Phi^{(m)}_{\geq-1}(W_{j})$.

The terminology is an analogy with monomial matrices.

######  Proposition 3.2.

Every element $\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ is monomial.

The idea is to give a combinatorial criterion to characterize when two
vertices are in the same block of the partition.

######  Lemma 3.3.

Let $\alpha^{k}\in\Phi^{(m)}_{\geq-1}(W_{i})$ and
$\beta^{\ell}\in\Phi^{(m)}_{\geq-1}(W_{j})$ with $1\leq i,j\leq r$. The
following conditions are equivalent:

  * •

$i\neq j$,

  * •

$\alpha^{k}\mathrel{\|}\beta^{\ell}$, and for all
$\gamma^{p}\in\Phi^{(m)}_{\geq-1}$ we have $\alpha^{k}\mathrel{\|}\gamma^{p}$
or $\beta^{\ell}\mathrel{\|}\gamma^{p}$.

###### Proof.

First, assume that $i\neq j$. We have $\alpha^{k}\mathrel{\|}\beta^{\ell}$ by
(9) and by definition of the join. Let
$\gamma^{p}\in\Phi^{(m)}_{\geq-1}(W_{\ell})$. We have
$\alpha^{k}\mathrel{\|}\gamma^{p}$ if $i\neq\ell$ and
$\beta^{\ell}\mathrel{\|}\gamma^{p}$ if $j\neq\ell$. Since $i\neq j$, at least
one of $\alpha^{k}\mathrel{\|}\gamma^{p}$ or
$\beta^{\ell}\mathrel{\|}\gamma^{p}$ thus holds. Therefore the second
condition in the lemma holds.

Now, assume $i=j$. If $\alpha^{k}\centernot{\|}\beta^{\ell}$, the second
condition in the lemma does not hold. So, we assume
$\alpha^{k}\mathrel{\|}\beta^{\ell}$. It remains to prove the following: there
exists $\gamma^{p}$ such that $\alpha^{k}\centernot{\|}\gamma^{p}$ and
$\beta^{\ell}\centernot{\|}\gamma^{p}$. By using using the map $\mathcal{R}$
on the irreducible factor $W_{i}$ and the invariance of compatibility, we can
assume that $\alpha\in-\Delta$ (and $k=1$). The construction of $\gamma^{p}$
is as follows.

  * •

If $\beta\in-\Delta$ (and $\ell=1$), let $\gamma\in\Phi_{+}$ be such that
$t_{\alpha}\in\operatorname{Supp}(t_{\gamma})$ and
$t_{\beta}\in\operatorname{Supp}(t_{\gamma})$ (it exists because $t_{\alpha}$
and $t_{\beta}$ are both in $W_{i}$ which is irreducible, and a finite
irreducible Coxeter group has reflections with full support). The color $p$
can be anything.

  * •

Otherwise, let $P\subset W_{i}$ be a standard parabolic subgroup such that $P$
is irreducible, $t_{\alpha}\in P$,
$P\mathrel{\cap}\operatorname{Supp}(t_{\beta})=\varnothing$, and $P$ contains
a neighbor of $\operatorname{Supp}(t_{\beta})$ in the Coxeter graph. (It
exists because $t_{\alpha}$ and $t_{\beta}$ are both in $W_{i}$ which is
irreducible, and these properties easily translate into properties of the
corresponding subset of $S$ or $\Delta$. We used the fact that
$t_{\alpha}\notin\operatorname{Supp}(t_{\beta})$, which holds since
$\alpha^{k}\mathrel{\|}\beta^{\ell}$). Let $\gamma\in\Delta$ such that
$\operatorname{Supp}(t_{\gamma})=P\cap S$, and $p$ is arbitrary. Now, we have:

    * –

$\alpha^{k}\centernot{\|}\gamma^{\ell}$ holds via (3), since $t_{\alpha}\in
P\mathrel{\cap}S=\operatorname{Supp}(t_{\gamma})$.

    * –

$\beta^{\ell}\centernot{\|}\gamma^{\ell}$ holds via (5), since
$\langle\beta|\gamma\rangle<0$. This is obtained as follows. Write

$\beta=\sum_{\rho\in\Delta,\;t_{\rho}\in\operatorname{Supp}(t_{\beta})}x_{\rho}\cdot\rho,\qquad\gamma=\sum_{\rho\in\Delta,\;t_{\rho}\in\operatorname{Supp}(t_{\gamma})}y_{\rho}\cdot\rho$

with $x_{\rho}>0$ and $y_{\rho}>0$. By expanding $\langle\beta|\gamma\rangle$,
the unique nonzero term is $x_{\sigma}y_{\tau}\langle\sigma|\tau\rangle$ where
$\sigma$ and $\tau$ are neighbors in the Coxeter graph, so that
$\langle\sigma|\tau\rangle<0$.

In both cases, we have found $\gamma^{p}$ with the desired properties. This
shows that if $i=j$, the second condition in the lemma doesn’t hold. ∎

###### Proof of Proposition 3.2.

Let $\alpha\in\Phi^{(m)}_{\geq-1}(W_{i})$ and
$\beta\in\Phi^{(m)}_{\geq-1}(W_{j})$ with $1\leq i,j\leq k$, and
$\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$. Because the second condition
in the previous lemma is invariant under $\mathcal{F}$, $\alpha$ and $\beta$
are in the same block of the vertex set partition if and only if
$\mathcal{F}(\alpha)$ and $\mathcal{F}(\beta)$ have the same property. So,
$\mathcal{F}$ stabilizes the vertex set partition in (11), which means that it
is a monomial automorphism. ∎

Let us describe one particular application of the previous proposition.

######  Remark 3.4.

Assume now that $W$ is irreducible, let $\alpha\in\Delta$, and let
$W^{\prime}=W_{\alpha}$ be the maximal standard parabolic subgroup of $W$
obtained by removing $\alpha$ in the Coxeter graph of $W$. Let
$\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)}(W))$. If
$\mathcal{F}(-\alpha^{1})=-\alpha^{1}$, the restriction of $\mathcal{F}$ to
the link of $-\alpha^{1}$ is also an automorphism that we denote
$\mathcal{F}^{\prime}\in\operatorname{Aut}(\Gamma^{(m)}(W^{\prime}))$. Because
every automorphism of $\Gamma^{(m)}(W^{\prime})$ is monomial, there is a type-
preserving permutation of the irreducible factors of $W^{\prime}$ underlying
$\mathcal{F}$. This will be helpful in understanding the stabilizer of
$-\alpha^{1}$ in $\operatorname{Aut}(\Gamma^{(m)}$).

##  4 Diagram automorphism

Let $\mathcal{D}:\Delta\to\Delta$ be an automorphism of the Coxeter graph of
$W$. There is an induced automorphism $W\to W$, and an induced self-bijection
$\Phi\to\Phi$. We keep the notation $\mathcal{D}$ for these induced maps.

###  4.1 Even diagram automorphisms

First, we assume that $\Delta_{\bullet}$ and $\Delta_{\circ}$ are preserved by
$\mathcal{D}$. A non-trivial such automorphism exists in types $A_{n}$ with
$n$ odd, $E_{6}$, $D_{n}$ with $n\geq 4$ (and it is unique except for type
$D_{4}$ where the group of even diagram automorphism is the symmetric group
$\mathfrak{S}_{3}$).

We extend $\mathcal{D}$ to $\Phi^{(m)}_{\geq-1}$ by
$\mathcal{D}(\alpha^{i})=\mathcal{D}(\alpha)^{i}$. It is straighforward to
check that the map $\mathcal{D}$ preserves the compatibility relation, so that
$\mathcal{D}\in\operatorname{Aut}(\Gamma^{(m)})$. It is called an even diagram
automorphism. Moreover, we have
$\mathcal{D}\mathcal{R}=\mathcal{R}\mathcal{D}$.

###  4.2 Odd diagram automorphisms

Now, assume that $\Delta_{\bullet}$ and $\Delta_{\circ}$ are exchanged by
$\mathcal{D}$. A non-trivial such automorphism exists in types $A_{n}$ with
$n$ even, $F_{4}$ and $I_{2}(k)$ (and it is unique).

We extend $\mathcal{D}$ to $\Phi^{(m)}_{\geq-1}$ by

$\mathcal{D}(\alpha^{i})=\begin{cases}\mathcal{D}(\alpha)^{i}&\text{ if
}\alpha\in-\Delta,\\\ \mathcal{D}(\alpha)^{m+1-i}&\text{ if
}\alpha\in\Phi_{+}.\end{cases}$

It is straightforward to check that this gives an automorphism
$\mathcal{D}\in\operatorname{Aut}(\Gamma^{(m)})$, which is called an odd
diagram automorphism. Moreover, we have
$\mathcal{D}\mathcal{R}=\mathcal{R}^{-1}\mathcal{D}$.

###  4.3 The canonical diagram automorphism

Let us recall standard facts of Coxeter theory. Conjugation by the longest
element $w_{0}$ acts on simple reflections as a symmetry of the Coxeter
diagram (see also Remark 2.4). It is the identity when all exponents are even,
and it is the unique nontrivial symmetry of order 2 in other cases ($A_{n}$,
$D_{n}$ with $n$ odd, $E_{6}$, $I_{2}(2k+1)$). If $h$ is even, we have
$c^{h/2}=w_{0}$. More precisely, for $\alpha\in\Delta$ we have
$-c^{h/2}(\alpha)\in\Delta$ and it is the image of $\alpha$ under the symmetry
of the Coxeter diagram.

######  Definition 4.1.

The canonical diagram automorphism $\mathcal{C}\in\operatorname{Diag}$ is the
element of $\operatorname{Diag}$ associated to the map $\Delta\to\Delta$,
$\alpha\mapsto-w_{0}(\alpha)$.

######  Lemma 4.2.

If $h$ is even, we have $\mathcal{R}^{(mh+2)/2}=\mathcal{C}$.

###### Proof.

By the definition of $\mathcal{R}$, we get

$\displaystyle\mathcal{R}^{m}(\alpha^{i})=\begin{cases}c(\alpha)^{i}&\text{ if
}\alpha\in\Phi_{+}\backslash\Delta_{\bullet},\\\ -c(\alpha)^{i-1}&\text{ if
}\alpha\in\Delta_{\circ}\text{ and }i>1,\\\ (-\alpha)^{1}&\text{ if
}\alpha\in\Delta_{\circ}\text{ and }i=1,\\\ c(\alpha)^{m}&\text{ if
}\alpha\in-\Delta_{\circ}\text{ and }i=1,\\\ (-\alpha)^{m}&\text{ if
}\alpha\in-\Delta_{\bullet}\text{ and }i=1.\end{cases}$ (12)

From this, it is straightforward to compute $\mathcal{R}^{mh/2}$ in the
different cases, and show that $\mathcal{R}^{mh/2+1}$ is the map
$\alpha^{i}\mapsto-c^{h/2}(\alpha)^{i}$. ∎

######  Proposition 4.3.

We have $\mathcal{C}\in\operatorname{Dih}$.

###### Proof.

Via the previous lemma, it remains only to consider the case where $h$ is odd,
i.e., $W$ has type $A_{2n}$. Using the combinatorial model of polygon
dissections, the result is clear. ∎

Recall from the introduction that
$\operatorname{Diag}\subset\operatorname{Aut}(\Gamma^{(m)})$ is defined as the
subgroup of diagram automorphisms. To complete this section, note that Lemma
6.2 below characterizes elements of $\operatorname{Diag}$ as the automorphisms
of $\Gamma^{(m)}$ that stabilize $-\Delta^{1}$ (setwise).

##  5 Involutive automorphisms

We define the two involutive automorphisms $\SS$ and $\mathcal{T}$, as
outlined in the introduction.

######  Definition 5.1.

The self-map $\SS$ on $\Phi^{(m)}_{\geq-1}$ is defined by:

$\displaystyle\SS(\alpha^{i})=\begin{cases}\alpha^{i}&\text{ if
}\alpha\in-\Delta_{\circ}\text{ and }i=1\text{ (i)},\\\ (-\alpha)^{m}&\text{
if }\alpha\in-\Delta_{\bullet}\text{ and }i=1\text{ (ii)},\\\
(-\alpha)^{1}&\text{ if }\alpha\in\Delta_{\bullet}\text{ and }i=m\text{
(ii')},\\\ \alpha^{m-i}&\text{ if }\alpha\in\Delta_{\bullet}\text{ and }1\leq
i\leq m-1\text{ (iii)},\\\ c_{\bullet}(\alpha)^{m+1-i}&\text{
if}\alpha\in\Phi_{+}\backslash\Delta_{\bullet}\text{ (iv)}.\end{cases}$ (13)

To gain some insight about this definition, we let the reader check how $\SS$
acts on the vertex partition in (7) as explained in Remark 2.8. Another
helpful verification is the following: in the combinatorial model of polygon
dissections (in type $A_{n}$), this map acts geometrically as a reflection of
the polygon and its diagonals.

######  Lemma 5.2.

We have $\SS^{2}=\mathcal{I}$.

###### Proof.

This is straightforward from the different cases in the definition and from
$c_{\bullet}$ being an involution on $\Phi_{+}\backslash\Delta_{\bullet}$. ∎

######  Proposition 5.3.

The map $\SS$ induces an automorphism of $\Gamma^{(m)}$.

###### Proof.

We show that the map $\SS$ sends a facet of $\Gamma^{(m)}$ to another facet,
using the description in terms of reduced factorization of the Coxeter element
(Section 2.2).

Let $f=(\alpha_{i}^{k_{i}})_{1\leq i\leq n}$ be a facet of $\Gamma^{(m)}$,
indexed as in Proposition 2.7, so that $c=t_{\alpha_{1}}\cdots
t_{\alpha_{n}}$. The canonical factorization of $c$ associated with a facet of
$\Gamma^{(m)}$ is:

$c=w_{\bullet}w_{1}\cdots w_{m}w_{\circ},$ (14)

defined as a coarsening of $t_{\alpha_{1}}\cdots t_{\alpha_{n}}$ where
$w_{\bullet}$ (respectively, $w_{j}$, $w_{\circ}$) contains the factors
$t_{\alpha_{i}}$ such that $\alpha_{i}$ is in $-\Delta_{\bullet}$
(respectively, $\Phi_{+}^{j}$, $-\Delta_{\circ}$). We further refine this by
writing

$w_{j}=w_{j,2}w_{j,1}$

where $w_{j,1}$ contains all factors $t_{\alpha_{i}}$ with
$\alpha_{i}\in\Delta_{\bullet}$.

It is helpful to first assume that $f^{\prime}:=\SS(f)$ is also a facet of
$\Gamma^{(m)}$, and compute what should be the associated canonical
factorization. Denote it:

$c=w^{\prime}_{\bullet}w^{\prime}_{1}\cdots w^{\prime}_{m}w^{\prime}_{\circ},$

and again we write

$w^{\prime}_{i}=w^{\prime}_{i,2}w^{\prime}_{i,1}$

where $w^{\prime}_{i,1}$ contains all factors $t_{\alpha}$ with
$\alpha\in\Delta_{\bullet}$. Now, the definition of $\SS$ gives necessary
relations between all the factors we just defined:

$\displaystyle w^{\prime}_{\circ}$ $\displaystyle=w_{\circ}\text{ (from
\eqref{eq:defS}, case (i))},$ $\displaystyle w^{\prime}_{i,1}$
$\displaystyle=w_{m-i,1}\;\text{where }1\leq i\leq m-1\text{ (from
\eqref{eq:defS}, case (iii))},$ $\displaystyle w^{\prime}_{\bullet}$
$\displaystyle=w_{m,1}\text{ (from \eqref{eq:defS}, case (ii))},$
$\displaystyle w^{\prime}_{m,1}$ $\displaystyle=w_{\bullet}\text{ (from
\eqref{eq:defS}, case (ii'))},$ $\displaystyle w^{\prime}_{i,2}$
$\displaystyle=c_{\bullet}w_{m+1-i,2}^{-1}c_{\bullet}\;\text{where }1\leq
i\leq m\text{ (from \eqref{eq:defS}, case (iv))}.$

About the latter equality, let us explain why we need to take the inverse. Let
$\alpha_{1}^{i},\dots,\alpha_{k}^{i}\in\Phi^{(m)}_{\geq-1}$ be the vertices of
$f$ that contributes to the factors of $w_{i,2}$, ordered so that
$w_{i,2}=t_{\alpha_{1}}\cdots t_{\alpha_{k}}$ (i.e., decreasingly with respect
to $\prec$). Let $\alpha^{\prime}_{i}=c_{\bullet}(\alpha_{i})$, so that
$t_{\alpha^{\prime}_{i}}=c_{\bullet}t_{\alpha_{i}}c_{\bullet}$. Note that
$w_{i,2}\leq c$ implies $c_{\bullet}w_{i,2}c_{\bullet}\leq
c_{\bullet}cc_{\bullet}=c^{-1}$, so that
$c_{\bullet}w_{i,2}^{-1}c_{\bullet}\leq c$ (it is well-known and easy to see
from the definition that the absolute order is invariant under conjugation).
The way to order the elements $t_{\alpha^{\prime}_{i}}$ and get an element
below $c$ in the absolute order is thus
$c_{\bullet}w_{i,2}^{-1}c_{\bullet}=t_{\alpha^{\prime}_{k}}\cdots
t_{\alpha^{\prime}_{1}}$.

Now, let us check that these are indeed the factors of a factorization of $c$:

$\displaystyle w^{\prime}_{\bullet}w^{\prime}_{1}\cdots
w^{\prime}_{m}w^{\prime}_{\circ}$
$\displaystyle=w^{\prime}_{\bullet}w^{\prime}_{1,2}w^{\prime}_{1,1}\cdots
w^{\prime}_{m,2}w^{\prime}_{m,1}w^{\prime}_{\circ}$
$\displaystyle=w_{m,1}\cdot c_{\bullet}w_{m,2}^{-1}c_{\bullet}\cdot
w_{m-1,1}\cdot c_{\bullet}w_{m-1,2}^{-1}c_{\bullet}\cdot w_{m-2,1}\cdots
c_{\bullet}w_{1,2}^{-1}c_{\bullet}\cdot w_{\bullet}w_{\circ}.$

Because $c_{\bullet}$ commutes with $w_{i,1}$, this gives:

$\displaystyle w^{\prime}_{\bullet}w^{\prime}_{1}\cdots
w^{\prime}_{m}w^{\prime}_{\circ}$ $\displaystyle=w_{m,1}\cdot c_{\bullet}\cdot
w_{m,2}^{-1}w_{m-1,1}w_{m-1,2}^{-1}w_{m-2,1}\cdots
w_{1,2}^{-1}c_{\bullet}\cdot w_{\bullet}w_{\circ}$
$\displaystyle=c_{\bullet}(w_{1}\cdots w_{m})^{-1}\cdot
c_{\bullet}w_{\bullet}w_{\circ}$
$\displaystyle=c_{\bullet}(w_{\circ}c^{-1}w_{\bullet})\cdot
c_{\bullet}w_{\bullet}w_{\circ}=c_{\bullet}w_{\circ}c_{\circ}c_{\bullet}w_{\bullet}c_{\bullet}w_{\bullet}w_{\circ}=c_{\bullet}w_{\circ}c_{\circ}w_{\circ}=c.$

The previous computations prove the proposition. Indeed, if
$f=\\{\alpha_{1}^{i_{1}},\dots,\alpha_{n}^{i_{n}}\\}$ and
$f^{\prime}=\\{\SS(\alpha_{1}^{i_{1}}),\dots,\SS(\alpha_{n}^{i_{n}})\\}$, the
factorization $c=w^{\prime}_{\bullet}w^{\prime}_{1}\cdots
w^{\prime}_{m}w^{\prime}_{\circ}$ can be refined as a reflection factorization
that proves $f^{\prime}\in\Gamma^{(m)}$ via Proposition 2.7.

An alternative proof would be check that $\SS$ preserves the compatibility
relation $\|$. This is straightforward, although a bit long because of the
various cases to consider. ∎

######  Lemma 5.4.

We have $\SS\mathcal{R}\SS=\mathcal{R}^{-1}$.

###### Proof.

We make the composition $\mathcal{SR}$ explicit. First,

$\mathcal{SR}(\alpha^{i})=\begin{cases}\SS(\alpha^{i+1})=\alpha^{m-i-1}&\text{
if }\alpha\in\Delta_{\bullet}\text{ and }1\leq i\leq m-2,\\\
\SS(\alpha^{i+1})=c_{\bullet}(\alpha)^{m-i}&\text{ if
}\alpha\in\Phi_{+}\backslash\Delta_{\bullet}\text{ and }1\leq i\leq
m-1,\end{cases}$

so $\mathcal{SR}$ is an involution on the elements considered in each case.
Second,

$\mathcal{SR}(\alpha^{i})=\begin{cases}\SS((\alpha)^{1})=(-\alpha)^{1}&\text{
if }\alpha\in\Delta_{\circ}\text{ and }i=m,\\\
\SS(c(\alpha)^{1})=\SS(c_{\bullet}(-\alpha)^{1})=(-\alpha)^{m}&\text{ if
}\alpha\in-\Delta_{\circ}\text{ and }i=1,\end{cases}$

and

$\mathcal{SR}(\alpha^{i})=\begin{cases}\SS((-\alpha)^{1})=(-\alpha)^{m-1}&\text{
if }\alpha\in-\Delta_{\bullet}\text{ and }i=1,\\\
\SS(\alpha^{m})=(-\alpha)^{1}&\text{ if }\alpha\in\Delta_{\bullet}\text{ and
}i=m-1,\end{cases}$

so $\SS\mathcal{R}$ is also an involution on these elements. Eventually, we
have:

$\SS\mathcal{R}(\alpha^{i})=\SS(c(\alpha)^{1})=c_{\circ}(\alpha)^{m}\quad\text{
if }\alpha\in\Phi_{+}\backslash\Delta_{\circ}\text{ and }i=m.$

(To check this, distinguish the cases $\alpha\in\Delta_{\bullet}$, $\alpha\in
c_{\circ}(\Delta_{\bullet})$, and
$\alpha\in\Phi_{+}\backslash(\Delta_{\circ}\cup\Delta_{\bullet}\cup
c_{\circ}(\Delta_{\bullet}))$). It is now clear that $\mathcal{SR}$ is an
involution. ∎

The previous lemma means that
$\langle\mathcal{R},\SS\rangle\subset\operatorname{Aut}(\Gamma^{(m)})$ is a
dihedral subgroup. Recall from the introduction that it is denoted
$\operatorname{Dih}$.

######  Definition 5.5.

The self-map $\mathcal{T}$ on $\Phi^{(m)}_{\geq-1}$ is defined by:

$\displaystyle\mathcal{T}(\alpha^{i})=\begin{cases}\alpha^{i}&\text{ if
}\alpha\in-\Delta_{\bullet}\text{ and }i=1\text{ (i)},\\\ -\alpha^{1}&\text{
if }\alpha\in\pm\Delta_{\circ}\text{ and }i=1\text{ (ii)},\\\
\alpha^{m+2-i}&\text{ if }\alpha\in\Delta_{\circ}\text{ and }2\leq i\leq
m\text{ (iii)},\\\ c_{\circ}(\alpha)^{m+1-i}&\text{ if
}\alpha\in\Phi_{+}\backslash\Delta_{\circ}\text{ (iv)}.\end{cases}$ (15)

######  Proposition 5.6.

We have $\mathcal{T}\in\operatorname{Aut}(\Gamma^{(m)})$, moreover
$\mathcal{T}=\iota\SS\iota$ (i.e., $\check{\SS}=\mathcal{T}$ and
$\check{\mathcal{T}}=\SS$).

The proof is straightforward, by black/white symmetry.

######  Remark 5.7.

When $m=1$, these maps $\SS$ and $\mathcal{T}$ are those considered by Fomin
and Zelevinsky in [10]. One can check that $\SS\mathcal{T}=\mathcal{R}^{m}$
(the map $\mathcal{R}^{m}$ is explicit in (12)). The two maps $\SS$ and
$\mathcal{T}$ are thus generators of $\operatorname{Dih}$ if $m$ is relatively
prime with the order of $\mathcal{R}$. This happens in the following two
situations:

  * •

the order of $\mathcal{R}$ is $mh+2$ and $m$ is relatively prime to $mh+2$
(i.e., $m$ is odd).

  * •

the order of $\mathcal{R}$ is $\frac{mh+2}{2}$ ($h$ is even in this case, so
that $m\frac{h}{2}+1$ is relatively prime to $m$).

Because this is not exhaustive, it is not possible to build the dihedral group
of symmetries with $\SS$ and $\mathcal{T}$ as generators, although they look
like a natural set of generators.

##  6 Stabilizer of a pair of vertices

Here and in the following sections, we take $\alpha,\beta\in\Delta$ such that
$\beta$ is the unique neighbor of some $\alpha$ in the Coxeter graph of $W$
($\alpha$ is a leaf). By symmetry (more precisely, using the isomorphism
between $\Gamma^{(m)}$ and $\check{\Gamma}^{(m)}$ from Section 2.3), we can
assume $\alpha\in\Delta_{\bullet}$ and $\beta\in\Delta_{\circ}$ in Proposition
6.1 and the other case follows. Let $W_{\beta}$ be the maximal standard
parabolic subgroup obtained by removing $\beta$ in the Coxeter graph of $W$,
and similarly for $\alpha$.

######  Proposition 6.1.

The (pointwise) stabilizer of $\\{-\alpha^{1},-\beta^{1}\\}$ in
$\operatorname{Aut}(\Gamma^{(m)})$ is generated by diagram automorphisms.

######  Lemma 6.2.

Assume that $W$ is irreducible, and its rank is at least $2$. If
$\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ is such that
$\mathcal{F}(-\rho^{1})=-\rho^{1}$ for all $\rho\in\Delta$, then
$\mathcal{F}=\mathcal{I}$.

###### Proof.

Consider the restriction of $\mathcal{F}$ to the link of a vertex $-\rho^{1}$,
for $\rho\in\Delta$. Using an induction hypothesis, we can apply Proposition
7.1 to the maximal standard parabolic subgroup $W_{\rho}$, and find that the
restriction of $\mathcal{F}$ on $\Gamma^{(m)}(W_{\rho})$ is the identity. In
particular, $\mathcal{F}(\rho^{i})$ for $\rho\in\Delta$ and $1\leq i\leq m$
(since $\rho^{i}$ is in the link of $-\tau^{1}$ if $\tau\in\Delta$ and
$\tau\neq\rho$).

To establish an induction showing that $\mathcal{F}=\mathcal{I}$, we consider
the block decomposition in (7).

  * •

We have shown that if the two neighbor blocks $-\Delta_{\circ}^{1}$ and
$-\Delta_{\bullet}^{1}$ contain only fixed points, this is also the case (in
particular) for the next blocks $\Delta_{\circ}^{1}$ and
$\Delta_{\bullet}^{m}$.

  * •

The automorphisms $\mathcal{R}$, $\SS$, and $\mathcal{T}$ can be used to send
any pair of neighbor blocks in (7) to another one. Therefore, the previous
property holds for any pair of consecutive blocks, not just
$-\Delta_{\circ}^{1}$ and $-\Delta_{\bullet}^{1}$.

We conclude that every element of $\Phi^{(m)}_{\geq-1}$ is fixed by
$\mathcal{F}$. ∎

In particular, the previous lemma settles the case of rank 2 in Proposition
6.1. We thus assume that the rank of $W$ is at least $3$ in the rest of this
section.

######  Lemma 6.3.

Suppose that the rank of $W$ is $3$. Write
$\Delta_{\bullet}=\\{\alpha,\gamma\\}$ and $\Delta_{\circ}=\\{\beta\\}$. There
exists no automorphism $\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ such
that $\mathcal{F}(-\alpha^{1})=-\alpha^{1}$ and
$\mathcal{F}(-\gamma^{1})=\gamma^{m}$.

###### Proof.

By the finite-type classification, we have
$t_{\alpha}t_{\beta}t_{\alpha}=t_{\beta}t_{\alpha}t_{\beta}$ or
$t_{\gamma}t_{\beta}t_{\gamma}=t_{\beta}t_{\gamma}t_{\beta}$. The statement is
symmetric in $\alpha$ and $\gamma$, because
$\mathcal{F}(-\alpha^{1})=-\alpha^{1}$ and
$\mathcal{F}(-\gamma^{1})=\gamma^{m}$ is equivalent to
$\SS\mathcal{F}(-\alpha^{1})=\alpha^{m}$ and
$\SS\mathcal{F}(-\gamma^{1})=-\gamma^{1}$. So, we assume
$t_{\gamma}t_{\beta}t_{\gamma}=t_{\beta}t_{\gamma}t_{\beta}$.

Consider the 1-dimensional face $f=\\{-\alpha^{1},-\gamma^{1}\\}$. We have:

$\forall\rho^{\ell}\text{ vertex of
}\operatorname{Link}(f),\;\operatorname{Link}(\\{\rho^{\ell}\\})\simeq\Gamma^{(m)}(A_{1}\times
A_{1}).$ (16)

Indeed, via (3), $\operatorname{Link}(f)$ is the $0$-dimensional complex with
vertices $\\{-\beta^{1},\beta^{1},\dots,\beta^{m}\\}$. These vertices are in
the $\mathcal{R}$-orbit of $-\beta^{1}$. So their links are all isomorphic to
the link of $-\beta^{1}$, which is
$\Gamma^{(m)}(W_{\beta})\simeq\Gamma^{(m)}(A_{1}\times A_{1}$).

Now, let $f^{\prime}=\\{-\alpha^{1},\gamma^{m}\\}$. Since the property in (16)
is invariant under automorphisms, it remains only to show that this property
doesn’t hold with $f^{\prime}$ in place of $f$ to conclude that no
automorphism sends $f$ to $f^{\prime}$. To do this, let
$\rho=t_{\gamma}(\beta)$. We check:

  * •

$f^{\prime}\cup\\{\rho^{m}\\}\in\Gamma^{(m)}$. The relation
$-\alpha^{1}\mathrel{\|}\rho^{m}$ is clear via (3). The relation
$\gamma^{m}\mathrel{\|}\rho^{m}$ holds via (5) because: i)
$t_{\rho}=t_{\gamma}t_{\beta}t_{\gamma}$ so that
$t_{\rho}t_{\gamma}=t_{\gamma}t_{\beta}\leq c$, and ii)
$\langle\rho|\gamma\rangle=\langle
t_{\gamma}(\beta)|\gamma\rangle=-\langle\beta|\gamma\rangle>0$.

  * •

$\operatorname{Link}(\\{\rho^{m}\\})\not\simeq\Gamma^{(m)}(A_{1}\times
A_{1})$. We have
$t_{\rho}=t_{\gamma}t_{\beta}t_{\gamma}=t_{\beta}t_{\gamma}t_{\beta}$ (by the
assumption at the beginning of this proof). Since $c_{\circ}=t_{\beta}$, it
follows that $c_{\circ}(\rho)=\gamma$, and
$\mathcal{T}(\rho^{m})=\gamma^{1}=\mathcal{R}(-\gamma^{1})$. So,
$\\{\rho^{m}\\}$ and $\\{-\gamma^{1}\\}$ have isomorphic links. Since $W$ is
irreducible, $W_{\gamma}\not\simeq A_{1}\times A_{1}$, and
$\operatorname{Link}(\\{\rho^{m}\\})\simeq\Gamma^{(m)}(W_{\gamma})\not\simeq\Gamma^{(m)}(A_{1}\times
A_{1})$ by Theorem 2.12.

So, (16) doesn’t hold with $f^{\prime}$ in place of $f$. ∎

###### Proof of Proposition 6.1.

Let $\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ with
$\mathcal{F}(-\alpha^{1})=-\alpha^{1}$ and
$\mathcal{F}(-\beta^{1})=-\beta^{1}$. Since
$\mathcal{F}(-\alpha^{1})=-\alpha^{1}$, the restriction of $\mathcal{F}$ to
the link of $-\alpha^{1}$ gives an element
$\mathcal{F}^{\prime}\in\operatorname{Aut}(\Gamma^{(m)}(W_{\alpha}))$. Since
$\\{-\alpha^{1},-\beta^{1}\\}\in\Gamma^{(m)}$, $-\beta^{1}$ is in the link of
$-\alpha^{1}$ and naturally identifies to a vertex
$-\beta^{1}\in\Gamma^{(m)}(W_{\alpha})$ which is fixed by
$\mathcal{F}^{\prime}$. Using an induction hypothesis, we can apply
Proposition 8.6.

Proposition 8.6 gives $\mathcal{F}^{\prime}=\SS^{\prime}\mathcal{D}^{\prime}$
or $\mathcal{F}^{\prime}=\mathcal{D}^{\prime}$, where $\mathcal{D}^{\prime}$
is a diagram automorphism of $\Gamma^{(m)}(W_{\alpha})$ fixing $-\beta^{1}$
(and $\mathcal{S}^{\prime}$ is the restriction of $\mathcal{S}$ in
$\operatorname{Aut}(\Gamma^{(m)}(W_{\alpha}))$). Note that
$\mathcal{D}^{\prime}$ is the restriction of a diagram automorphism
$\mathcal{D}$ of $\Gamma^{(m)}(W)$ fixing $-\alpha^{1}$ and $-\beta^{1}$ (this
is easily checked at the level of Coxeter graphs). Our goal is to show that
$\mathcal{F}=\mathcal{D}$. Let $\mathcal{G}:=\mathcal{F}\mathcal{D}^{-1}$.

By construction $\mathcal{G}$ fixes $-\alpha^{1}$ and $-\beta^{1}$, moreover
its restriction on $\Gamma^{(m)}(W_{\alpha})$, denoted $\mathcal{G}^{\prime}$,
is $\SS^{\prime}$ or the identity. By way of contradiction, assume that
$\mathcal{G}^{\prime}=\SS^{\prime}$. Let $W_{I}$ be the standard parabolic
subgroup of $W$ with simple roots $\beta$ and its neighbors in the Coxeter
graph. If $\gamma\in\Delta$ is at distance 2 from $\beta$, the vertex
$-\gamma^{1}$ is fixed by $\SS^{\prime}$ (because $\gamma\in\Delta_{\circ}$,
just like $\beta$), and consequently it is also fixed by $\mathcal{G}$. It
follows that the restriction of $\mathcal{G}$ gives an automorphism
$\mathcal{H}\in\operatorname{Aut}(\Gamma^{(m)}(W_{I}))$. The situation can be
summarized as follows.

  * •

By definition of $W_{I}$, its Coxeter graph is a star centered at $\beta$. By
the finite-type classification, either it has rank 3, or it is of type
$D_{4}$.

  * •

The vertices $-\alpha^{1}$ and $-\beta^{1}$ are fixed by
$\mathcal{H}\in\operatorname{Aut}(\Gamma^{(m)}(W_{I}))$. The other vertices
$\gamma$ and possibly $\delta$ (if $W_{I}$ has type $D_{4}$) are such that
$\mathcal{H}(-\gamma^{1})=\gamma^{m}$ and
$\mathcal{H}(-\delta^{1})=\delta^{m}$.

If the rank of $W_{I}$ is $3$, Lemma 6.3 gives a contradiction, so that such
$\mathcal{H}$ doesn’t exist. In the case where $W_{I}$ has type $D_{4}$,
consider the composition $\SS\mathcal{H}$: $-\beta^{1}$, $-\gamma^{1}$,
$-\delta^{1}$ are fixed, and the image of $-\alpha^{1}$ is $\alpha^{m}$. By
considering the link of $-\delta^{1}$, we see that the nonexistence of such
$\mathcal{H}$ in type $D_{4}$ follows from the nonexistence in type $A_{3}$.

At this point, we have proved that $\mathcal{G}^{\prime}$ is the identity.
This proves that $\mathcal{G}$ fixes all the vertices $-\alpha^{1}$ for
$\alpha\in\Delta$. By Lemma 6.2, it follows that $\mathcal{G}=\mathcal{I}$, so
$\mathcal{F}=\mathcal{D}$. (Note that the assumption on the rank in Lemma 6.2
holds, because we assumed that the rank of $W$ is at least 3.) ∎

##  7 Stabilizer of vertices

Recall that $\alpha\in\Delta_{\bullet}$ and $\beta\in\Delta_{\circ}$ are such
that $\beta$ is the unique neighbor of $\alpha$ in the Coxeter graph of $W$.
The goal of this section is to describe the stabilizer of the vertex
$-\beta^{1}$ in $\operatorname{Aut}(\Gamma^{(m)})$.

######  Proposition 7.1.

Let $\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ be such that
$\mathcal{F}(-\beta^{1})=-\beta^{1}$. Then we have either
$\mathcal{F}=\mathcal{D}$ or $\mathcal{F}=\SS\mathcal{D}$, where $\mathcal{D}$
is an even diagram automorphism such that
$\mathcal{D}(-\beta^{1})=-\beta^{1}$.

######  Lemma 7.2.

Let $\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ be such that
$\mathcal{F}(-\beta^{1})=-\beta^{1}$. If $\mathcal{F}$ stabilizes the vertex
set $\\{-\alpha^{1},\alpha^{1},\dots,\alpha^{m}\\}$, then it also stabilizes
the vertex set $\\{-\alpha^{1},\alpha^{m}\\}$.

###### Proof.

In this proof, a tuple of 4 elements of $\Phi^{(m)}_{\geq-1}$ is called a
square if their induced subgraph in the compatibility graph (the $1$-skeleton
of $\Gamma^{(m)}$) is a cycle of length 4. The idea is to characterize the
pair $\\{-\alpha^{1},\alpha^{m}\\}$ among all pairs of elements in
$\\{-\alpha^{1},\alpha^{1},\dots,\alpha^{m}\\}$ via certain squares containing
$-\beta^{1}$. Note that $-\beta^{1}$ is compatible with elements in
$\\{-\alpha^{1},\alpha^{1},\dots,\alpha^{m}\\}$ (via (3)), and
$\\{-\alpha^{1},\alpha^{1},\dots,\alpha^{m}\\}$ doesn’t contain a compatible
pair (via Lemma 2.9). We check the following properties:

  * •

$\\{-\beta^{1},-\alpha^{1},\alpha^{i}\\}$ can be completed to form a square if
$i<m$. We take $\beta^{m}$ as the fourth vertex. We have
$\beta^{m}\mathrel{\|}\alpha^{i}$ via (5) ($t_{\alpha}t_{\beta}\leq c$ can be
seen by taking a subword of $c=c_{\bullet}c_{\circ}$).

  * •

$\\{-\beta^{1},\alpha^{i},\alpha^{j}\\}$ can be completed to form a square if
$1\leq i<j\leq m$. Define $\gamma=t_{\alpha}(\beta)$, so that
$t_{\gamma}t_{\alpha}=t_{\alpha}t_{\beta}\leq c$. If $i>1$, this shows that we
can take $\gamma^{1}$ as the fourth vertex of the square (via (5)). If $i=1$,
note that we also have
$\langle\gamma|\alpha\rangle=-\langle\beta|\alpha\rangle>0$ (we have
$\langle\beta|\alpha\rangle<0$ since $\alpha$ and $\beta$ are neighbors in the
Coxeter graph). Again, it follows that $\gamma^{1}$ can be chosen as the
fourth vertex of a square (via (4)).

  * •

$\\{-\beta^{1},-\alpha^{1},\alpha^{m}\\}$ cannot be completed to form a
square. Assume otherwise, and let $\gamma^{i}$ be the fourth vertex.

    * –

From $-\alpha^{1}\mathrel{\|}\gamma^{i}$, we get $t_{\alpha}t_{\gamma}\leq c$.
From $\alpha^{m}\mathrel{\|}\gamma^{i}$, we get $t_{\gamma}t_{\alpha}\leq c$.
So $t_{\alpha}t_{\gamma}=t_{\gamma}t_{\alpha}$ (this follows from the fact
that elements below $c$ in the absolute order is a lattice, because
$t_{\alpha}t_{\gamma}$ and $t_{\gamma}t_{\alpha}$ are rank 2 elements that
covers the rank 1 elements $t_{\alpha}$ and $t_{\gamma}$). Thus, we get
$\langle\alpha|\gamma\rangle=0$.

    * –

Write $\gamma=\sum_{\rho\in\Delta}x_{\rho}\cdot\rho$ with real coefficients
$x_{\rho}\geq 0$. Since $\gamma^{i}$ is the fourth side of the square, we have
$-\beta^{1}\centernot{\|}\gamma^{i}$, so $x_{\beta}>0$. Moreover
$-\alpha^{1}\mathrel{\|}\gamma^{i}$ by definition of the square, so
$c_{\alpha}=0$. Now,
$\langle\alpha,\gamma\rangle=\sum_{\rho\in\Delta}x_{\rho}\langle\alpha|\rho\rangle=x_{\beta}\langle\alpha|\beta\rangle<0$
(since $\alpha$ and $\beta$ are neighbors in the Coxeter graph), so that
$\langle\alpha|\gamma\rangle<0$.

This gives a contradiction and completes this point.

Since squares are preserved by the automorphism $\mathcal{F}$, the triple
$\\{-\beta^{1},-\alpha^{1},\alpha^{m}\\}$ is preserved as the unique one that
cannot be completed to form a square among the triples considered above. The
result follows. ∎

###### Proof of Proposition 7.1.

The decomposition of $W_{\beta}$ in irreducible components is written

$W_{\beta}\simeq W_{1}\times\dots\times W_{k}.$

Up to reindexing, assume that $W_{1}$ is the factor that contains the
reflection $t_{\alpha}$. By Proposition 3.2, the restriction of $\mathcal{F}$
on $\Gamma^{(m)}(W_{\beta})$ permutes the irreducible factor of type $A_{1}$
in $W_{\beta}$. These irreducible factor of type $A_{1}$ correspond to leaves
of the Coxeter graph that are neighbors of $\beta$. Clearly, any permutation
of these leaves can be realized by an automorphism of the Coxeter graph that
stabilizes $\beta$. So, there is a diagram automorphism $\mathcal{D}$ such
that $\mathcal{F}\mathcal{D}$ stabilizes
$\Gamma^{(m)}(W_{1})=\\{-\alpha^{1}\alpha^{1},\dots,\alpha^{m}\\}$ (setwise)
and $-\beta^{1}$.

By Lemma 7.2, $\mathcal{F}\mathcal{D}$ stabilizes
$\\{-\alpha^{1},\alpha^{m}\\}$. If $\mathcal{F}\mathcal{D}$ stabilizes
$-\alpha^{1}$ and $-\beta^{1}$, it follows from Lemma 6.1 that it is a diagram
automorphism. Otherwise, $\SS\mathcal{F}\mathcal{D}$ stabilizes $-\alpha^{1}$
and $-\beta^{1}$, and it follows from Lemma 6.1 that
$\SS\mathcal{F}\mathcal{D}$ is a diagram automorphism. This completes the
proof. ∎

##  8 Automorphism groups

######  Lemma 8.1.

We have
$\operatorname{Aut}(\Gamma^{(m)})=\operatorname{Dih}\cdot\operatorname{Diag}$.

###### Proof.

Let $\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$, and
$\alpha,\beta\in\Delta$ as in the previous sections. There is an integer $i$
such that $\mathcal{R}^{i}\mathcal{F}(-\beta^{1})\in-\Delta^{1}$.

If $\mathcal{R}^{i}\mathcal{F}(-\beta^{1})=-\beta^{1}$, by Proposition 7.1 we
obtain $\mathcal{R}^{i}\mathcal{F}=\SS\mathcal{D}$ or
$\mathcal{R}^{i}\mathcal{F}=\mathcal{D}$, where
$\mathcal{D}\in\operatorname{Diag}$. It follows
$\mathcal{F}\in\operatorname{Dih}\cdot\operatorname{Diag}$.

In the general case, let $\gamma\in\Delta$ be such that $\beta\neq\gamma$ and
$\mathcal{R}^{i}\mathcal{F}(-\beta^{1})=-\gamma^{1}$. The links at
$-\beta^{1}$ and $-\gamma^{1}$ are isomorphic since
$\mathcal{R}^{i}\mathcal{F}$ is an automorphism of $\Gamma^{(m)}$. By Theorem
2.12, it follows that the Coxeter graphs of $W_{\beta}$ and $W_{\gamma}$ are
isomorphic. So, there is a diagram automorphism $\mathcal{D}$ such that
$\mathcal{D}(-\gamma^{1})=-\beta^{1}$ (this is easily checked at the level of
Coxeter graphs). So $\mathcal{G}=\mathcal{R}^{i}\mathcal{F}\mathcal{D}$ is
such that $\mathcal{G}(-\gamma^{1})=-\gamma^{1}$. If
$\gamma\in\Delta_{\circ}$, we obtain
$\mathcal{G}\in\operatorname{Dih}\cdot\operatorname{Diag}$ by Proposition 7.1,
and $\mathcal{F}\in\operatorname{Dih}\cdot\operatorname{Diag}$ follows.
Otherwise, by black/white symmetry there is an analog statement (with
$\mathcal{T}$ in place of $\SS$), and we get
$\mathcal{G}\in\operatorname{Dih}\cdot\operatorname{Diag}$ again. ∎

######  Lemma 8.2.

We have
$\operatorname{Dih}\mathrel{\cap}\operatorname{Diag}=\langle\mathcal{C}\rangle$.

###### Proof.

Note that $\mathcal{C}\in\operatorname{Dih}\mathrel{\cap}\operatorname{Diag}$,
by Proposition 4.3. It remains to check that there is no other diagram
automorphism in $\operatorname{Dih}$.

Suppose that $\mathcal{R}^{i}\in\operatorname{Diag}$ for some integer $i$. By
Proposition 2.3, for any $\rho\in\Delta$ we have
$\mathcal{R}^{i}(-\rho^{1})=-\rho^{1}$ or
$\mathcal{R}^{i}(-\rho^{1})=w_{0}(\rho)^{1}$. It follows that
$\mathcal{R}^{i}$ is either $\mathcal{I}$ or $\mathcal{C}$. It remains only to
show that $\mathcal{R}^{i}\SS$ cannot be a diagram automorphism other than
$\mathcal{I}$ or $\mathcal{C}$.

Let us first consider the case where $h$ is odd, i.e., $W$ has type $A_{2n}$.
(It is treated separately because this is the only case where
$\mathcal{C}=\mathcal{R}^{i}\SS$ for some integer $i$.) Since
$\operatorname{Diag}=\\{\mathcal{I},\mathcal{C}\\}$, there is nothing to
prove. We assume that $h$ is even in the rest of this proof.

It remains to show $\mathcal{R}^{i}\SS\notin\operatorname{Diag}$. Assume
otherwise and let $\rho\in\Delta_{\circ}$. We get
$\mathcal{R}^{i}(-\rho^{1})=\mathcal{R}^{i}\SS(-\rho^{1})\in-\Delta^{1}$.
Since $h$ is even, by Lemma 4.2 it follows that $i\equiv 0\mod\frac{mh+2}{2}$.
But we then have $\mathcal{R}^{i}=\mathcal{I}$ or
$\mathcal{R}^{i}=\mathcal{C}$, i.e., $\mathcal{R}^{i}\in\operatorname{Diag}$.
This is not possible since $\SS\notin\operatorname{Diag}$. ∎

######  Lemma 8.3.

We have $\operatorname{Dih}\triangleleft\operatorname{Aut}(\Gamma^{(m)})$.

###### Proof.

If $|{\operatorname{Diag}}|=2$, we have
$[\,\operatorname{Aut}(\Gamma^{(m)}):\operatorname{Dih}\,]\leq 2$ by the
previous proposition and the result follows. This covers all cases except
$D_{4}$.

If $\operatorname{Diag}$ contains only even diagram automorphisms, it commutes
with $\operatorname{Dih}$ and the result follows from the previous lemma. This
cover all cases except $F_{4}$, $A_{n}$ ($n$ even), $I_{2}(k)$. ∎

######  Theorem 8.4.

We have:

$\operatorname{Aut}(\Gamma^{(m)})=\operatorname{Dih}\rtimes(\operatorname{Diag}/\langle\mathcal{C}\rangle).$

###### Proof.

First note that $\langle\mathcal{C}\rangle\triangleleft\operatorname{Diag}$,
because its index is at most 2 (in all types except $D_{4}$) or because it is
the trivial subgroup (to include $D_{4}$).

If $\langle\mathcal{C}\rangle$ is the trivial subgroup, the result follows
from the previous lemmas. If $\operatorname{Diag}=\langle\mathcal{C}\rangle$,
we get $\operatorname{Aut}(\Gamma^{(m)})=\operatorname{Dih}$ and the result is
clear. All cases are covered. ∎

######  Corollary 8.5.

Let $\omega=|{\operatorname{Diag}}|$. We have:

$\big{|}\operatorname{Aut}(\Gamma^{(m)})\big{|}=(mh+2)\omega.$

###### Proof.

By the previous theorem, we have

$\big{|}\operatorname{Aut}(\Gamma^{(m)})\big{|}=\frac{|{\operatorname{Dih}}|}{|\mathcal{C}|}\omega.$

Moreover, $|{\operatorname{Dih}}|=2|\mathcal{R}|$. We have
$|{\mathcal{R}}|=\frac{mh+2}{2}|{w_{0}}|$ by (2), and clearly
$|{w_{0}}|=|{\mathcal{C}}|$. Putting this together completes the proof. ∎

The final step is the following generalization of Proposition 7.1, without
restriction on the chosen vertex of the Coxeter diagram. Recall that it was
used in the proof of Proposition 6.1 (where it was assumed to hold via an
induction hypothesis).

######  Proposition 8.6.

Let $\rho\in\Delta_{\circ}$ and
$\mathcal{F}\in\operatorname{Aut}(\Gamma^{(m)})$ be such that
$\mathcal{F}(-\rho^{1})=-\rho^{1}$. Then we have either
$\mathcal{F}=\mathcal{D}$ or $\mathcal{F}=\SS\mathcal{D}$, where $\mathcal{D}$
is an even diagram automorphism such that $\mathcal{D}(-\rho^{1})=-\rho^{1}$.
(And by black/white symmetry, there is a similar statement with
$\Delta_{\bullet}$ and $\mathcal{T}$ in place of $\Delta_{\circ}$ and $\SS$.)

###### Proof.

We use the previous theorem and the orbit-stabilizer theorem. This suffices to
show that the stabilizer contains only the listed elements. In most cases, the
stabilizer has order 2 and it follows that it is $\\{\mathcal{I},\SS\\}$. We
only give details about the other cases:

  * •

If $\omega=2$ and the orbit of $-\rho^{1}$ has cardinality $\frac{mh+2}{2}$:
it means that the stabilizer has order 4, moreover $-\rho^{1}$ is fixed by the
nontrivial element $\mathcal{D}\in\operatorname{Diag}$. It follows that the
stabilizer is $\\{\mathcal{I},\SS,\mathcal{D},\SS\mathcal{D}\\}$ (we have
$\SS\mathcal{D}=\mathcal{D}\SS$ since $\mathcal{D}$ is even).

  * •

In the the case of $D_{4}$, $\omega=6$ and the orbit of $-\rho^{1}$ has
cardinality $\frac{mh+2}{2}$. The stabilizer has cardinality $4$ if $\rho$ is
a leaf of the Coxeter diagram and $12$ if it is the central vertex. There are
2 elements of $\operatorname{Diag}$ stabilizing $-\rho^{1}$ in the first case,
and 6 in the second case. This completes the proof.

All cases have been covered. ∎

##  9 Final remarks

Understanding the automorphism group of a combinatorial object is certainly
interesting on its own, but let us mention some problems that motivate the
present work.

Cluster parking functions were recently introduced in a joint work with
Douvropoulos [7]. They can be seen combinatorially as faces of $\Gamma^{(m)}$
with additional structure (essentially a labeling by a coset in $W$). It is
conjectured that symmetries of $\Gamma^{(m)}$ have a counterpart in cluster
parking functions (for example, see [7, Open problem 9.3]). This is completely
apparent in the combinatorial models in types $A$ and $B$, and a general
Coxeter-theoretic result in this direction would be very interesting.

As briefly mentioned in the introduction, the generalized cluster complex has
a natural representation-theoretic interpretation (via an orbit category in
the derived category of certain algebras, see [4] for a survey). In this
context, $\mathcal{R}$ naturally identifies with a shift functor. It would be
very interesting to also give a representation-theoretic interpretation of the
involutive automorphisms in this context.

## References

  * [1] Ibrahim Assem, Ralf Schiffler, Vasilisa Shramchenko: Cluster automorphisms. Proc. Lond. Math. Soc. (3) 104, No. 6, 1271–1302 (2012).
  * [2] Ibrahim Assem, Ralf Schiffler, Vasilisa Shramchenko: Cluster automorphisms and compatibility of cluster variables. Glasg. Math. J. 56, No. 3, 705–720 (2014).
  * [3] Tom Brady, Colum Watt: Non-crossing partition lattices in finite real reflection groups. Trans. Amer. Math. Soc. 360(4) (2008), 1983–2005.
  * [4] Aslak Bakke Buan: An introduction to higher cluster categories. Bull. Iran. Math. Soc. 37, No. 2, Part 2, 137–157 (2011).
  * [5] Wen Chang, Bin Zhu: Cluster automorphism groups of cluster algebras of finite type. J. Algebra 447, 490–515 (2016).
  * [6] Wen Chang, Bin Zhu: Cluster automorphism groups and automorphism groups of exchange graphs. Pac. J. Math. 307, No. 2, 283–302 (2020).
  * [7] Theo Douvropoulos, Matthieu Josuat-Vergès: The generalized cluster complex: refined enumeration of faces and related parking spaces. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 19 (2023), 069, 40 pages.
  * [8] Theo Douvropoulos, Matthieu Josuat-Vergès: Cluster parking functions, Preprint, https://arxiv.org/abs/2402.03052.
  * [9] Sergey Fomin, Nathan Reading: Generalized cluster complexes and Coxeter combinatorics. Int. Math. Res. Notices 2005, no. 44, 2709–2757.
  * [10] Sergey Fomin, Andrei Zelevinsky: Y-systems and generalized associahedra. Ann. Math. (2) 158(3) (2003), 977–1018.
  * [11] Siyang Liu, Fang Li: Periodicities in cluster algebras and cluster automorphism groups. Algebra Colloq. 28, No. 4, 601–624 (2021).
  * [12] Christian Stump, Hugh Thomas, Nathan Williams: Cataland: why the Fuß ? Discrete Math. Theor. Comput. Sci., Proc., 1123-1134 (2020). Preprint, https://arxiv.org/abs/1503.00710.
  * [13] Hugh Thomas: Defining an m-cluster category. J. Algebra 318(1) (2007), 37–46.
  * [14] Eleni. Tzanaki: Faces of generalized cluster complexes and noncrossing partitions. SIAM J. on Disc. Math. 22 (2008), 15–30.
  * [15] Bin Zhu: Generalized cluster complexes via quiver representations. J. Algebr. Comb. 27(1) (2008), 35–54.