# Cluster parking functions II: dihedral symmetry

Matthieu Josuat-Vergès

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Matthieu Josuat-Vergès. Cluster parking functions II: dihedral symmetry. 2025. hal-05311421

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# Cluster parking functions II: dihedral symmetry

Matthieu Josuat-Vergès

IRIF, CNRS, Université Paris-Cité

October 13, 2025

## Abstract

In a previous work, we defined a dihedral automorphism group of the generalized cluster complex. We show that this dihedral automorphism group preserves a natural face invariant, namely the parabolic type. As a consequence, we find that the dihedral action can be extended to cluster parking functions (a simplicial complex that we defined in a previous work as a fiber product of the generalized cluster complex with parking functions). An interesting open problem is to compute the homology character of cluster parking functions, taking the dihedral action into account. We discuss (mostly speculative) connections between $q$, $t$-combinatorics and dihedral sieving phenomena.

# 1 Introduction

## 1.1 Parabolic types and dihedral symmetry in the generalized cluster complex

The generalized cluster complex $\Gamma^{(m)}$ associated to a finite real reflection group $W$ is a simplicial complex with many combinatorial properties. It was originally defined by Fomin and Reading [9], then reformulated in the context of representation theory by Thomas [21] and Zhu [23]. We refer to the thorough monograph by Stump, Thomas and Williams [18].

In a joint work with Douvropoulos [6, 7], we defined the *parabolic type* of each face $f \in \Gamma^{(m)}$. For example, in type $A_{n-1}$ and $m = 1$, we associate an integer partition of $n$ to each dissection of the regular $(n+2)$-gon (these dissections form the combinatorial realization of the cluster complex in this case). Via a connection with parking functions, we get an exact enumeration formula for the number of faces of each parabolic type, see [6, Corollary 7.3].

The automorphism group of $\Gamma^{(m)}$ was described in [14]. We will consider here the dihedral automorphism group generated by two elements $R$ and $S$ that satisfy $R^{mh+2} = (RS)^2 = S^2 = I$ (see the definitions in Section 3).

**Theorem 1.1.** For each (parabolic) conjugacy class $C$, the set of faces $f \in \Gamma^{(m)}$ of parabolic type $C$ is stable under the action of the dihedral group $\langle R, S \rangle$.

This can be illustrated as follows: in the case of a dissection of the regular $(n+2)$-gon, the associated integer partition $\lambda = (\lambda_1, \lambda_2, \dots)$ is defined by the rule that $\lambda_i$ is the number of inner $(i+2)$-gons inside the dissection. Clearly, $\lambda$ is invariant under the dihedral group of symmetry of the regular $(n+2)$-gon.
Rao and Suk [16] introduced a notion of dihedral sieving phenomenon, encoding a dihedral action on a combinatorial set. The previous theorem suggests investigating dihedral sieving in each set $\Gamma_C^{(m)}$, defined as the faces of $\Gamma^{(m)}$ of parabolic type $\mathcal{C}$. In the case $\mathcal{C} = \{e\}$ (the conjugacy class of the neutral element), $\Gamma_C^{(m)}$ contains all clusters (facets of $\Gamma^{(m)}$) and dihedral sieving phenomena have been investigated by Stier, Wellman, Xu [20] through $q, t$-Catalan numbers. Cyclic sieving phenomena for the action of $\mathcal{R}$ on the sets $\Gamma_C^{(m)}$ have been obtained by Pouillart [15] in many cases.

## 1.2 Extension of the dihedral symmetry to cluster parking functions

The enumeration formula for the sets $\Gamma_C^{(m)}$ mentioned above can be interpreted using certain characters (in the parabolic Burnside ring of $W$) that are related to parking functions. This led us to define cluster parking functions. They form a simplicial complex $\text{CPF}^{(m)}$ acted on by $W$, and by design their homology character is a parking character twisted by the sign character. See [7] for details. Concretely, an element of $\text{CPF}^{(m)}$ is pair $(f, wW_f)$ where $f \in \Gamma^{(m)}$ and $wW_f$ is a parking function, seen as a coset modulo a parabolic subgroup $W_f$ that depends on $f$. The action of $W$ is given by

$$u \cdot (f, wW_f) = (f, uwW_f).$$

Theorem 1.1 suggests to extend the dihedral action on $\Gamma^{(m)}$ to cluster parking functions (in a way that is compatible with the action of $W$ and the simplicial structure). Combinatorial models in types A, B, and I also suggest that this is possible (see [7, Open problem 10.3]). The proof of Theorem 1.1 actually provides an explicit answer to this question.

**Theorem 1.2.** The automorphisms $\mathbf{R}, \mathbf{S} \in \text{Aut}(\text{CPF}^{(m)})$ defined in Section 6 are such that:

* $p \circ \mathbf{R} = \mathcal{R} \circ p$ and $p \circ \mathbf{S} = \mathbf{S} \circ p$, where $p$ is the projection $p(f, wW_f) = f$,

* $\mathbf{R}$ and $\mathbf{S}$ commute with the action of $W$.

* $\mathbf{R}^{mh+2} = (\mathbf{R}\mathbf{S})^2 = \mathbf{S}^2 = \mathbf{I}$ (i.e., they satisfy the same relations as $\mathcal{R}$ and $\mathbf{S}$).

Assuming that the rank of $W$ is at least 2, another pair $(\mathbf{R}', \mathbf{S}')$ satisfying the same conditions is such that $\mathbf{R}^{-1} \circ \mathbf{R}'$ (respectively $\mathbf{S}^{-1} \circ \mathbf{S}'$) is the action of a central element of $W$.

Recall that the center of $W$ is either trivial (so that $\mathbf{R}, \mathbf{S}$ are unique) or is $\{e, w_0\}$ where $w_0$ is the long element.

A natural question is to investigate the action of $\langle \mathbf{R}, \mathbf{S} \rangle \times W$ on the homology of $\text{CPF}^{(m)}$ (which is nonzero only in degree $n-1$). If we only consider the action of $W$, the homology character is

$$w \mapsto \epsilon(w) \cdot (mh + 1)^{\dim(\text{Fix}(w))} \qquad (1)$$

where $\epsilon$ is the sign character, $h$ is the Coxeter number, and $\text{Fix}(w)$ is the fixed point subspace of $w$ in the geometric representation (see Section 4 for details). Up to the sign character, this is precisely the character from the rich theory of parking spaces [1, 17]. But a discrepancy comes from the fact that $\mathbf{R}$ has order dividing $mh + 2$, while parking spaces bear an action of $\mathbb{Z}_{mh} \times W$ (for example, when $m = 1$ the order is $h$ because this comes from the Coxeter element acting by conjugation).
## 1.3 Speculative connections to $q, t$-combinatorics

Haiman's *diagonal coinvariants* [12] give a natural construction of a bigraded representation of the symmetric group $S_n$ of dimension $(n+1)^{n-1}$. The associated Hilbert series in two variables $q$ and $t$ is subject to many combinatorial developments. There is an extension to other finite Weyl groups by Gordon [11]. In the case $m=1$, the character in (1) is exactly that of (ungraded) diagonal coinvariants. It is natural to ask if we can find a bigrading on the homology of CPF that matches the bigrading on diagonal coinvariants (see [7, Open problem 10.7]).

In the context of the present work, it is natural to ask whether dihedral sieving phenomena can be related with $q, t$-combinatorics beyond the $q, t$-Catalan numbers mentioned above. The drawback here is that the dihedral sieving phenomenon has an especially nice formulation only when the parameter of the dihedral group is odd. This is illustrated by the fact that in $I_2(m)$ with odd $m$, there is only one conjugacy class of reflections so that the eigenvalues in the geometric representation characterize a conjugacy class. As a consequence, dihedral sieving for $I_2(m)$ with odd $m$ can be encoded by symmetric polynomials in two variables, and one might expect a connection with $q, t$-combinatorics. The formulation of dihedral sieving with even $m$ is slightly different, see [16, 20] for details. (Note also that $mh+2$ is odd only and in type $A_n$ with $n$ even and $m$ odd. But there are other cases where we can look for a dihedral sieving with odd parameter: if the long element of $W$ is central then $\mathcal{R}$ has order $\frac{mh+2}{2}$ rather than $mh+2$, so that we have an action of the dihedral group $I_2(\frac{mh}{2}+1)$ on $\Gamma^{(m)}$ and it is in order to investigate all cases where $\frac{mh}{2}+1$ is odd.)

See Section 8 for an instance of a dihedral sieving phenomenon on cluster parking functions of type $A_{n-1}$ with $n$ odd.

# 2 The generalized cluster complex

Let $W$ be a finite real reflection group of rank $n$ with $n \ge 2$, $V \simeq \mathbb{R}^n$ its geometric representation, and $\Phi$ the corresponding root system. We choose a set $\Delta$ of simple roots and denote by $\Phi_+$ the corresponding set of positive roots.

## 2.1 Steinberg's construction

We fix a partition $\Delta = \Delta_0 \uplus \Delta_\bullet$, where $\Delta_0$ (respectively, $\Delta_\bullet$) contains pairwise orthogonal roots. Let

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

The bipartite Coxeter element is $c = c_\bullet c_\circ$.

In a famous paper, Steinberg [19] describes a recipe to compute all roots in terms of simple roots. A consequence is that the whole root system can be decomposed as a disjoint union of $2h$ blocks

$$\Phi = \bigcup_{i=1}^{2h} \Delta_{(i)}, \qquad (2)$$

such that:

$$\bullet \quad \Phi_+ = \bigcup_{i=1}^h \Delta_{(i)}, \quad \Delta_\bullet = \Delta_{(1)}, \quad \Delta_\circ = \Delta_{(h)},$$
• $c_{\bullet}(\Delta_{(i)}) = \Delta_{(h+2-i)}$, $c_{\circ}(\Delta_{(i)}) = \Delta_{(h-i)}$, $-\Delta_{(i)} = \Delta_{(i+h)}$ (where indices are taken modulo $2h$).

Moreover, this decomposition has a nice interpretation when we project $\Phi$ onto the Coxeter plane: the roots in each $\Delta_{(i)}$ have their projection in a common half-line issued from the origin. (This projection on the Coxeter plane is often used to illustrate high-dimensional root systems.)

## 2.2 Brady and Watt on the associahedron

The set of almost-positive roots is:

$$ \Phi_{\ge -1} := \Phi_+ \oplus (-\Delta) = \bigcup_{i=0}^{h+1} \Delta_{(i)}. $$

(Note that $\Delta_{(0)} = -\Delta_{\circ}$ and $\Delta_{(h+1)} = -\Delta_{\bullet}$.) When $\Phi$ is crystallographic, this set naturally appears in the context of Fomin and Zelevinsky's finite type cluster algebras: $\Phi_{\ge -1}$ indexes cluster variables and it is the vertex set of the cluster complex [8]. Accordingly, this set also indexes the facets of a generalized associahedron. Brady and Watt [3, Section 8] used Steinberg's construction and minimal factorizations of the Coxeter element to give a purely Coxeter-theoretic derivation of the cluster complex $\Gamma$.

For $f \subset \Phi_{\ge -1}$ and $0 \le i \le h+1$, let us denote

$$ f_{(i)} := \prod_{\alpha \in f \cap \Delta_{(i)}} t_{\alpha} $$

where $t_{\alpha}$ is the reflection associated with the root $\alpha$. The factors are pairwise commuting, so that the product is well-defined and $f_{(i)}^2 = e$.

Let $\mathrm{NC}_k$ denote the set of rank $k$ noncrossing partitions (see Section 4). Faces of the $(n-1)$-dimensional simplicial complex $\Gamma$ are characterized by the following, which can be taken as a definition.

**Proposition 2.1** (Brady and Watt [3]). Let $0 \le k \le n$ and $f$ be a $k$-tuple of elements in $\Phi_{\ge -1}$. Then $f \in \Gamma$ iff $f_{(h+1)} \cdots f_{(1)}f_{(0)} \in \mathrm{NC}_k$.

In particular, facets of $\Gamma$ are the $n$-tuples $f$ in $\Phi_{\ge -1}$ such that $f_{(h+1)} \cdots f_{(1)}f_{(0)} = c$.

## 2.3 The Fuß extension

We fix an integer $m \ge 1$ (it's called a Fuß parameter, because the analog of Catalan numbers counting facets of $\Gamma$ are called the Fuß-Catalan numbers). A colored root is a pair $(\alpha, i)$ (denoted by $\alpha^i$ for brevity) with $1 \le i \le m$. A colored root $\alpha^i$ is almost positive if either $\alpha \in \Phi_+$, or $i = 1$ and $-\alpha \in \Delta$. The set of almost positive colored roots is denoted by $\Phi_{\ge -1}^{(m)}$. This is the vertex set of the generalized cluster complex introduced by Fomin and Reading [9].

Tzanaki [22] extended Brady and Watt's result to this setting. Let us describe the block decomposition of $\Phi_{\ge -1}^{(m)}$. Denote $\Delta_{\circ}^i := \{\alpha^i : \alpha \in \Delta_{\circ}\}$, and similarly for $\Delta_{\bullet}$, $-\Delta_{\circ}$, $-\Delta_{\bullet}$, $\Delta_{(j)}$ in place of $\Delta_{\circ}$. We have

$$ \Phi_{\ge -1}^{(m)} = \bigcup_{i=0}^{mh+1} \Delta_{(i)} \qquad (3) $$

where
• $\Delta_{(0)} := (-\Delta_{\circ})^1$, $\Delta_{(mh+1)} := (-\Delta_{\bullet})^1$

• $\Delta_{ih+j} := \Delta_{(j)}^{m-i}$ if $0 \le i \le m-1$ and $1 \le j \le h$,

i.e.,

$$ (\Delta_{(0)}, \dots, \Delta_{mh+1}) = (-\Delta_{\circ}^{1}, \Delta_{\bullet}^{m}, \dots, \Delta_{\circ}^{m}, \Delta_{\bullet}^{m-1}, \dots, \Delta_{\circ}^{m-1}, \dots, \\ \Delta_{\bullet}^{2}, \dots, \Delta_{\circ}^{2}, \Delta_{\bullet}^{1}, \dots, \Delta_{\circ}^{1}, -\Delta_{\bullet}^{1}) \quad (4) $$

It will always be understood that indices are taken modulo $mh + 2$.

For each $f \subset \Phi_{\ge -1}^{(m)}$, we define

$$ f_{(i)} := \prod_{\alpha \in f \cap \Delta_{(i)}} t_{\alpha}. $$

(Of course, the reflection associated with a colored root is obtained by forgetting its color.) The factors are again pairwise commuting, so that $f_{(i)}^2 = e$.

**Proposition 2.2** (Tzanaki [22]). Let $f$ be a $k$-tuple of elements in $\Phi_{\ge -1}^{(m)}$. Then $f \in \Gamma^{(m)}$ iff $f_{(mh+1)} \cdots f_{(0)} \in \mathrm{NC}_k$. In particular, an $n$-tuple $f$ is a facet of $\Gamma^{(m)}$ iff $f_{(mh+1)} \cdots f_{(0)} = c$.

There is thus a noncrossing partition associated to each face of $\Gamma^{(m)}$:

**Definition 2.3.** For each $f \in \Gamma^{(m)}$, we define

$$ \prod f := f_{(mh+1)} \cdots f_{(0)} \in \mathrm{NC}. $$

**Remark 2.4.** It is well-known that there is a bijection between clusters $f \in \Gamma = \Gamma^{(1)}$ and noncrossing partitions, both counted by Catalan numbers. Note the difference with the map $\Gamma^{(m)} \to \mathrm{NC}$ that we use here: $\prod f = c$ if $f$ is a cluster, and the map is also defined on faces of $\Gamma$ of smaller dimension.

# 3 Dihedral symmetry of the generalized cluster complex

We review the dihedral automorphism group of the generalized cluster complex, as described in [14]. This group is generated by two elements $R, S$ such that $R^{mh+2} = S^2 = (RS)^2 = I$ (the identity of Aut($\Gamma^{(m)}$)). Note also that the dihedral automorphism group might not be the full automorphism group Aut($\Gamma^{(m)}$) (because there might exist diagram automorphisms which are not in $\langle R, S \rangle$, see [14] for details).

**Definition 3.1** (Fomin and Reading [9]). The self-map $\mathcal{R} : \Phi_{\ge -1}^{(m)} \to \Phi_{\ge -1}^{(m)}$ is defined by

$$ \mathcal{R}(\alpha^i) := \begin{cases} \alpha^{i+1} & \text{if } \alpha \in \Phi_+ \text{ and } i < m, \\ (-\alpha)^1 & \text{if } \alpha \in \Delta_\circ \text{ and } i = m, \text{ or } -\alpha \in \Delta_\bullet \text{ and } i = 1, \\ c(\alpha)^1 & \text{if } \alpha \in \Phi_+ \setminus \Delta_\circ \text{ and } i = m, \text{ or } -\alpha \in \Delta_\circ \text{ and } i = 1. \end{cases} \quad (5) $$

The map $\mathcal{R}$ was used in [9] to define $\Gamma^{(m)}$. By design, we have $\mathcal{R} \in \mathrm{Aut}(\Gamma^{(m)})$. Note that the map $\mathcal{R}$ induces a simple action on the block decomposition (3):

$$ \mathcal{R}(\Delta_{(i)}) = \Delta_{(i-h)}, \quad (6) $$

where indices are taken modulo $mh + 2$.
**Definition 3.2** ([14]). The self-map $\mathcal{S}$ on $\Phi_{\ge -1}^{(m)}$ is defined by:

$$ \mathcal{S}(\alpha^i) = \begin{cases} \alpha^i & \text{if } \alpha \in -\Delta_{\circ} \text{ and } i = 1 \text{ (i)}, \\ (-\alpha)^m & \text{if } \alpha \in -\Delta_{\bullet} \text{ and } i = 1 \text{ (ii)}, \\ (-\alpha)^1 & \text{if } \alpha \in \Delta_{\bullet} \text{ and } i = m \text{ (ii')}, \\ \alpha^{m-i} & \text{if } \alpha \in \Delta_{\bullet} \text{ and } 1 \le i \le m-1 \text{ (iii)}, \\ c_{\bullet}(\alpha)^{m+1-i} & \text{if } \alpha \in \Phi_{+} \setminus \Delta_{\bullet} \text{ (iv)}. \end{cases} \qquad (7) $$

It is also proved in [14] that $\mathcal{S} \in \text{Aut}(\Gamma^{(m)})$. Note that

$$ \mathcal{S}(\Delta_{(i)}) = \Delta_{(-i)}. \qquad (8) $$

The composition $\mathcal{SR} \in \text{Aut}(\Gamma^{(m)})$ can be made explicit:

$$ \mathcal{SR}(\alpha^i) = \begin{cases} \alpha^{m-i-1} & \text{if } \alpha \in \Delta_{\bullet} \text{ and } 1 \le i \le m-2 \text{ (i)}, \\ c_{\bullet}(\alpha)^{m-i} & \text{if } \alpha \in \Phi_{+} \setminus \Delta_{\bullet} \text{ and } 1 \le i \le m-1 \text{ (ii)}, \\ (-\alpha)^1 & \text{if } \alpha \in \Delta_{\circ} \text{ and } i = m \text{ (iii)}, \\ (-\alpha)^m & \text{if } \alpha \in -\Delta_{\circ} \text{ and } i = 1 \text{ (iii')}, \\ (-\alpha)^{m-1} & \text{if } \alpha \in -\Delta_{\bullet} \text{ and } i = 1 \text{ (iv)}, \\ (-\alpha)^1 & \text{if } \alpha \in \Delta_{\bullet} \text{ and } i = m-1 \text{ (iv')}, \\ c_{\circ}(\alpha)^m & \text{if } \alpha \in \Phi_{+} \setminus \Delta_{\circ} \text{ and } i = m \text{ (v)}. \end{cases} \qquad (9) $$

By checking all cases, it is straightforward to check that $\mathcal{SR}$ is an involution. By (6) and (8), we have:

$$ \mathcal{SR}(\Delta_{(i)}) = \Delta_{(h-i)}. \qquad (10) $$

# 4 Parking functions

Let us review *noncrossing parking functions*, following Armstrong, Reiner and Rhoades [1]. This will be useful for cluster parking functions, but also to highlight the notion of *parabolic type* for faces of $\Gamma^{(m)}$ in the next section.

It is well-known that *noncrossing partitions* can be defined as the interval $[e, c]$ between the neutral element and the Coxeter element for the absolute order on $W$. They form a ranked lattice denoted $NC \subset W$, moreover $NC_k \subset NC$ denotes the set of rank $k$ elements in $NC$. The parabolic subgroup associated to $\pi \in NC$ is $W_{\pi}$, where

$$ \forall v \in W, W_v := \{w \in W : \text{Fix}(v) \subset \text{Fix}(w)\} $$

and $\text{Fix}(w) = \ker(e - w) \subset V$. (By Steinberg's theorem, $W_v$ is indeed parabolic.) Note that if $u, v \in W$ we have

$$ W_{uvu^{-1}} = uW_vu^{-1}. \qquad (11) $$

**Definition 4.1.** A *parking function* is a coset $wW_{\pi}$ where $w \in W$ and $\pi \in NC$. Let PF denote the poset of parking functions, where the order is reverse inclusion.
The group $W \times \langle c \rangle$ acts on PF by:

$$ (w_1, c^i) \cdot w_2 W_\pi = w_1 w_2 W_\pi c^i. $$

This element is also a parking function because it can be written:

$$ (w_1, c^i) \cdot w_2 W_\pi = w_1 w_2 W_\pi c^i = w_1 w_2 c^i (c^{-i} W_\pi c^i) = w_1 w_2 c^i W_{c^{-i} \pi c^i}, $$

and $\pi \in \text{NC}$ implies $c^{-i}\pi c^i \in \text{NC}$. Let where $\omega = e^{2i\pi/h}$ be a primitive $h$th root of 1. By [1], the character of PF is:

$$ (w, c^k) \mapsto \lim_{q \to \omega^k} \frac{\det(e - q^{h+1}w)}{\det(e - qw)} = (h+1)^{\dim \ker(e - \omega^k w)}. $$

If we focus on the action of $W$, the character of PF is, from the combinatorial definition:

$$ \sum_{\pi \in \text{NC}} \text{Ind}_{W_\pi}^{W}(1) $$

where $\text{Ind}_{W_\pi}^{W}(1)$ means the trivial character of $W_\chi$ induced to $W$. It is natural to gather the terms $\text{Ind}_{W_\pi}^{W}(1)$ where $\pi$ are conjugate, because the characters are the same. The *parabolic type* of $\pi \in \text{NC}$ is simply its conjugacy class. Computing the character of PF thus amounts to counting noncrossing partitions of each parabolic type. This counting problem and the relationship with the generalized cluster complex is discussed in [6].

# 5 The parabolic type of a face

Following the construction on noncrossing partitions and the associated noncrossing parking functions, we associate a parabolic type to each face $f \in \Gamma^{(m)}$. For this we need the *Kreweras complement*, a lattice anti-automorphism $K : \text{NC} \to \text{NC}$ defined by

$$ K(\pi) := \pi^{-1}c. $$

**Definition 5.1 ([6]).** For $f \in \Delta^{(m)}$, define

$$ \underline{f} := K(\prod f) = (\prod f)^{-1} c \in \text{NC}. \qquad (12) $$

The *parabolic type* of $f$ is defined as the conjugacy class of $\underline{f}$.

(Another definition of $K$ and $\underline{f}$ was used in [6], but both lead to the same conjugacy class.) In our previous work [6, 7], we showed that the parabolic type is an important invariant of faces $f \in \Gamma^{(m)}$. A prominent result is an exact enumeration formula for faces refined by their parabolic type [6, Corollary 7.3]. It was also proved in [6, Proposition 11.1] that the parabolic type is invariant under the action of $\mathcal{R}$. We review this here and complete the picture by proving Theorem 1.1.
## 5.1 The map $\mathcal{R}$

We revisit the proof of [6, Proposition 11.1]. Let $f' := \mathcal{R}(f)$. Using (6), we have:

$$f'_{(i)} = \prod_{\alpha \in \mathcal{R}(f) \cap \Delta_{(i)}} t_{\alpha} = \prod_{\alpha \in f \cap \Delta_{(i+h)}} t_{\mathcal{R}(\alpha)}.$$

From the definition of $\mathcal{R}$ in (5) (and $t_{w(\alpha)} = wt_{\alpha}w^{-1}$ for $\alpha \in \Phi_+$, $w \in W$), we have:

$$t_{\mathcal{R}(\alpha)} = \begin{cases} t_{\alpha} & \text{if } \alpha \in \Delta_{(i)} \text{ with } h \le i \le mh + 1, \\ c \cdot t_{\alpha} \cdot c^{-1} & \text{if } \alpha \in \Delta_{(i)} \text{ with } 0 \le i \le h - 1. \end{cases}$$

From the previous two equations, we get

$$f'_{(i)} = \begin{cases} f_{(i+h)} & \text{if } 0 \le i \le (m-1)h + 1, \\ c \cdot f_{(i+h)} \cdot c^{-1} & \text{if } (m-1)h + 2 \le i \le mh + 1. \end{cases} \qquad (13)$$

So,

$$\prod f' = f'_{(mh+1)} \cdots f'_{(0)} \qquad (14)$$

$$= c \cdot (f_{(h-1)} \cdots f_{(0)}) \cdot c^{-1} \cdot f_{(mh+1)} \cdots f_{(h)}. \qquad (15)$$

and it follows

$$\underline{f}' = (f_{(h)} \cdots f_{(mh+1)}) \cdot c \cdot (f_{(0)} \cdots f_{(h-1)}) \qquad (16)$$

$$= (f_{(h-1)} \cdots f_{(0)}) \cdot \underline{f} \cdot (f_{(0)} \cdots f_{(h-1)}). \qquad (17)$$

This shows that $\underline{f}'$ and $\underline{f}$ are conjugate.

## 5.2 The map $\mathcal{S}$

As for $\mathcal{S}$, the computations are similar. Let $f' := \mathcal{S}(f)$. From (8), we have:

$$f'_{(i)} = \prod_{\alpha \in \mathcal{S}(f) \cap \Delta_{(i)}} t_{\alpha} = \prod_{\alpha \in f \cap \Delta_{(-i)}} t_{\mathcal{S}(\alpha)}.$$

From the definition of $\mathcal{S}$ in (7), we have:

$$f'_{(i)} = \begin{cases} f_{(i)} & \text{if } i = 0, \\ c_{\bullet} \cdot f_{(-i)} \cdot c_{\bullet} & \text{otherwise.} \end{cases} \qquad (18)$$

From the previous two equations, we get:

$$\prod f' = c_{\bullet} \cdot (f_{(1)} \cdots f_{(mh+1)}) \cdot c_{\bullet} \cdot f_{(0)} \qquad (19)$$

$$= c_{\bullet} \cdot f_{(0)} \cdot (\prod f)^{-1} \cdot c_{\bullet} \cdot f_{(0)} \qquad (20)$$

so that

$$\underline{f}' = f_{(0)} \cdot c_{\bullet} \cdot (\prod f) \cdot f_{(0)} \cdot c_{\circ} \qquad (21)$$

$$= f_{(0)} \cdot c_{\bullet} \cdot c \cdot \underline{f}^{-1} \cdot f_{(0)} \cdot c_{\circ} \qquad (22)$$

$$= f_{(0)} \cdot c_{\circ} \cdot \underline{f}^{-1} \cdot f_{(0)} \cdot c_{\circ}. \qquad (23)$$
Note that

$$f(0) \cdot c_o = \prod_{\alpha \in \Delta_o - f(0)} t_\alpha,$$

so $f(0) \cdot c_o$ is an involution as a product of pairwise commuting reflections. Moreover, each element of W is conjugate to its inverse. So, the previous equation shows that $f'$ and $f$ are conjugate.

## 5.3 The map $\mathcal{SR}$

Although the previous two sections prove Theorem 1.1, it is useful to make the explicit computations regarding the composition $\mathcal{SR}$. The goal is to have an explicit conjugation equation between $f$ and $\mathcal{SR}(f)$. Let $f' := \mathcal{SR}(f)$. By (10) we have:

$$f'_{(i)} = \prod_{\alpha \in \mathcal{SR}(f) \cap \Delta_{(i)}} t_{\alpha} = \prod_{\alpha \in f \cap \Delta_{(h-i)}} t_{\mathcal{SR}(\alpha)}.$$

Now, using the various cases in (9) we get

$$f'_{(i)} = \begin{cases} c_{\bullet} \cdot f_{(h-i)} \cdot c_{\bullet} & \text{if } h+1 \le h-i \le mh+1, \\ c_{\circ} \cdot f_{(h-i)} \cdot c_{\circ} & \text{if } 0 \le h-i \le h. \end{cases} \qquad (24)$$

And it follows

$$\prod f' = c_{\bullet} \cdot (f_{(h+1)} \cdots f_{(mh+1)}) \cdot c_{\bullet} \cdot c_{\circ} \cdot (f_{(0)} \cdots f_{(h)}) \cdot c_{\circ} \qquad (25)$$

and

$$\underline{f}' = c_{\circ} \cdot (f_{(h)} \cdots f_{(0)}) \cdot c_{\circ} \cdot c_{\bullet} \cdot (f_{(mh+1)} \cdots f_{(h+1)}) \cdot c_{\circ} \qquad (26)$$

$$= c_{\circ} \cdot (f_{(h)} \cdots f_{(0)}) \cdot c_{\circ} \cdot c_{\bullet} \cdot (\prod f) \cdot (f_{(0)} \cdots f_{(h)}) \cdot c_{\circ} \qquad (27)$$

$$= c_{\circ} \cdot (f_{(h)} \cdots f_{(0)}) \cdot \underline{f}^{-1} \cdot (f_{(0)} \cdots f_{(h)}) \cdot c_{\circ}. \qquad (28)$$

## 6 Dihedral symmetry on CPF

Cluster parking functions are the pairs $(f, wW_f) \subset \Gamma^{(m)} \times PF$, and the product order endows them with a partial order which can be interpreted as the inclusion of faces of a simplicial complex denoted $\text{CPF}^{(m)}$. The faces of a simplex $(f, wW_f) \in \text{CPF}^{(m)}$ are the elements $(f', wW_{f'})$ where $f' \subset f$ (in particular, they are naturally in correspondence with faces of $f$). We refer to [7] for details.

Let $\Gamma_n$ (respectively, $\text{CPF}_n$) denote the complex $\Gamma^{(m)}$ (respectively, $\text{CPF}^{(m)}$) in type $A_{n-1}$ with $m=1$. A face $f \in \Gamma_n$ is a dissection of the (regular) $(n+2)$-gon, i.e., a set of pairwise noncrossing diagonals. For example,

<figure><img src="image_2.png" /></figure>
A face of $\mathrm{CPF}_n$ is obtained by putting labels $1, \dots, n$ in the inner polygons defined by this dissection, with the rule that a $(k+2)$-gon receives a set of $k$ (unordered) labels. For example,

<figure><img src="image_3.png" /></figure>

The symmetric group acts by natural relabeling. As a consequence of the unicity of $\mathbf{R}, \mathbf{S}$ in Theorem 1.2, the dihedral group $\langle \mathbf{R}, \mathbf{S} \rangle$ naturally identifies to the group of symmetry of the regular $(n+2)$-gon. See also [7] for combinatorial models in types $B$ and $I$.

For each $f \in \Gamma^{(m)}$, let us define:

$$ \kappa_f := f_{(0)} \cdots f_{(h-1)}, \qquad (29) $$

$$ \lambda_f := f_{(0)} \cdot c_o, \qquad (30) $$

$$ \mu_f := (f_{(0)} \cdots f_{(h)}) \cdot c_o, \qquad (31) $$

so that the computations in Section 3 give:

$$ \underline{\mathcal{R}(f)} = \kappa_f^{-1} \cdot \underline{f} \cdot \kappa_f, \qquad (32) $$

$$ \underline{\mathcal{S}(f)} = \lambda_f^{-1} \cdot \underline{f}^{-1} \cdot \lambda_f, \qquad (33) $$

$$ \underline{\mathcal{SR}(f)} = \mu_f^{-1} \cdot \underline{f}^{-1} \cdot \mu_f. \qquad (34) $$

By (11), the same relations also hold for associated parabolic subgroups:

$$ W_{\mathcal{R}(f)} = \kappa_f^{-1} \cdot W_{\underline{f}} \cdot \kappa_f, \qquad (35) $$

$$ W_{\mathcal{S}(f)} = \lambda_f^{-1} \cdot W_{\underline{f}} \cdot \lambda_f, \qquad (36) $$

$$ W_{\mathcal{SR}(f)} = \mu_f^{-1} \cdot W_{\underline{f}} \cdot \mu_f. \qquad (37) $$

(Note that $W_{\underline{f}^{-1}} = W_{\underline{f}}$, since an element and its inverse have the same 1-eigenspace.)

**Definition 6.1.** The two self-maps $\mathbf{R}$ and $\mathbf{S}$ on $\mathrm{CPF}^{(m)}$ are defined by:

$$ \mathbf{R}(f, wW_{\underline{f}}) = (\mathcal{R}(f), wW_{\underline{f}} \cdot \kappa_f), \qquad (38) $$

$$ \mathbf{S}(f, wW_{\underline{f}}) = (\mathcal{S}(f), wW_{\underline{f}} \cdot \lambda_f). \qquad (39) $$

Alternatively (via (35) and (36)), these can be written:

$$ \mathbf{R}(f, wW_{\underline{f}}) = (\mathcal{R}(f), w\kappa_f W_{\mathcal{R}(f)}), \qquad (40) $$

$$ \mathbf{S}(f, wW_{\underline{f}}) = (\mathcal{S}(f), w\lambda_f W_{\mathcal{S}(f)}). \qquad (41) $$

The latter expressions makes it clear that the right-hand sides are in $\mathrm{CPF}^{(m)}$, but the former are more clearly nonambiguous.

**Lemma 6.2.** For $f \in \Gamma^{(m)}$, we have $\mu_f = \kappa_f \cdot \lambda_{\mathcal{R}(f)}$.

*Proof.* Let $f' = \mathcal{R}(f)$. From (13), we have $f'_{(0)} = f_{(h)}$. From (30), we get $\lambda_{f'} = f'_{(0)} \cdot c_o = f_{(h)} \cdot c_o$. From (29), we get $\kappa_f \cdot \lambda_{f'} = (f_{(0)} \cdots f_{(h)}) \cdot c_o$. By (31), we obtain the result. $\square$
We have:

$$
\begin{align*}
\mathbf{S}(\mathbf{R}(f, wW_f)) &= \mathbf{S}(\mathcal{R}(f), wW_f \cdot \kappa_f) \\
&= (\mathcal{SR}(f), wW_f \cdot \kappa_f \cdot \lambda_{\mathcal{R}(f)}) \\
&= (\mathcal{SR}(f), wW_f \cdot \mu_f),
\end{align*}
$$

which means that the composition **SR** is the natural map that can be defined by analogy
with (38) and (39).

**Lemma 6.3.** *The maps S and SR are involutions.*

Proof. Let’s begin with **S**. Let $f \in \Gamma^{(m)}$ and $f' = \mathcal{S}(f)$. By (18), we have $f'_{(0)} = f_{(0)}$, so that:

$$
\lambda_{\mathcal{S}(f)} = \lambda_f.
$$

Then note that

$$
\lambda_f = f_{(0)}c_o = \prod_{\alpha \in \Delta_o - f} t_\alpha
$$

is involutive as a product of pairwise commuting reflections. So, $\lambda_f \lambda_{f'} = \lambda_f^2 = e$. We then
have:

$$
\mathbf{S}^2(f, w\mathbf{W}_f) = \mathbf{S}(f', w\mathbf{W}_f \cdot \lambda_f) = (\mathcal{S}^2(f), w\mathbf{W}_f \cdot \lambda_f \cdot \lambda_{f'}) = (f, w\mathbf{W}_f).
$$

So, **S** is an involution.

Let's do the same for **SR**. Let $f' = \mathcal{SR}(f)$. By (24) and the definition of $\mu_f$, we have

$$
\mu_{f'} = (c_{\circ} f_{(h)} c_{\circ}) \cdots (c_{\circ} f_{(0)} c_{\circ}) \cdot c_{\circ} = c_{\circ} \cdot (f_{(h)} \cdots f_{(0)}) = \mu_{f}^{-1}.
$$

It follows:

$$
(\mathbf{SR})^2(f, w\mathbf{W}_f) = \mathbf{SR}(\mathcal{SR}(f), w\mathbf{W}_f \mu_f) = (f, w\mathbf{W}_f \mu_f \mu_{f'}) = (f, w\mathbf{W}_f).
$$

So, **SR** is also an involution.

<figure><img src="image_4.png" /></figure>

**Lemma 6.4.** *For any $f \in \Delta^{(m)}$, we have:*

$$
\kappa_f \cdot \kappa_{\mathcal{R}(f)} \cdots \kappa_{\mathcal{R}^{mh+1}(f)} = \underline{f}^h. \tag{42}
$$

*Proof.* Let us define a sequence $(u_i)_{0 \le i \le (mh+2)h-1}$ by the rule

$$
u_{(mh+2)i+j} = c^i \cdot f_{(j)} \cdot c^{-i}
$$

if $0 \le i < h$ and $0 \le j < mh + 2$ are integers. Note that we have

$$
u_{i+(mh+2)} = c \cdot u_i \cdot c^{-1}, \tag{43}
$$

for any $i$ (indices are modulo $(mh+2)h$). We first compute

$$
\begin{align*}
u_{(mh+2)i} \cdot u_{(mh+2)i+1} \cdots u_{(mh+2)(i+1)-1} &= (c^i \cdot f_{(0)} \cdot c^{-i}) \cdots (c^i \cdot f_{(mh+1)} \cdot c^{-i}) \\
&= c^i \cdot (\prod f)^{-1} \cdot c^{-i},
\end{align*}
$$
so that the whole product is

$$u_0 u_1 \cdots u_{(mh+2)h-1} = (\prod f)^{-1} \cdot (c \cdot (\prod f)^{-1} \cdot c^{-1}) \cdots (c^{h-1} \cdot (\prod f)^{-1} \cdot c^{-h+1}) \quad (44)$$

$$= (\prod f)^{-1} \cdot c)^h = \underline{f}^h, \quad (45)$$

where we simplified by $c^{-i}c^{i+1} = c$ and $c^{-h+1} = c$. To complete the proof, we will compute this product by gathering factors in another way.

First, we show that

$$\mathcal{R}^i(f)_{(j)} = u_{hi+j}, \quad (46)$$

where $0 \le i, j \le mh + 1$. This is done by induction on $i$. The case $i = 0$ is clear because $u_i = f_{(i)}$ if $0 \le i \le mh + 1$, by definition. The induction from $i$ to $i + 1$ comes from (13). More precisely, if $0 \le j \le (m-1)h + 1$ then the first case of (13) gives:

$$\mathcal{R}^i(f)_{(j)} = \mathcal{R}^{i-1}(f)_{(j+h)} = u_{h(i-1)+j+h} = u_{hi+j}.$$

And if $(m-1)h + 2 \le j \le mh + 1$, the second case of (13) gives:

$$\mathcal{R}^i(f)_{(j)} = c \cdot \mathcal{R}^{i-1}(f)_{(j+h-mh-2)} \cdot c^{-1} = c \cdot u_{hi+j-mh-2} \cdot c^{-1}.$$

(Equation (13) is written with indices modulo $mh+2$, here $j+h-mh-2$ is the representative of $j+h$ in $\{0, \dots mh+1\}$). By (43), the previous equation proves (46). From the definition of $\kappa_f$ in (29), we then obtain:

$$\kappa_{\mathcal{R}^i(f)} = u_{hi} \cdot u_{hi+1} \cdots u_{h(i+1)-1}. \quad (47)$$

So, by gathering the factors we have:

$$u_0 \cdots u_{(mh+2)h-1} = \kappa_f \cdot \kappa_{\mathcal{R}(f)} \cdots \kappa_{\mathcal{R}^{mh+1}(f)}.$$

This completes the proof.

<figure><img src="image_5.png" /></figure>

**Proposition 6.5.** We have $\mathbf{R}^{mh+2} = \mathbf{I}$.

*Proof.* By iterating the definition of $\mathbf{R}$, we have

$$\mathbf{R}^i(f, w\underline{W}_f) = (\mathcal{R}^i(f), w\underline{W}_f \cdot \kappa_f \cdot \kappa_{\mathcal{R}(f)} \cdots \kappa_{\mathcal{R}^{i-1}(f)}).$$

Therefore, this follows from $\mathcal{R}^{mh+2} = \mathcal{I}$ and the previous lemma. Note that $\underline{f} \in W_f$, so that $W_f \cdot \underline{f}^h = W_f$. □

<figure><img src="image_6.png" /></figure>

The proof of Theorem 1.2 is almost complete: the first two points are clear from the definition, the third point follows from the lemmas in this section. Uniqueness (up to the actions of central elements) will be proved in the next section.
# 7 Automorphisms of CPF

Let's first describe some edges in the adjacency graph of the facets of $CPF^{(m)}$. Let's fix a cluster $f \in \Gamma^{(m)}$. For each $\alpha \in f$, let $W_{f-\{\alpha\}} = \{e, t_{\beta}\}$ where $\beta$ implicitly depends on $f$ and $\alpha$. The 1-codimensional faces of $(f, \{w\})$ are: $(f - \{\alpha\}, \{w, wt_{\beta}\})$, for $\alpha \in f$. In the facet adjacency graph, the neighbors of $(f, \{w\})$ through $(f - \{\alpha\}, \{w, wt_{\beta}\})$ are:

* one neighbor $(f, \{wt_{\beta}\})$,

* for each cluster $f' = (f - \{\alpha\}) \cup \{\alpha'\}$ (a neighbor of $f$ in $\Gamma^{(m)}$), two neighbors $(f', \{w\})$ and $(f', \{wt_{\beta}\})$.

**Lemma 7.1.** With the notation above, the $n$ values of $\beta$ can be ordered as a sequence $\beta_1, \dots, \beta_n$ so that $c = \beta_1 \cdots \beta_n$.

*Proof.* By Proposition 2.2, the vertices of $f$ can be ordered as $\alpha_1, \dots, \alpha_n$ so that $c = t_{\alpha_1} \cdots t_{\alpha_n}$. We have:

$$ \underline{f - \{\alpha_i\}} = (\prod f - \{\alpha_i\})^{-1} c = (t_{\alpha_1} \cdots t_{\alpha_{i-1}} t_{\alpha_{i+1}} \cdots t_{\alpha_n})^{-1} t_{\alpha_1} \cdots t_{\alpha_n} = t_{\alpha_n} \cdots t_{\alpha_i} \cdots t_{\alpha_n}. $$

It follows that $\underline{f - \{\alpha_i\}} = \{e, t_{\alpha_n} \cdots t_{\alpha_n} \cdots t_{\alpha_n}\}$ (this is a rank 1 parabolic subgroup containing $\underline{f - \{\alpha_i\}}$). So, let us define $\beta_i = t_{\alpha_n} \cdots t_{\alpha_i} \cdots t_{\alpha_n}$. We get: $\beta_1 \cdots \beta_n = t_{\alpha_n} \cdots t_{\alpha_1} = c^{-1}$. $\square$

**Lemma 7.2.** Let $F \in \mathrm{Aut}(\mathrm{CPF}^{(m)})$ such that $p \circ F = p$. If the rank of $W$ is at least 2, there exists $u \in W$ such that $F$ is the action of $u$ on $\mathrm{CPF}^{(m)}$.

*Proof.* Let's fix a facet $f \in \Gamma^{(m)}$. We will first prove:

$$ \forall w \in W, \quad \mathbf{F}(f, \{w\}) = (f, \{uw\}), \qquad (48) $$

for some $u \in W$ that we can define by $\mathbf{F}(f, \{e\}) = (f, \{u\})$. This is true for $w = e$ by definition, and we show that if this is true for $w$ it is for $wt_{\beta}$ too (for each value of $\beta$). Let's fix $\alpha$ and $\beta$. Note that there is only one facet of the form $(f, \{w'\})$ which is adjacent to $(f, \{w\})$ through $(f - \{\alpha\}, wW_{f-\{\alpha\}})$, namely $(f, \{wt_{\beta}\})$. Since $F$ is an automorphism, it follows that $\mathbf{F}(f, \{wt_{\beta}\})$ is the unique facet of the form $(f, \{w'\})$ which is adjacent to $\mathbf{F}(f, \{w\}) = (f, \{uw\})$ through $\mathbf{F}(f - \{\alpha\}, wW_{f-\{\alpha\}}) = (f - \{\alpha\}, uwW_{f-\{\alpha\}})$. So $\mathbf{F}(f, \{wt_{\beta}\}) = (f, \{uw_{\beta}\})$. This shows that (48) is also true for $wt_{\beta}$. A consequence of Lemma 7.1 is that $t_{\beta_1}, \dots, t_{\beta_n}$ generate $W$, and this completes the proof of (48).

It remains to prove that the element $u$ defined via (48) actually doesn't depend on $f$. Let $f' = (f - \{\alpha\}) \cup \alpha'$ be a neighbor facet of $f$ (and $\beta$ as before). We clearly have $\mathbf{F}(f - \{\alpha\}, \{w, wt_{\beta}\}) = (f - \{\alpha\}, \{uw, uwt_{\beta}\})$. Since $F$ preserves incidence, we have either $\mathbf{F}(f', \{w\}) = (f', \{uw\})$ or $\mathbf{F}(f', \{w\}) = (f', \{uwt_{\beta}\})$. But we already know that (48) holds with $f'$ and some $u'$ in place of $f$ and $u$. Since the rank of $W$ is at least 2, $W$ is noncommutative and it is straightforward to see that the only coherent possibility is $\mathbf{F}(f', \{w\}) = (f', \{uw\})$ for all $w$. Now, the result follows from connectedness of the adjacency graph of facets of $\Gamma^{(m)}$ (which holds because $\Gamma^{(m)}$ is a shellable complex). $\square$
*Proof of Theorem 1.2.* It remains only to prove the last statement about other pairs $(\mathbf{R}', \mathbf{S}')$. Suppose that $(\mathbf{R}', \mathbf{S}')$ satisfy the same conditions as $(\mathbf{R}, \mathbf{S})$ in the theorem.

We have $p \circ \mathbf{R}' \circ \mathbf{R}^{-1} = \mathcal{R} \circ p \circ \mathbf{R}^{-1} = \mathcal{R} \circ \mathcal{R}^{-1} \circ p = p$. By the previous lemma (using the hypothesis that the rank is at least 2), $\mathbf{R}' \circ \mathbf{R}^{-1}$ acts on $\text{CPF}^{(m)}$ as the action of some $u \in W$. But $\mathbf{R}' \circ \mathbf{R}^{-1}$ commutes with the action of $W$ by hypothesis on $\mathbf{R}$ and $\mathbf{R}'$. It follows that $u$ is central. Similarly, $\mathbf{S}^{-1} \circ \mathbf{S}'$ is the action of a central element $u'$. $\square$

# 8 A (homological) dihedral sieving phenomenon

In order to support the ideas in Section 1.3, let us present one result in this direction. It is essentially a consequence of what is already present in the literature. Through this section, let $n$ be an odd integer and $\text{CPF}_n$ be the complex of cluster parking functions of type $A_{n-1}$ (with Fuß parameter $m = 1$).

for $\mathbf{F} \in \langle \mathbf{R}, \mathbf{S} \rangle \subset \text{Aut}(\text{CPF}_n)$, let $\text{CPF}_n^{(\mathbf{F})}$ denote the set of fixed points of $\mathbf{F}$. It is easily seen that this is a subcomplex of $\text{CPF}_n$. We denote by $\Lambda(\text{CPF}_n, \mathbf{F})$ its Euler characteristic:

$$ \Lambda(\text{CPF}_n, \mathbf{F}) = \chi(\text{CPF}_n^{(\mathbf{F})}). $$

It is called a *Lefschetz number*. It can also be defined via the action of $\mathbf{F}$ on the homology of $\text{CPF}_n$ (see [2]).

Let $H(q, t)$ be the bi-graded Hilbert series of diagonal coinvariants for the symmetric group $S_n$. It's a symmetric polynomial in $q$ and $t$ such that $H(1, 1) = (n+1)^{n-1}$. See [12] for a precise definition.

**Proposition 8.1.** We have:

$$ \begin{aligned} (-1)^{\frac{n+1}{2}} \Lambda(\text{CPF}_n, \mathbf{S}) &= H(1, -1) = E_n, \\ -\Lambda(\text{CPF}_n, \mathbf{R}^d) &= H(e^{\frac{2i\pi d}{n+2}}, e^{-\frac{2i\pi d}{n+2}}) = 1, \end{aligned} $$

where $d$ is a strict divisor of $n+2$. The integer $E_n$ is a tangent number, defined by $\tan(z) = \sum_{n \ge 0} E_{2n+1} \frac{z^{2n+1}}{(2n+1)!}$.

*Proof.* Let's begin with the evaluations of $H$. The identity $H(1, -1) = E_n$ has been investigated in [4] (see also references therein). As for the second equation, from [13, Proposition 3.6] we have:

$$ H(q, q^{-1}) = q^{-\binom{n}{2}} \left( \frac{1 - q^{n+1}}{1 - q} \right)^{n-1} $$

It is straightforward to specialize at $q = e^{\frac{2i\pi d}{n+2}}$.

It remains to evaluate the Lefschetz numbers. The second one is clear: when $\mathbf{R}^d$ is a nontrivial rotation of the polygon, there are no fixed points and the subcomplex is empty (so that the Euler characteristic is $-1$). For the case of $\mathbf{S}$, the fixed points are faces whose diagonal are parallel to a fixed side of the polygon, for example:

<figure><img src="image_7.png" /></figure>
By reading labels from top to bottom, we get set compositions whose first part has odd length and others have even length. In the example, we get 23(10)|157(11)|4689. We leave as an exercise the signed enumeration to obtain $(-1)^{\frac{n+1}{2}} E_n$. □

**Remark 8.2.** The homology of $\mathrm{CPF}_n^{(S)}$ is acted on by $\mathfrak{S}_n$. The previous proposition shows that its dimension is $E_n$. In fact, it is possible to see that the Frobenius characteristic of this representation is a certain skew Schur function considered by Foulkes [10].

Together with $H(1, 1) = (n+1)^{n-1} = -\chi(\mathrm{CPF}_n)$, the previous proposition gives an instance of a *dihedral sieving phenomenon*. This amounts to remarking that $(1, -1)$ (respectively, $(e^{\frac{2i\pi d}{n+2}}, e^{-\frac{2i\pi d}{n+2}})$, and $(1, 1)$) are the eigenvalues of $\mathbf{S}$ (respectively, $\mathbf{R}^d$, and $\mathbf{I}$) in the 2-dimensional geometric representation of the dihedral group $\langle \mathbf{R}, \mathbf{S} \rangle \simeq I_2(n+2)$. This is summarized as follows.

**Proposition 8.3.** Let $H^\bullet(q, t) = -(qt)^{\frac{n-1}{2}} H(q, t)$. For $\mathbf{F} \in \langle \mathbf{R}, \mathbf{S} \rangle$, we have:

$$ \Lambda(\mathrm{CPF}_n, \mathbf{F}) = H^\bullet(\lambda_1, \lambda_2) $$

where $\lambda_1, \lambda_2$ are the eigenvalues of $\mathbf{F}$ in the 2-dimensional geometric representation of $\langle \mathbf{R}, \mathbf{S} \rangle$.

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