# The generalized cluster complex: refined enumeration of faces and related
parking spaces

Theo Douvropoulos  University of Massachusetts at Amherst, USA  Matthieu
Josuat-Vergès  IRIF, CNRS, Université Paris-Cité, FRANCE

###### Abstract

The generalized cluster complex was introduced by Fomin and Reading, as a
natural extension of the Fomin-Zelevinsky cluster complex coming from finite
type cluster algebras. In this work, to each face of this complex we associate
a parabolic conjugacy class of the underlying finite Coxeter group. We show
that the refined enumeration of faces (respectively, positive faces) according
to this data gives an explicit formula in terms of the corresponding
characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents).
This characteristic polynomial originally comes from the theory of hyperplane
arrangements, but it is conveniently defined via the parabolic Burnside ring.
This makes a connection with the theory of parking spaces: our results
eventually rely on some enumeration of chains of noncrossing partitions that
were obtained in this context. As consequences of our enumeration of faces, we
completely explain some observations by Fomin and Reading about $f$-vectors,
we obtain identities that essentially refine the transformation between
$f$-vectors and $h$-vectors, and introduce new kinds of parking spaces that we
call cluster parking spaces.

##  1 Introduction

The cluster complex of a finite type cluster algebra was introduced by Fomin
and Zelevinsky [17]. It is a simplicial complex, which can be built using
almost positive roots as its vertex set. It can be seen as the dual of a
corresponding associahedron. A natural extension is the generalized cluster
complex, defined by Fomin and Reading [16] via colored almost-positive roots.
Although there is no related cluster algebra nor associahedron in this case,
it can be given a representation theoretic interpretation via quiver
representations [35, 39]. Most importantly, this is a simplicial complex with
nice enumerative and topological properties [4, 5, 17, 34, 36]. In particular,
its number of facets is the Fuß-Catalan number

$\operatorname{Cat}^{(m)}(W):=\frac{1}{|W|}\prod_{i=1}^{n}(mh+e_{i}+1),$

and its number of positive facets is the positive Fuß-Catalan number

$\operatorname{Cat}^{(m)}_{+}(W):=\frac{1}{|W|}\prod_{i=1}^{n}(mh+e_{i}-1).$

Here, $W$ is a finite and irreducible real reflection group, $h$ is its
Coxeter number, $e_{i},\dots,e_{n}$ is the sequence of exponents. Moreover,
$m$ is the number of different colors that a positive root can have. See
Section 6 for details.

Consider a flat, i.e., an element $X$ in the intersection lattice $L(W)$
generated by the reflecting hyperplanes of $W$. There is an associated
hyperplane arrangement on $X$ called the restricted arrangement (a general
reference on this subject is Orlik and Terao [25]). It turns out to be a free
arrangement, so that its characteristic polynomial $\chi_{X}(t)$ is factorized
in the form $\prod_{i=1}^{\dim(X)}(t-b_{i}^{X})$ where the roots $b_{i}^{X}$
are positive integers called the Orlik-Solomon exponents. (Two flats that are
congruent under the action of $W$ have the same characteristic polynomial and
we consider that indices are orbits of flats.) It has been established that
these characteristic polynomials can be used to refine the enumeration of some
Catalan families. In particular, Sommers [31] considers certain ideals in Lie
theory (that correspond to the combinatorial notion of nonnesting partitions).
He showed in [31, Theorem 5.7] that the number of such ideals (respectively,
positive ideals) is the Catalan number
$\operatorname{Cat}(W):=\operatorname{Cat}^{(1)}(W)$ (respectively, the
positive Catalan number
$\operatorname{Cat}_{+}(W):=\operatorname{Cat}^{(1)}_{+}(W)$). Moreover, he
showed in [31, Proposition 6.6] that the number of ideals (respectively,
positive ideals) associated to the orbit of a flat $X$ under a certain natural
map is

$\frac{\chi_{X}(h+1)}{[N(W_{X}):W_{X}]},\quad\text{
respectively,}\quad\frac{\chi_{X}(h-1)}{[N(W_{X}):W_{X}]}.$

Here, the denominator is the index of the parabolic subgroup $W_{X}$ in its
normalizer, see Section 3 for details. In particular, the numbers in the
previous equation add up to $\operatorname{Cat}(W)$ and
$\operatorname{Cat}_{+}(W)$, respectively.

Another Catalan family consists of noncrossing partitions (see [8] for a
recent survey). They are particularly important here because of the close
connection to the cluster complex. A refined enumeration of these objects akin
to Sommers’ exists: indeed Athanasiadis and Reiner [6, Theorem 6.3] had
previously shown that such refined enumerations of nonnesting and noncrossing
partitions coincide (without giving the explicit formulas in terms of
characteristic polynomials). However, this coincidence is proved via a case-
by-case check using the finite type classification and finding a more
conceptual explanation is still an open problem. The most promising attempt in
this direction is parking space theory, introduced by Armstrong, Reiner and
Rhoades [2]: among various other features, it gives a representation theoretic
framework to prove this kind of refined enumeration. In particular, the
characteristic polynomials $\chi_{X}(t)$ naturally appear there via an
identity in the parabolic Burnside ring (see Section 3, and Orlik and Solomon
[24]), rather than via hyperplane arrangements. This theory was also extended
in the Fuß-Catalan setting by Rhoades [27]: $m$-element chains of noncrossing
partitions are counted by $\operatorname{Cat}^{(m)}(W)$, and a natural
refinement gives the numbers:

$\frac{\chi_{X}(mh+1)}{[N(W_{X}):W_{X}]}.$

See Theorem 3.8 for the precise statement.

The main goal of this work is to give a refined enumeration of faces in the
generalized cluster complex. In Corollary 7.3, we show that the number of
faces (respectively, positive faces) associated to the orbit of a flat $X$
under a certain natural map is

$\displaystyle(-1)^{\dim(X)}\frac{\chi_{X}(-mh-1)}{[N(W_{X}):W_{X}]},\quad\text{
respectively, }\quad(-1)^{\dim(X)}\frac{\chi_{X}(-mh+1)}{[N(W_{X}):W_{X}]}.$
(1)

A notable difference with the refined enumeration of nonnesting and
noncrossing partitions is that these numbers don’t add up to the Fuß-Catalan
numbers: rather, $\operatorname{Cat}^{(m)}(W)$ and
$\operatorname{Cat}^{(m)}_{+}(W)$ appear as particular cases when $X$ is the
minimal flat of the intersection lattice (which corresponds to the enumeration
of facets and positive facets). Our derivation is case-free, but eventually
rely on Theorem 3.8 mentioned above (for which the only known proof is via a
case-by-case check).

There are a few preliminaries that are interesting on their own, as well as
nice consequences. Let us outline those both by giving the detailed
organization of this article:

  * •

Section 2 contains some background material.

  * •

Parking space theory is reviewed in Section 3, following [2, 27]. The aim is
to present Theorems 3.8 and 3.9, on which our results rely. Sections 4 and 5
contain proofs of the latter theorem (more precisely, we show it is a
consequence of the former). Eventually, it reoccurs in Section 9 which
contains a representation theoretic interpretation of our refined enumeration
in (1) via cluster parking functions. These form a simplicial complex whose
(equivariant) Euler characteristic is what we call a cluster parking space.

  * •

The generalized cluster complex is reviewed in Section 6, following [10, 16,
17, 36]. This section also contains the definition of the natural map from
faces to orbits of flats. The formulas in terms of characteristic polynomials
is obtained Section 7, via combinatorial reciprocities which make a link with
chains of noncrossing partitions. It is also proved in Section 8 via a
bijection, again making a link with certain chains of noncrossing partitions.
In Section 10, we give some consequences concerning the $f$-vectors and
$h$-vectors of the generalized cluster complex. In particular, an identity can
be seen as a refinement of the relations between $f$\- and $h$-vectors.
Eventually, Section 11 provides a recursion satisfied by the left-hand side of
(1), proved via the combinatorics of the generalized cluster complex.

  * •

Minimal factorizations of the Coxeter element are ubiquitous in the context of
noncrossing partitions and cluster complexes. In Section 12, we get a formula
for a $q$-enumeration of certain minimal factorizations (where one factor is
in a given parabolic conjugacy class and the others are reflections). This is
related to the enumeration of faces of the generalized cluster complex, via a
recursion which is equivalent to that in Section 11.

  * •

Two order relations on noncrossing partitions, denoted $\sqsubset$ and $\ll$,
are used throughout. We introduced them in [9], and in some sense they refine
the absolute order (used to define the lattice structure on noncrossing
partitions). They are useful to prove an identity on parking spaces (Section
5), to give bijections between faces of the generalized cluster complex and
certain chains of noncrossing partitions (Section 8), and to define the
$q$-statistic in our $q$-enumeration of minimal factorizations mentioned above
(Section 12).

### Acknowledgements

We thank Frédéric Chapoton for suggesting us to investigate the generating
function $\phi$ (defined in Section 12), which was a motivation for the whole
project. We also thank Philippe Biane for our fruitful discussion throughout.

##  2 Preliminary definitions

Through this work, $W$ is a finite real reflection group of rank $n$. We don’t
assume that it be irreducible, unless stated otherwise. Its geometric
representation is an $n$-dimensional Euclidean space $V\simeq\mathbb{R}^{n}$.
Let $T\subset W$ denote the set of reflections, and $S\subset T$ a set of
simple reflections that we write $S=\\{s_{1},\dots,s_{n}\\}$. The standard
parabolic subgroup $W_{I}$ for $I\subset S$ is the subgroup of $W$ generated
by $I$. Any subgroup conjugated to some $W_{I}$ for $I\subset S$ is called a
parabolic subgroup. The support of $w\in W$ is:

$\operatorname{supp}(w):=\min\big{\\{}I\subset S\;:\;w\in W_{I}\big{\\}}$

where the minimum is taken with respect to inclusion, and $w$ is said to have
full support if $\operatorname{supp}(w)=S$.

###  2.1 The intersection lattice

######  Definition 2.1.

For each $w\in W$, we denote

$\operatorname{Fix}(w):=\ker(w-I)=\big{\\{}v\in V\;:\;w(v)=v\big{\\}}.$

The _intersection lattice_ of $W$ is defined as

$L(W):=\big{\\{}\operatorname{Fix}(w)\;:\;w\in W\big{\\}}.$

By convention, the order relation on $L(W)$ is reversed inclusion.

This is indeed a lattice, and the join operation is given by intersection of
subspaces. It is order-isomorphic to the lattice of parabolic subgroups of $W$
(where the order is inclusion) via

$\displaystyle X\mapsto W_{X}:=\\{w\in
W\;:\;X\subset\operatorname{Fix}(w)\\}.$ (2)

There is a natural action of $W$ on $L(W)$ induced by the action of $W$ on
$V$, and the corresponding action on parabolic subgroups is by conjugation.
Let $\sim$ denote the equivalence class given by the orbit decomposition.

The apparent clash of notation between $W_{I}$ and $W_{X}$ is dealt with by
identifying the powerset of $S$ with a subposet of $L(W)$, via

$I\mapsto\operatorname{Fix}\big{(}\prod_{s\in I}s\big{)}.$

(The order of the product is irrelevant.) Implicitly, it is assumed that
$I,J,K,\dots$ are in this subposet, while $X,Y,Z,\dots$ are general flats in
$L(W)$. In particular, $I\sim J$ means that the standard parabolic subgroups
$W_{I}$ and $W_{J}$ are conjugated. This convention will be used when some
objects are indexed sometimes by $I,J,\dots$ and sometimes by $X,Y,\dots$.

###  2.2 The noncrossing partition lattice

Noncrossing partitions are defined with respect to a standard Coxeter element
$c$, which is the product of all the simple reflections in $S$. By reindexing
the set $S=\\{s_{1},\dots,s_{n}\\}$, we can assume $c=s_{1}\cdots s_{n}$.

######  Definition 2.2.

The reflection length of $w\in W$ is defined by

$\ell(w):=n-\dim(\operatorname{Fix}(w)).$

It is also the minimal integer $k$ such that $w$ is a product of $k$
reflections, and any such factorization $w=t_{1}\cdots t_{k}$ is called
minimal.

######  Definition 2.3.

The absolute order of $W$ is defined by $w_{1}\leq w_{2}$ if
$\ell(w_{1})+\ell(w_{1}^{-1}w_{2})=\ell(w_{2})$. Alternatively, $w_{1}\leq
w_{2}$ iff a minimal reflection factorization of $w_{1}$ is a subword of a
minimal reflection factorization of $w_{2}$. The noncrossing partition lattice
of $W$ (with respect to a standard Coxeter element $c$), denoted
$\operatorname{NC}(W)$, is defined as the order ideal containing elements
below $c$ in the absolute order.

This is a widely studied object, and we refer to Baumeister et. al. [8] for a
recent survey. Note that we have a map $w\mapsto\operatorname{Fix}(w)$ from
$\operatorname{NC}(W)$ to $L(W)$. This map is injective, increasing and rank-
preserving. Moreover $w_{1},w_{2}\in\operatorname{NC}(W)$ are conjugated (in
$W$) iff $\operatorname{Fix}(w_{1})$ and $\operatorname{Fix}(w_{2})$ are in
the same orbit under the action of $W$.

######  Lemma 2.4.

The following two sets are in bijection:

  * •

_parabolic conjugacy classes_ , i.e., conjugacy classes $\mathcal{X}\subset W$
such that $\mathcal{X}\cap\operatorname{NC}(W)\neq\varnothing$,

  * •

$L(W)/W$, i.e., orbits of flats under the action of $W$.

###### Proof.

We refer to [24] (see Lemma (3.4), Lemma (3.5), and the lines thereafter). Let
us briefly describe the explicit bijections.

To each parabolic conjugacy class $\mathcal{X}$, we associate the orbit of
$\operatorname{Fix}(w)$ for some arbitrary $w\in\mathcal{X}$. In the other
direction, consider a flat $X$ defined up to the action of $W$. There is
$I\subset S$ such that $W_{X}$ is conjugated to the standard parabolic
subgroup $W_{I}$, and to the orbit of $X$ we associate the conjugacy class of
$\prod_{s\in I}s$ (the order of the product is irrelevant). ∎

For $X\in L(W)$, the parabolic conjugacy class corresponding to its orbit in
$L(W)/W$ (via the previous bijection) is denoted $\mathcal{X}$. We sometimes
use the bijection implicitly and the notation makes clear what are the
objects. For example, if $w\in\operatorname{NC}(W)$ then the condition
$w\in\mathcal{X}$ is equivalent to $\operatorname{Fix}(w)\sim X$.

######  Lemma 2.5.

For each $w\in\operatorname{NC}(W)$, the parabolic subgroup
$W_{\operatorname{Fix}(w)}$ is the minimal parabolic subgroup of $W$
containing $w$.

######  Proposition 2.6.

A noncrossing partition $w\in\operatorname{NC}(W)$ is a standard Coxeter
element of $W_{\operatorname{Fix}(w)}$. This means there is a factorization
$w=t_{1}\cdots t_{k}$ (called the _canonical factorization_ , unique up to
commutations among the factors) where the elements $t_{1},\dots,t_{k}$ are the
simple generators of $W_{\operatorname{Fix}(w)}$.

This has been observed by several authors. We refer to [9, Proposition 3.1]
for a discussion.

###  2.3 A tale of two orders

The lattice structure of $\operatorname{NC}(W)$ (the absolute order) can be
refined: we introduced in [9] two partial order relations $\sqsubset$ and
$\ll$ with many combinatorial properties. In particular, they are useful to
deal with the combinatorics of the cluster complex.

######  Definition 2.7 ([9]).

Let $w\in\operatorname{NC}(W)$, and write its canonical factorization
$w=t_{1}\cdots t_{k}$. We define $\sqsubset$ and $\ll$ on
$\operatorname{NC}(W)$ by:

  * •

$v\sqsubset w$ if $v$ can be written as a subword of $t_{1}\cdots t_{k}$ (so
that $v\leq w$, in particular),

  * •

$v\ll w$ if $v\leq w$ and $v$ has full support in $W_{\operatorname{Fix}(w)}$,
i.e., each $t_{i}$ for $1\leq i\leq k$ appear at least once in any
factorization $w=t_{i_{1}}t_{i_{2}}\cdots$.

Note that these partial orders are such that $u\sqsubset v\Rightarrow u\leq v$
and $u\ll v\Rightarrow u\leq v$.

Another characterization of these partial orders makes a close connection with
the Bruhat order, denoted $\leq_{B}$. This will be used in Section 12. It
states that the cover relations for $\sqsubset$ and $\ll$ are such that:

$u\sqsubset\cdot\hskip 9.6739ptv\;\Leftrightarrow\;u\lessdot v\text{ and
}u\leq_{B}v,\qquad u\ll\cdot\hskip 7.11317ptv\;\Leftrightarrow\;u\lessdot
v\text{ and }u\geq_{B}v.$

Let us give some other properties, mostly taken from [9].

######  Proposition 2.8.

For each $u,w\in\operatorname{NC}(W)$ such that $u\leq w$, there exists a
unique $v\in\operatorname{NC}(W)$ such that $u\ll v\sqsubset w$.

###### Proof.

Let us first consider the case where $w$ is maximal, i.e., it is the Coxeter
element $c$. In this case, let $I\subset S$ be the support of $u$ and $v$ be
the unique $v\sqsubset c$ which is the product of $s_{i}$ for $i\in I$. It is
easily checked that it satisfies $u\ll v\sqsubset w$, and it is unique. The
general case follows by doing the same procedure in the parabolic subgroup
$W_{\operatorname{Fix}(w)}$. ∎

######  Proposition 2.9 ([9, Corollary 4.10]).

Let $w\in\operatorname{NC}(W)$. We have:

  * •

The two orders $\leq$ and $\sqsubset$ agree on the set
$\\{v\in\operatorname{NC}(W)\;:\;v\sqsubset w\\}$. The resulting poset is a
boolean lattice of order $2^{\ell(w)}$, containing all elements than can be
written as subwords of the canonical factorization of $w$.

  * •

The two orders $\leq$ and $\ll$ agree on the set
$\\{v\in\operatorname{NC}(W)\;:\;v\gg w\\}$. The resulting poset is a boolean
lattice of order $2^{\\#\operatorname{supp}(w)-\ell(w)}$, and its maximal
element is the unique $w^{\prime}$ such that $w\ll w^{\prime}\sqsubset c$
(given by the previous proposition).

##  3 Parking spaces and their characters

The goal of this section is to introduce some background about the parking
space theory, and introduce a “prime” analog of the parking space. The main
result about this prime parking space is Theorem 3.9.

###  3.1 The parabolic Burnside ring

We first need some preliminaries about characters of $W$, due to Orlik and
Solomon [24]. See also Geck and Pfeifer [18, Chapter 2.4].

Generically, we will use bold symbols to denote characters. In particular,
${\bf 1}$ denote the trivial character (of a group which is clear from the
context).

######  Definition 3.1.

For $I\subset S$, let
$\operatorname{\boldsymbol{\Phi}}_{I}:=\mathbf{ind}_{W_{I}}^{W}(\bf 1)$ (the
trivial character of $W_{I}$ induced to $W$). The parabolic Burnside ring
$R(W)$ of $W$ is the ring linearly generated by
$(\operatorname{\boldsymbol{\Phi}}_{I})_{I\subset S}$ (as a subring of the
character ring of $W$).

It is not obvious that the linear span of
$(\operatorname{\boldsymbol{\Phi}}_{I})_{I\subset S}$ is indeed a ring. We
refer to [18, Section 2.4.3]. It also follows from loc. cit. that a basis of
$R(W)$ is $(\operatorname{\boldsymbol{\Phi}}_{I})_{I\in\Theta}$, where
$\Theta$ is a set of representatives of subsets of $S$ modulo the equivalence
relation $\sim$ as in Lemma 2.4. Using implicitly one of the bijections from
Lemma 2.4, we identify $\Theta$ with a set of representatives for the quotient
$L(W)/W$. We thus write $\operatorname{\boldsymbol{\Phi}}_{X}$ in place of
$\operatorname{\boldsymbol{\Phi}}_{I}$ for $X\in L(W)$ such that $W_{X}$ and
$W_{I}$ are conjugated.

Note that $\operatorname{\boldsymbol{\Phi}}_{S}$ is the trivial character of
$W$, and the unit of $R(W)$. Also, it can be seen that
$\operatorname{\boldsymbol{\Phi}}_{I}$ is the character of the representation
$\mathbb{C}^{W/W_{I}}$ (the linearization of the group action on $W/W_{I}$
where $W$ acts by left multiplication on the cosets).

######  Remark 3.2.

Let us mention that $R(W)$ is a subring of the Burnside ring of $W$ (which is
linearly generated by characters of the representations
$\mathbb{C}^{W/W^{\prime}}$ where $W^{\prime}$ is any subgroup of $W$). The
terminology comes from the fact that here we only consider parabolic
subgroups.

######  Remark 3.3.

The algebra $\mathbb{Q}\otimes R(W)$ is the space of functions
$\chi:W\to\mathbb{Q}$ such that the value $\chi(w)$ only depends on the orbit
of $\operatorname{Fix}(w)$ in $L(W)/W$. This characterization is essentially
due to Orlik and Solomon [24].

The sign character of $W$, denoted $\boldsymbol{\epsilon}$, is defined by

$\boldsymbol{\epsilon}(w):=(-1)^{\ell(w)}=(-1)^{n-\dim\operatorname{Fix}(w)}.$

This character is usually defined with the Coxeter length rather than the
reflection length, but they have the same parity as each reflection has odd
Coxeter length. The following lemma says how it acts on $R(W)$ by
multiplication (note that taking $J=\varnothing$ shows that
$\boldsymbol{\epsilon}\in R(W)$, which can also be seen via the previous
remark).

######  Lemma 3.4 (Solomon [29]).

For any $J\subset S$, we have

$\boldsymbol{\epsilon}\otimes\operatorname{\boldsymbol{\Phi}}_{J}=\sum_{I\subset
J}(-1)^{\\#I}\operatorname{\boldsymbol{\Phi}}_{I}.$ (3)

Now, define a class function
$\operatorname{\boldsymbol{\Psi}}_{t}:W\to\mathbb{Z}$ for each integer $t$ by

$\operatorname{\boldsymbol{\Psi}}_{t}(w)=t^{\dim\operatorname{Fix}(w)}.$

In particular,
$\operatorname{\boldsymbol{\Psi}}_{-1}=(-1)^{n}\boldsymbol{\epsilon}$. Note
that $\operatorname{\boldsymbol{\Psi}}_{t}\in\mathbb{Q}\otimes R(W)$ for each
$t\in\mathbb{Z}$. This is easily seen from the characterization of
$\mathbb{Q}\otimes R(W)$ stated in Remark 3.3. In the more general context of
complex reflection groups, Ito and Okada [22] gave a classification of
positive integers $t$ such that $\operatorname{\boldsymbol{\Psi}}_{t}$ is in
the positive integral span of
$(\operatorname{\boldsymbol{\Phi}}_{I})_{I\in\Theta}$ (i.e.,
$\operatorname{\boldsymbol{\Psi}}_{t}$ is the character of some representation
of $W$). The important cases for us are the following: if $t=mh+1$ with $m$ a
nonnegative integer, or $t=mh-1$ with $m$ a positive integer, this property
holds for $\operatorname{\boldsymbol{\Psi}}_{t}$.

######  Remark 3.5.

The results of Ito and Okada [22] also show the following. For any finite
complex reflection group with Coxeter number $h$, it is still true that
$\operatorname{\boldsymbol{\Psi}}_{mh+1}$ ($m\geq 0$) is the genuine character
of a group representation. But this does not hold for
$\operatorname{\boldsymbol{\Psi}}_{mh-1}$ ($m>0$) beyond the real case.

In the next statement, we treat $t$ as a formal variable rather than an
integer. Also, recall from the introduction that $N(W_{X})$ is the normalizer
of $W_{X}$ in $W$.

######  Proposition 3.6 (Orlik and Solomon [24]).

In $\mathbb{Q}[t]\otimes R(W)$, there is an expansion:

$\operatorname{\boldsymbol{\Psi}}_{t}=\sum_{X\in\Theta}\frac{\chi_{X}(t)}{[N(W_{X}):W_{X}]}\operatorname{\boldsymbol{\Phi}}_{X}$
(4)

where $\chi_{X}(t)$ is a polynomial in $t$ called the _characteristic
polynomial_ of $X$. It can be defined using the Möbius function $\mu$ of
$L(W)$ by

$\chi_{X}(t)=\sum_{Y\in L(W),\;Y\geq X}\mu(X,Y)t^{\dim(Y)}$ (5)

and can be factorized in the form:

$\chi_{X}(t)=\prod_{i=1}^{\dim(X)}(t-b_{i}^{X})$

where the roots $b_{i}^{X}$ are positive integers called the _Orlik-Solomon
exponents_ of $X$.

See [25] for tables containing the Orlik-Solomon exponents for all irreducible
$W$ in the finite type classification. When $X$ is the minimal element of
$L(W)$ (i.e., the full-dimensional subspace of the geometric representation
$V$ of $W$), the associated Orlik-Solomon exponents are the exponents
$e_{1},\dots,e_{n}$ of $W$ (classically defined by considering eigenvalues of
the Coxeter element, see [20]).

From the previous proposition, we see that there are at least two ways to
compute the characteristic polynomials (or the Orlik-Solomon exponents, by
taking their roots):

  * •

we can use the intersection lattice and its Möbius function via Equation (5),

  * •

we can use Equation (4) and some character calculations.

Let us make the second point more explicit. The values of the characters
$\operatorname{\boldsymbol{\Phi}}_{I}$ on parabolic conjugacy classes can be
organized in a square matrix (with rows and columns indexed by $\Theta$)
called the parabolic table of marks. An algorithm to compute it is given by
Geck and Pfeiffer [18, Section 2.4]. Since the values of the character
$\operatorname{\boldsymbol{\Psi}}_{t}$ is explicit, we can get the
coefficients $([N(W_{I}):W_{I}]^{-1}\chi_{I}(t))$ by inverting the parabolic
table of marks.

Another way to compute these characteristic polynomials is given by Sommers in
[30, Propositions 4.7 and 5.1]. We will give another method below, by giving a
recursion satisfied by the numbers $\gamma(W,\mathcal{X},m)$ (see Section 11).

The denominator $[N(W_{X}):W_{X}]$ that appear above can be written
differently. Orlik and Solomon [24] showed that:

$\frac{1}{[N(W_{X}):W_{X}]}=\frac{\nu(X)}{\prod_{i=1}^{k}(b_{i}^{X}+1)}$ (6)

where $\nu(X)$ is the number of $J\subset S$ such that $W_{J}\sim W_{X}$.
Indeed, this follows by plugging $t=-1$ in the previous proposition, and using
Lemma 3.4. As a check of what happens in the case of the symmetric group (type
A), let $\lambda$ be a partition of $n$ and $\mathfrak{S}_{\lambda}$ the
corresponding Young subgroup of $\mathfrak{S}_{n}$. Let $\mu_{i}$ be the
multiplicity of $i$ in $\lambda$. The normalizer $N(\mathfrak{S}_{\lambda})$
is a semidirect product
$\mathfrak{S}_{\lambda}\rtimes(\prod_{i}\mathfrak{S}_{\mu_{i}})$, and it
follows
$[N(\mathfrak{S}_{\lambda}):\mathfrak{S}_{\lambda}]=\prod_{i}\mu_{i}!$.

Eventually, let us mention the following (see also [19, Theorem 7.4.2]):

######  Proposition 3.7.

If $t>0$ is such that $\operatorname{\boldsymbol{\Psi}}_{t}$ is the character
of a representation, the multiplicity of the trivial character in
$\operatorname{\boldsymbol{\Psi}}_{t}$ is the _rational Catalan number_ :

$\frac{1}{|W|}\prod_{i=1}^{n}(t+e_{i}).$

###### Proof.

The trivial character is orthogonal to the other irreducible characters, and
it follows that this multiplicity is:

$\frac{1}{|W|}\sum_{w\in W}\operatorname{\boldsymbol{\Psi}}_{t}(w).$

The result then follows from the Shephard-Todd formula [28]:

$\sum_{w\in W}t^{\dim\operatorname{Fix}(w)}=\prod_{i=1}^{n}(t+e_{i}).$

∎

In the two cases which are relevant to this work, we get that the multiplicity
of the trivial character in $\Psi_{mh+1}$ and $\Psi_{mh-1}$ are respectively
$\operatorname{Cat}^{(m)}(W)$ and $\operatorname{Cat}_{+}^{(m)}(W)$.

###  3.2 Parking spaces

We review the theory of parking spaces, introduced by Armstrong, Reiner and
Rhoades [2]. This is a very brief account, which mostly aims at some
enumeration formulas for chains of noncrossing partitions. In particular, we
simply consider characters of $W$ whereas the general theory deals with
characters of $W$ times a cyclic group. Also, we focus on the noncrossing
parking space but other kinds of parking spaces exist.

Assume that $W$ is irreducible, and let $h$ be its Coxeter number. A parking
space is a representation of $W$ having
$\operatorname{\boldsymbol{\Psi}}_{h+1}$ as its character. The name comes from
the fact that in type $A_{n}$, the symmetric group $\mathfrak{S}_{n+1}$ acting
on parking functions of length $n+1$ is such a parking space. More generally,
a $t$-parking space (for an integer $t\geq 1$) is a representation of $W$
having $\operatorname{\boldsymbol{\Psi}}_{t}$ as its character. In the case
where $W$ is a Weyl group, there is a root lattice $Q$ acted on by $W$. It
follows from Sommers [30, Proposition 3.9] that the quotient $Q/tQ$ is a
$t$-parking space for the natural action of $W$, upon some conditions on $t$
(being coprime to $h$ is sufficient). For the symmetric group
$\mathfrak{S}_{n}$, other $t$-parking spaces are given by the action on
rational parking functions (see [3] for details).

Two $t$-parking spaces are isomorphic as representations of $W$, since by
definition they have the same character. However, it might be difficult to
find an explicit isomorphism and it is therefore interesting to consider
various kinds of parking spaces. The noncrossing parking space from [2] is
defined in terms of noncrossing partitions. Rhoades’ generalization [27] in
the Fuß-Catalan settings is:

$\bigoplus_{\begin{subarray}{c}w_{1},\dots,w_{m}\in\operatorname{NC}(W)\\\
w_{1}\leq\dots\leq
w_{m}\end{subarray}}\mathbb{C}^{W/W_{\operatorname{Fix}(w_{1})}}.$ (7)

It can be seen as the linearization of the $W$-set of $m$-parking functions:

$\displaystyle\operatorname{PF}(W,m):=\biguplus_{\begin{subarray}{c}w_{1},\dots,w_{m}\in\operatorname{NC}(W)\\\
w_{1}\leq\dots\leq w_{m}\end{subarray}}\\{(w_{1},\dots,w_{m})\\}\times
W/W_{\operatorname{Fix}(w_{1})}$ (8)

(where $W$ acts on the second factor). Rather than the representation itself,
we mostly consider its character:

$\displaystyle\mathbf{park}_{W,m}:=\sum_{X\in\Theta}\kappa(W,X,m)\operatorname{\boldsymbol{\Phi}}_{X}\in
R(W)$ (9)

where

$\displaystyle\kappa(W,X,m):=\\#\Big{\\{}(w_{1},\dots,w_{m})\in\operatorname{NC}(W)^{m}\;:\;w_{1}\leq\dots\leq
w_{m}\text{ and }\operatorname{Fix}(w_{1})\sim X\Big{\\}}.$ (10)

######  Theorem 3.8 ([27], except for $E_{7}$, $E_{8}$).

Assume that $W$ is irreducible, with Coxeter number $h$. Then, the
representation in (7) is a $mh+1$-parking space, i.e.,
$\mathbf{park}_{W,m}=\operatorname{\boldsymbol{\Psi}}_{mh+1}$. Equivalently:

$\forall
X\in\Theta,\quad\kappa(W,X,m)=\frac{\chi_{X}(mh+1)}{[N(W_{X}):W_{X}]}.$ (11)

###### Proof.

Note that the equivalence between the two statements follows from Proposition
3.6. Since the infinite families of the finite type classification have been
treated (see Table 3 in [27]), there remains only to deal with a finite number
of exceptional groups. This can be done by computer. ∎

Let us give some comments about the computer verification mentioned in this
proof. First note that the computation is possible because we have here a
finite statement about polynomials in $m$, whereas Rhoades’ general conjecture
involves a cyclic group of order $mh+1$ and is therefore of a different
nature. A naive computation of the set $\operatorname{NC}(W)$ might be
lengthy, but an efficient way to get all noncrossing partitions is to use the
characterization in [9, Section 4.5]. It says that there is a binary relation
$\between$ on $T$ (that depends on $c$) such that each
$w\in\operatorname{NC}(W)$ can be identified with a set of pairwise-related
reflections (explicitly, this is the set of simple reflections of
$W_{\operatorname{Fix}(w)}$). It gives a quick way to get all noncrossing
partitions, and for each $w\in\operatorname{NC}(W)$ we also have the canonical
factorization $w=t_{1}\cdots t_{k}$ into simple reflections of the parabolic
subgroup $W_{\operatorname{Fix}(w)}$. Now, the right-hand side of (10) can be
rewritten

$\sum_{w\in\mathcal{X}}\operatorname{Cat}^{(m-1)}(W_{\operatorname{Fix}(w^{-1}c)}).$

Using the formula for Fuß-Catalan numbers in terms of exponents (which we can
use here since we have the Coxeter type of $W_{\operatorname{Fix}(w^{-1}c)}$
via the canonical factorization of $w^{-1}c$), we can compute this sum in
reasonable time even for $E_{7}$ and $E_{8}$.

Now, Let $\operatorname{NC}^{\prime}(W)\subset\operatorname{NC}(W)$ denote the
subset of element with full support. The “prime” analog of the noncrossing
parking space is now defined as

$\bigoplus_{\begin{subarray}{c}w_{1},\dots,w_{m-1}\in\operatorname{NC}(W),\;w_{m}\in\operatorname{NC}^{\prime}(W)\\\
w_{1}\leq\dots\leq
w_{m}\end{subarray}}\mathbb{C}^{W/W_{\operatorname{Fix}(w_{1})}}.$ (12)

Again, we mostly consider its character:

$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}:=\sum_{X\in\Theta}\kappa^{+}(W,X,m)\operatorname{\boldsymbol{\Phi}}_{X}$
(13)

where

$\displaystyle\begin{split}\kappa^{+}(W,X,m):=\\#\Big{\\{}(w_{1},\dots,w_{m})\in\operatorname{NC}(W)^{m-1}\times\operatorname{NC}^{\prime}(W)\;:\;w_{1}\leq\dots\leq
w_{m}\\\ \text{ and }\operatorname{Fix}(w_{1})\sim X\Big{\\}}.\end{split}$
(14)

In complete analogy with Theorem 3.8, we have the following one. Even though
it does not explicitly appeared in the literature, it is not particularly
surprising: some known enumeration formulas clearly suggest that the parameter
$mh-1$ should appear in the present situation (see [5], concerning the
positive part of the generalized cluster complex, and [31], concerning
strictly positive ideals in the root poset).

######  Theorem 3.9.

Assume that $W$ is irreducible, with Coxeter number $h$. The representation in
(12) is a $mh-1$-parking space, i.e.,
$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}=\operatorname{\boldsymbol{\Psi}}_{mh-1}$.
Equivalently:

$\forall
X\in\Theta,\quad\kappa^{+}(W,X,m)=\frac{\chi_{X}(mh-1)}{[N(W_{X}):W_{X}]}.$
(15)

Theorem 3.9 can be approached along the same path as Theorem 3.8. Rather than
doing that, we will prove that Theorems 3.8 and 3.9 are equivalent. More
precisely, we give two proofs of this equivalence, that follow a similar
pattern:

  * •

In Section 4, we show that the characters
$\operatorname{\boldsymbol{\Psi}}_{mh+1}$ and
$\operatorname{\boldsymbol{\Psi}}_{mh-1}$ satisfy a kind of inclusion-
exclusion (which is straightforward for the characters $\mathbf{park}_{W,m}$
and $\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$).

  * •

In Section 5, we show that the characters $\mathbf{park}_{W,m}$ and
$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$ satisfy
$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}=(-1)^{n}\boldsymbol{\epsilon}\otimes\mathbf{park\mathrlap{{}^{\prime}}}_{W,-m}$
(the corresponding identity for $\operatorname{\boldsymbol{\Psi}}_{mh+1}$ and
$\operatorname{\boldsymbol{\Psi}}_{mh-1}$ being clear).

In each case, the equivalence of the two theorems follows. These two proofs
are case-free (they do not rely in the finite type classification), so it
might happen that both have a role to play in a fully combinatorial and case-
free approach to parking space theory.

##  4 Proof of Theorem 3.9 via the W-Laplacian

We begin this section by stating a property of the characters
$\mathbf{park}_{W,m}$ and $\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$.

######  Proposition 4.1.

We have the “inclusion-exclusion” formulas:

$\displaystyle\mathbf{park}_{W,m}$ $\displaystyle=\sum_{I\subset
S}\mathbf{ind}_{W_{I}}^{W}(\mathbf{park\mathrlap{{}^{\prime}}}_{W_{I},m}),$
$\displaystyle\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$
$\displaystyle=\sum_{I\subset
S}(-1)^{\\#S-\\#I}\mathbf{ind}_{W_{I}}^{W}(\mathbf{park}_{W_{I},m}).$

###### Proof.

First note that if $I\subset J\subset S$, we have
$\mathbf{ind}_{W_{J}}^{W}(\mathbf{ind}_{W_{I}}^{W_{J}}({\bf
1}))=\mathbf{ind}_{W_{J}}^{W}({\bf 1})$. This transitivity property follows
from the general theory of group characters. With this at hand, these
identities follow from the same lines as the classical inclusion-exclusion
principle. ∎

Our goal is to show that the characters
$\operatorname{\boldsymbol{\Psi}}_{mh+1}$ and
$\operatorname{\boldsymbol{\Psi}}_{mh-1}$ satisfy the same relations. First
note that these two characters are only defined in the irreducible case, and
we need to extend them in the natural multiplicative way. For these we refer
to the abstract parking spaces, as opposed to other kinds of parking spaces
which are defined as characters of some explicit representation of $W$.

######  Definition 4.2.

If $W$ is irreducible with Coxeter number $h$, we define
$\mathbf{park}^{\operatorname{abs}}_{W,m}:=\operatorname{\boldsymbol{\Psi}}_{mh+1}$
and
$\mathbf{park^{\prime}}^{\operatorname{abs}}_{W,m}:=\operatorname{\boldsymbol{\Psi}}_{mh-1}$.
Otherwise, write $W=\prod_{i=1}^{r}W_{i}$ its decomposition in irreducible
factors. Note that there is an isomorphism

$R(W)\simeq\bigotimes_{i=1}^{r}R(W_{i})$

and define:

$\displaystyle\mathbf{park}^{\operatorname{abs}}_{W,m}$
$\displaystyle:=\mathbf{park}^{\operatorname{abs}}_{W_{1},m}\otimes\cdots\otimes\mathbf{park}^{\operatorname{abs}}_{W_{r},m},$
$\displaystyle\mathbf{park^{\prime}}^{\operatorname{abs}}_{W,m}$
$\displaystyle:=\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{1},m}\otimes\cdots\otimes\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{r},m}.$

More explicitly, there is a natural identification
$L(W)\simeq\prod_{i=1}^{r}L(W_{i})$. So $w=(w_{1},\dots,w_{r})\in W$
determines flats $Z_{i}\in L(W_{i})$ by:

$\operatorname{Fix}(w)=\prod_{i=1}^{r}Z_{i}.$

We then have:

$\displaystyle\mathbf{park}^{\operatorname{abs}}_{W,m}(w)$
$\displaystyle=\prod_{i=1}^{r}(mh_{i}+1)^{\dim(Z_{i})},$
$\displaystyle\mathbf{park^{\prime}}^{\operatorname{abs}}_{W,m}(w)$
$\displaystyle=\prod_{i=1}^{r}(mh_{i}-1)^{\dim(Z_{i})}.$

where $h_{i}$ is the Coxeter number of $W_{i}$.

Since the values of these characters at $w\in W$ only depend on
$\operatorname{Fix}(w)$, we can describe their parabolic inductions in terms
of the geometry of hyperplane arrangements.

######  Lemma 4.3.

For $X\in L(W)$, $w\in W$ and $Z=\operatorname{Fix}(w)$, we have:

$\Big{(}\mathbf{ind}_{W_{X}}^{W}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}\Big{)}(w)=[N(X):W_{X}]\cdot\sum_{\begin{subarray}{c}Y\in
L(W),\\\ Y\sim X\text{ and }Y\subset
Z\end{subarray}}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{Y},m}(w).$

###### Proof.

By the general formula for induction of characters, we have

$\Big{(}\mathbf{ind}_{W_{X}}^{W}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}\Big{)}(w)=\sum_{y\in
W/W_{X},\;y^{-1}wy\in
W_{X}}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}(y^{-1}wy).$

Note that we have:

$\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{(y\cdot
X)},m}(w)=\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}(y^{-1}wy),$

as can be seen by using the inner automorphism $w\mapsto y^{-1}wy$ (and its
extension to $L(W)$ which the action of $y$). Moreover, we have:

$y^{-1}wy\in
W_{X}\;\Leftrightarrow\;X\subset\operatorname{Fix}(y^{-1}wy)\;\Leftrightarrow\;X\subset
y^{-1}\cdot\operatorname{Fix}(w)\;\Leftrightarrow\;y\cdot X\subset Z.$

We can thus rewrite the sum and get:

$\Big{(}\mathbf{ind}_{W_{X}}^{W}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}\Big{)}(w)=\sum_{y\in
W/W_{X},\;y\cdot X\subset
Z}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{(y\cdot X)},m}(w).$

By letting $Y=y\cdot X$, this can be rewritten:

$\displaystyle\Big{(}\mathbf{ind}_{W_{X}}^{W}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}\Big{)}(w)=[N(W_{X}):W_{X}]\cdot\sum_{\begin{subarray}{c}Y\in
L(W)\\\ Y\sim X\text{ and }Y\subset
Z\end{subarray}}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{Y},m}(w).$

To get the equality, it suffices to check that $[N(W_{X}):W_{X}]$ is the
number of $y\in W/W_{X}$ such that $y\cdot X=Y$, for each $Y\sim X$. This is
the orbit-stabilizer theorem, indeed $N(W_{X})$ is the subgroup of $W$
containing $w\in W$ such that $w\cdot X=X$. ∎

######  Remark 4.4.

The previous lemma holds with $\mathbf{park}^{\operatorname{abs}}$ in place of
$\mathbf{park^{\prime}}^{\operatorname{abs}}$, with a completely similar
proof.

Every parabolic subgroup $W_{X}$ of $W$ is a possibly reducible reflection
group and we would like a simpler notation of the values of the parking
characters $\mathbf{park}^{\operatorname{abs}}_{W_{X},m}$ and
$\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}$. For this reason, we
introduce the following notation.

######  Notation 4.5 (multisets of Coxeter numbers).

Let $W$ be an irreducible reflection group, and $X,Z\in L(W)$ such that
$X\subset Z$. Assume that $W_{X}=W_{1}\times\cdots\times W_{r}$ is the
decomposition into irreducible factors. Write $Z=\prod_{i=1}^{r}Z_{i}$, using
as above $L(W_{X})\simeq\prod_{i=1}^{r}L(W_{i})$. Note that this isomorphism
is such that

$\dim(Z)-\dim(X)=\sum_{i=1}^{r}\dim(Z_{i}),$

which can be seen by comparing the rank functions on each side. Then we write
the _multiset of Coxeter numbers_ associated to $X$ and $Z$ as:

$(h_{i}(X,Z))_{1\leq
i\leq\dim(Z)-\dim(X)}:=(\underbrace{h_{1},\dots,h_{1}}_{\operatorname{dim}(Z_{1})\text{
times}},\dots,\underbrace{h_{r},\dots,h_{r}}_{\operatorname{dim}(Z_{r})\text{
times}}),$

where $h_{i}$ is the Coxeter number of $W_{i}$. With this notation, we can
write the parking space characters for parabolic subgroups: for each $w\in
W_{X}$ with $\operatorname{Fix}(w)=Z$, we have:

$\displaystyle\mathbf{park}^{\operatorname{abs}}_{W_{X},m}(w)$
$\displaystyle=\prod_{i=1}^{\operatorname{dim}(Z)-\operatorname{dim}(X)}(mh_{i}(X,Z)+1),$
$\displaystyle\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}(w)$
$\displaystyle=\prod_{i=1}^{\dim(Z)-\dim(X)}(mh_{i}(X,Z)-1).$

What we need is a relation involving the Coxeter numbers of $W$ and those of
its parabolic subgroups that generalizes the one which appeared in [12,
Theorem 8.8]. Recall that this was given as follows: if $W$ is irreducible
with Coxeter number $h$,

$(h+t)^{n}=\sum_{X\in L(W)}\Big{(}\prod_{i=1}^{n-\dim(X)}h_{i}(X)\Big{)}\cdot
t^{\operatorname{dim}(X)}$ (16)

where $h_{i}(X)$ for $1\leq i\leq n$ is the particular case of
$(h_{i}(X,Z))_{1\leq i\leq\dim(Z)-\dim(X)}$ when $Z$ is the minimal element of
$L(W)$. Such a generalization (see [15, §4] for details) is the following:

######  Proposition 4.6 (Corollary of the restricted Laplacian recursion).

For any $X,Z\in L(W)$ such that $X\subset Z$, we have the following relation
between Coxeter numbers:

$\prod_{i=1}^{\dim(Z)-\dim(X)}\big{(}h_{i}(X,Z)+t\big{)}=\sum_{X\subset
Y\subset
Z}\Bigg{(}\prod_{i=1}^{\dim(Z)-\dim(Y)}h_{i}(Y,Z)\Bigg{)}t^{\dim(Y)-\dim(X)}.$
(17)

###### Proof.

For $X,Z\in L(W)$ such that $X\subset Z$, consider the set

$\mathcal{A}^{X,Z}:=\Big{\\{}Y/X\;:\;X\subset Y\subset
Z,\;\dim(Y)=\dim(Z)-1\Big{\\}}$

which is a hyperplane arrangement in the quotient $Z/X$ (the essentialization
of the restricted localization $\mathcal{A}_{X}^{Z}$). When $X$ is the
$0$-dimensional subspace, we just denote $\mathcal{A}^{Z}=\mathcal{A}^{X,Z}$.

Generalizing [12], to this hyperplane arrangement there is an associated
laplacian $\mathcal{L}_{X,Z}$ for which each hyperplane
$K\in\mathcal{A}^{X,Z}$ is taken with multiplicity equal to $h_{1}(K,Z)$. Its
characteristic polynomial is the left-hand side of (17). This follows
analogously to [12, Proposition 3.13].

Now, the laplacian recursion [12, Proposition 8.3] relates the characteristic
polynomial of $\mathcal{L}_{X,Z}$ to the determinants of $\mathcal{L}_{Y,Z}$
for $X\subset Y\subset Z$, and give precisely the above identity. ∎

The hyperplane arrangement $\mathcal{A}^{Z}$ introduced above is called a
restricted arrangement, see [24, 25]. The number $r(\mathcal{A}^{Z})$ of
regions in the complement of this arrangement is given by

$r(\mathcal{A}^{Z})=(-1)^{\dim(Z)}\chi_{Z}(-1)=\nu(Z)\cdot[N(W_{Z}):W_{Z}].$
(18)

What we need is the latter equality, see [24, Equations (4.1) and (4.2)]. It
can also be obtained by plugging $t=-1$ in (4), using (6) and (3) in the case
$J=S$.

######  Proposition 4.7.

Assume that $W$ is irreducible, with Coxeter number $h$. For any $Z\in L(W)$,
we have the following relation between Coxeter numbers:

$(mh+1)^{\dim(Z)}=\sum_{X\subset
Z}r(\mathcal{A}^{X})\cdot\prod_{i=1}^{\dim(Z)-\dim(X)}(mh_{i}(X,Z)-1).$

###### Proof.

Consider Equation (17) in the case where $X$ is the $0$-dimensional subspace,
so that $h_{i}(X,Z)=h$. By replacing $t$ with $t/m$ and multiplying both sides
by $m^{\operatorname{dim}(Z)}$, we get

$(mh+t)^{\dim(Z)}=\sum_{Y\subset
Z}\Big{(}\prod_{i=1}^{\dim(Z)-\dim(Y)}mh_{i}(Y,Z)\Big{)}\cdot t^{\dim(Y)}.$

By plugging $t^{\dim(Y)}=\sum_{X\subset Y}\chi(\mathcal{A}^{X},t)$ (which is
the inverse of (5)), we obtain

$\displaystyle(mh+t)^{\dim(Z)}$ $\displaystyle=\sum_{X\subset Y\subset
Z}\Big{(}\prod_{i=1}^{\operatorname{dim}(Z)-\operatorname{dim}(Y)}mh_{i}(Y,Z)\Big{)}\cdot\chi_{X}(t)$
$\displaystyle=\sum_{X\subset Z}\Big{(}\sum_{X\subset Y\subset
Z}\prod_{i=1}^{\dim(Z)-\dim(Y)}mh_{i}(Y,Z)\Big{)}\cdot\chi_{X}(t).$

Again using (17) (with $t=1$), this gives:

$(mh+t)^{\operatorname{dim}(Z)}=\sum_{X\subset
Z}\Big{(}\prod_{i=1}^{\operatorname{dim}(Z)-\operatorname{dim}(X)}\big{(}mh_{i}(X,Z)+1\big{)}\Big{)}\cdot\chi_{X}(t).$

Now, replacing $(m,t)$ with $(-m,-1)$ and getting rid of the signs, this
becomes

$(mh+1)^{\operatorname{dim}(Z)}=\sum_{X\subset
Z}\Big{(}\prod_{i=1}^{\operatorname{dim}(Z)-\operatorname{dim}(X)}\big{(}mh_{i}(X,Z)-1\big{)}\Big{)}\cdot(-1)^{\dim(X)}\chi_{X}(-1).$

This is precisely what we needed to prove. ∎

######  Theorem 4.8.

We have:

$\mathbf{park}^{\operatorname{abs}}_{W,m}=\sum_{I\subset
S}\mathbf{ind}_{W_{I}}^{W}(\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{I},m}).$
(19)

###### Proof.

We assume that $W$ is irreducible, and let $h$ be its Coxeter number. We let
the reader check that the reducible case follows.

Let $\operatorname{\bf R}$ denote the character in the right-hand side of
(19). Using the notation $\nu(X)$ (see Equation (6)), we have:

$\operatorname{\bf
R}=\sum_{X\in\Theta}\nu(X)\cdot\mathbf{ind}_{W_{X}}^{W}(\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}).$

Let $w\in W$ and $Z=\operatorname{Fix}(w)$. Lemma 4.3 tells us that the
evaluation of the character is:

$\displaystyle\operatorname{\bf
R}(w)=\sum_{X\in\Theta}\nu(X)\cdot[N(X):W_{X}]\cdot\sum_{\begin{subarray}{c}X^{\prime}\in
L(W),\\\ X^{\prime}\sim X\text{ and }X^{\prime}\subset
Z\end{subarray}}\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X^{\prime}},m}(w).$

This can be rewritten as a single sum over $X^{\prime}$, and we get:

$\displaystyle\operatorname{\bf R}(w)$
$\displaystyle=\sum_{\begin{subarray}{c}X\in L(W),\\\ X\subset
Z\end{subarray}}\nu(X)\cdot[N(X):W_{X}]\cdot\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}(w)$
$\displaystyle=\sum_{X\subset
Z}r(\mathcal{A}^{X})\cdot\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{X},m}(w).$

Indeed, recall
$\nu(X)\cdot[N(X):W_{X}]=(-1)^{\dim(X)}\chi_{X}(-1)=r(\mathcal{A}^{X})$. From
Proposition 4.6, this becomes $\operatorname{\bf R}(w)=(mh+1)^{\dim(Z)}$, so
that $\operatorname{\bf R}=\mathbf{park}^{\operatorname{abs}}_{W,m}$. ∎

######  Theorem 4.9.

We have:

$\mathbf{park^{\prime}}^{\operatorname{abs}}_{W,m}=\sum_{I\subset
S}(-1)^{\\#S-\\#I}\mathbf{ind}_{W_{I}}^{W}(\mathbf{park}^{\operatorname{abs}}_{W_{I},m}).$

###### Proof.

It is straightforward to adapt the proof of the previous theorem. We can also
see it as a consequence: substitute $m$ to $-m$ in (19) and tensor with
$\boldsymbol{\epsilon}$, this gives:

$\displaystyle\mathbf{park^{\prime}}^{\operatorname{abs}}_{W,m}$
$\displaystyle=(-1)^{n}\sum_{I\subset
S}\boldsymbol{\epsilon}\otimes\mathbf{ind}_{W_{I}}^{W}(\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{I},-m})$
$\displaystyle=(-1)^{n}\sum_{I\subset
S}\mathbf{ind}_{W_{I}}^{W}(\boldsymbol{\epsilon}\otimes\mathbf{park^{\prime}}^{\operatorname{abs}}_{W_{I},-m})$
$\displaystyle=(-1)^{n}\sum_{I\subset
S}\mathbf{ind}_{W_{I}}^{W}((-1)^{\\#I}\mathbf{park}^{\operatorname{abs}}_{W_{I},m}).$

We used the identity

$\boldsymbol{\epsilon}\otimes\mathbf{ind}_{W_{I}}^{W}(\chi)=\mathbf{ind}_{W_{I}}^{W}(\boldsymbol{\epsilon}\otimes\chi)$

(where the latter $\boldsymbol{\epsilon}$ denote the sign character of
$W_{I}$, which is also the restriction of the sign character of $W$, and
$\chi$ is any character of $W_{I}$), which easily follows the general general
formula for induction:

$\mathbf{ind}_{W_{I}}^{W}(\boldsymbol{\epsilon}\otimes\chi)(w)=\frac{1}{|W_{I}|}\sum_{x\in
W,\;x^{-1}wx\in
W_{I}}(\boldsymbol{\epsilon}\otimes\chi)(x^{-1}wx)=\frac{\boldsymbol{\epsilon}(w)}{|W_{I}|}\sum_{x\in
W,\;x^{-1}wx\in W_{I}}\chi(x^{-1}wx).$

∎

We have thus proved that $\mathbf{park}^{\operatorname{abs}}_{W,m}$ and
$\mathbf{park^{\prime}}^{\operatorname{abs}}_{W,m}$ satisfy the same
identities as $\mathbf{park}_{W,m}$ and
$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$ in Proposition 4.1.

###### Proof of Theorem 3.9.

We have $\mathbf{park}_{W_{I},m}=\mathbf{park}^{\operatorname{abs}}_{W_{I},m}$
for all $I\subset S$ by Theorem 3.8. From the second identity in Proposition
7.2 and the previous theorem, we immediately get
$\mathbf{park^{\prime}}^{\operatorname{abs}}_{W,m}=\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$.
∎

##  5 Proof of Theorem 3.9 via combinatorial matrices

The goal of this section is to prove Identity (20) below, as the proof of
Theorem 3.9 will immediately follow.

######  Proposition 5.1.

We have:

$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}=\operatorname{\boldsymbol{\Psi}}_{-1}\otimes\;\mathbf{park}_{W,-m}.$
(20)

###### Proof of Theorem 3.9.

If $W$ is irreducible, we have
$\mathbf{park}_{W,m}=\operatorname{\boldsymbol{\Psi}}_{mh+1}$ by Theorem 3.8.
The right-hand side of (20) is then
$\operatorname{\boldsymbol{\Psi}}_{-1}\otimes\operatorname{\boldsymbol{\Psi}}_{-mh+1}=\operatorname{\boldsymbol{\Psi}}_{mh-1}$.
We thus get
$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}=\operatorname{\boldsymbol{\Psi}}_{mh-1}$.
The reducible case follows immediately. ∎

The proof of Proposition 5.1 relies on combinatorial properties of the two
order relations $\sqsubset$ and $\ll$ on $\operatorname{NC}(W)$. We begin with
a few definitions and lemmas.

Below, we identify elements in $R(W)$ with column vectors indexed by $\Theta$.
We also use matrices with rows and columns indexed by $\Theta$. In both cases,
we use bold fonts as we do for elements of $R(W)$. Note that $\otimes$ still
denote the product of $R(W)$, but we use no sign for the matrix products.

######  Definition 5.2.

We define three matrices $\operatorname{\bf Q}$, $\operatorname{\bf R}$, and
$\operatorname{\bf N}$ by:

$\displaystyle\operatorname{\bf Q}_{X,Y}$
$\displaystyle=\\#\Big{\\{}v\in\operatorname{NC}(W,c)\;:\;\operatorname{Fix}(v)\sim
X\text{ and }v\leq w\Big{\\}},$ $\displaystyle\operatorname{\bf R}_{X,Y}$
$\displaystyle=\\#\Big{\\{}v\in\operatorname{NC}(W,c)\;:\;\operatorname{Fix}(v)\sim
X\text{ and }v\ll w\Big{\\}},$ $\displaystyle\operatorname{\bf N}_{X,Y}$
$\displaystyle=\\#\Big{\\{}v\in\operatorname{NC}(W,c)\;:\;\operatorname{Fix}(v)\sim
X\text{ and }v\sqsubset w\Big{\\}},$

where $w$ is a fixed element of $\operatorname{NC}(W)$ such that
$\operatorname{Fix}(w)\sim Y$. Also we define $\operatorname{\bf D}$ as the
diagonal matrix such that, for all $X\in\Theta$:

$\operatorname{\bf D}_{X,X}=(-1)^{n-\dim X}.$

Moreover $\operatorname{\bf I}$ denote the identity matrix.

It turns out that these matrices don’t depend on the chosen element $w$. Note
that the matrices $\operatorname{\bf Q}$, $\operatorname{\bf R}$, and
$\operatorname{\bf N}$ are unitriangular (upon ordering $\Theta$ so that
dimension is increasing).

######  Lemma 5.3.

For any $\operatorname{\bf X}\in R(W)$, we have:
$\boldsymbol{\epsilon}\otimes\operatorname{\bf X}=\operatorname{\bf
D}\operatorname{\bf N}\operatorname{\bf X}$.

###### Proof.

This is a rewriting of Lemma 3.4. ∎

######  Lemma 5.4.

We have $(\operatorname{\bf D}\operatorname{\bf N})^{2}=\operatorname{\bf I}$.

###### Proof.

This follows from the previous proposition, since
$\boldsymbol{\epsilon}\otimes\boldsymbol{\epsilon}={\bf 1}$. Alternatively,
one can see the relation $\operatorname{\bf N}^{-1}=\operatorname{\bf
D}\operatorname{\bf N}\operatorname{\bf D}$ as a reformulation of the
inclusion-exclusion principle on subsets of $S$. ∎

######  Lemma 5.5.

We have $\operatorname{\bf R}\operatorname{\bf N}=\operatorname{\bf Q}$.

###### Proof.

A combinatorial expansion of the product $\operatorname{\bf
R}\operatorname{\bf N}$ shows that the $X,Y$-coefficient is the number of
$u,v$ such that $u\ll v\sqsubset w$ and $\operatorname{Fix}(u)\sim X$, where
$w$ is a fixed element such that $\operatorname{Fix}(W)\sim Y$. So this is an
immediate consequence of Proposition 2.8. ∎

######  Lemma 5.6.

We have $\operatorname{\bf Q}^{-1}=(\operatorname{\bf D}\operatorname{\bf
N})\operatorname{\bf Q}(\operatorname{\bf D}\operatorname{\bf N})$.

###### Proof.

A combinatorial expansion of the product $(\operatorname{\bf
R}\operatorname{\bf D})^{2}$ shows that the coefficient of $X,Y$ is:

$\sum_{\begin{subarray}{c}u,v\in\operatorname{NC}(W)\\\ u\ll v\ll
w,\;\operatorname{Fix}(u)\sim X\end{subarray}}(-1)^{\ell(v)+\ell(w)}$

where $w\in\operatorname{NC}(W)$ is such that $\operatorname{Fix}(w)\sim Y$.
By Lemma 2.9, we get an alternating sum over a Boolean lattice. So this is $0$
if $X\neq Y$, and $1$ if $X=Y$ (the two signs cancel each other out). We thus
have $(\operatorname{\bf R}\operatorname{\bf D})^{2}=\operatorname{\bf I}$.

Since $\operatorname{\bf R}=\operatorname{\bf Q}\operatorname{\bf N}^{-1}$ by
Lemma 5.5, we get $(\operatorname{\bf Q}\operatorname{\bf
N}^{-1}\operatorname{\bf D})^{2}=\operatorname{\bf I}$, so that:

$\operatorname{\bf Q}^{-1}=\operatorname{\bf N}^{-1}\operatorname{\bf
D}\operatorname{\bf Q}\operatorname{\bf N}^{-1}\operatorname{\bf D}.$

We have $\operatorname{\bf N}^{-1}\operatorname{\bf D}=\operatorname{\bf
D}\operatorname{\bf N}$ from Lemma 5.4, and the result follows. ∎

######  Definition 5.7.

Let $\operatorname{\bf U}$ denote the column vector that correspond to the
trivial character ${\bf 1}\in R(W)$ under our convention. Explicitly, it is
given by:

$\operatorname{\bf U}_{X}=\begin{cases}1&\text{if $X$ is the maximal element
of $L(W)$},\\\ 0&\text{otherwise}.\end{cases}$

Indeed, by (2) we have $W_{X}=W$ if $X$ is the $0$-dimensional subspace. It
follows that
$\operatorname{\boldsymbol{\Phi}}_{X}=\mathbf{ind}_{W_{X}}^{W}({\bf 1})={\bf
1}$.

######  Proposition 5.8.

We have:

$\displaystyle\mathbf{park}_{W,m}$ $\displaystyle=\operatorname{\bf
Q}^{m}\operatorname{\bf U},$ (21)
$\displaystyle\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$
$\displaystyle=\operatorname{\bf Q}^{m-1}\operatorname{\bf R}\operatorname{\bf
U}.$ (22)

###### Proof.

By a combinatorial expansion of the product $\operatorname{\bf
Q}^{m}\operatorname{\bf U}$, we find that the coefficient of
$\operatorname{\boldsymbol{\Phi}}_{X}$ is the number of chains $w_{1}\leq
w_{2}\leq\dots\leq w_{m}\leq c$ in $\operatorname{NC}(W)$ with
$\operatorname{Fix}(w_{1})\sim X$. Doing the same in the product
$\operatorname{\bf Q}^{m-1}\operatorname{\bf R}\operatorname{\bf U}$, we find
that the coefficient of $\operatorname{\boldsymbol{\Phi}}_{X}$ is the number
of chains $w_{1}\leq w_{2}\leq\dots\leq w_{m}\ll c$ in $\operatorname{NC}(W)$
with $\operatorname{Fix}(w_{1})\sim X$. We thus recover the combinatorial
definitions of $\mathbf{park}_{W,m}$ and
$\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$ in (9) and (13). ∎

###### Proof of Proposition 5.1.

By (21) and Lemma 5.3, we have:

$\boldsymbol{\epsilon}\otimes\mathbf{park}_{-m}^{W}=\operatorname{\bf
D}\operatorname{\bf N}\operatorname{\bf Q}^{-m}\operatorname{\bf U}.$

We have $\operatorname{\bf Q}^{-1}=(\operatorname{\bf D}\operatorname{\bf
N})\operatorname{\bf Q}(\operatorname{\bf D}\operatorname{\bf N})^{-1}$ by
Lemma 5.6, so we get:

$\boldsymbol{\epsilon}\otimes\mathbf{park}_{-m}^{W}=(\operatorname{\bf
D}\operatorname{\bf N})^{2}\operatorname{\bf Q}^{m}(\operatorname{\bf
D}\operatorname{\bf N})^{-1}\operatorname{\bf U}.$

We have $(\operatorname{\bf D}\operatorname{\bf N})^{2}=\operatorname{\bf I}$
by Lemma 5.4, so we get:

$\boldsymbol{\epsilon}\otimes\mathbf{park}_{-m}^{W}=\operatorname{\bf
Q}^{m}\operatorname{\bf N}^{-1}\operatorname{\bf D}\operatorname{\bf U}.$

We have $\operatorname{\bf Q}\operatorname{\bf N}^{-1}=\operatorname{\bf R}$
by Lemma 5.5, and $\operatorname{\bf D}\operatorname{\bf
U}=(-1)^{n}\operatorname{\bf U}$ is easily checked. We thus get
$(-1)^{n}\operatorname{\bf Q}^{m-1}\operatorname{\bf R}\operatorname{\bf U}$
on the right-hand side of the previous equation. By the previous proposition,
this is $(-1)^{n}\mathbf{park\mathrlap{{}^{\prime}}}_{W,m}$. This completes
the proof since
$(-1)^{n}\boldsymbol{\epsilon}=\operatorname{\boldsymbol{\Psi}}_{-1}$. ∎

##  6 The generalized cluster complex

We review Fomin and Reading’s generalized cluster complex, introduced in [16],
and Tzanaki’s characterization from [36]. Afterwards, we introduce in
Definition 6.13 the natural way to associate a parabolic conjugacy class to
each face of this complex. This will be the basis of the refined enumeration
of faces.

Through this section, we assume that $c$ is a _bipartite Coxeter element_ : we
can write $S=S_{+}\uplus S_{-}$ where $S_{\pm}$ contains pairwise commuting
reflections, and $c=c_{+}c_{-}$ where $c_{\pm}$ is the product of elements in
$S_{\pm}$.

Let $\Phi\subset V$ be a _root system_ for $W$ (in the sense of Coxeter
groups), $\Phi_{+}\subset\Phi$ a set of positive roots, and
$\Delta\subset\Phi_{+}$ the corresponding set of simple roots. We denote
$R:\Phi\to T$ the natural map that sends $\alpha$ to the unique $t\in T$ such
that $t(\alpha)=-\alpha$. Eventually, let $\Delta_{+}$ and $\Delta_{-}$ the
set of simple roots such that $R(\Delta_{+})=S_{+}$ and $R(\Delta^{-})=S_{-}$.

###  6.1 Colored almost-positive roots

Let $[m]=\\{1,\dots,m\\}$. We think of it as a set of “colors” that we use to
define colored roots.

######  Definition 6.1 (Fomin and Reading [16]).

An $m$-colored root is a pair $(\alpha,i)$, denoted $\alpha^{i}$ for short,
where $\alpha\in\Phi$ and $i\in[m]$. It is positive if $\alpha\in\Phi_{+}$. It
is almost-positive if either $\alpha\in\Phi_{+}$, or $\alpha\in-\Delta$ and
$i=1$. We denote $\Phi^{(m)}_{\geq-1}$ the set of almost-positive $m$-colored
roots, and $\Phi^{(m)}_{+}$ the set of positive $m$-colored roots.

When $m=1$, we can ignore colors and $\Phi^{1}_{\geq-1}$ is thus identified
with $\Phi_{\geq-1}:=\Phi_{+}\cup(-\Delta)$. It is the set of almost-positive
roots introduced by Fomin and Zelevinsky to define the cluster complex [17].
In general, we also identify $\rho\in-\Delta$ with the colored root $\rho^{1}$
(as no confusion can arise).

Fomin and Zelevinsky [17] defined a rotation $\mathcal{R}$ on $\Phi_{\geq-1}$
by $\mathcal{R}=\mathcal{R}_{+}\circ\mathcal{R}_{-}$, where:

$\mathcal{R}_{+}(\rho)=\begin{cases}\rho&\text{ if }\rho\in-\Delta_{-},\\\
c_{+}(\rho)&\text{otherwise},\end{cases}\qquad\mathcal{R}_{-}(\rho)=\begin{cases}\rho&\text{
if }\rho\in-\Delta_{+},\\\ c_{-}(\rho)&\text{otherwise}.\end{cases}$

Explicitly, we can check that:

$\mathcal{R}(\rho)=\begin{cases}-\rho&\text{if
}\rho\in(-\Delta_{+})\cup\Delta_{-},\\\ c(\rho)&\text{otherwise}.\end{cases}$

Following Fomin and Reading [16], the higher order rotation $\mathcal{R}_{m}$
acting on $\Phi_{\geq-1}^{(m)}$ is defined as follows:

$\mathcal{R}_{m}(\alpha^{i})=\begin{cases}\alpha^{i+1}&\text{ if
}\alpha\in\Phi_{+}\text{ and }i<m,\\\ \mathcal{R}(\alpha)^{1}&\text{
otherwise.}\end{cases}$ (23)

It will be useful to make second case more explicit:

$\displaystyle\mathcal{R}_{m}(\alpha^{m})$
$\displaystyle=\begin{cases}(-\alpha)^{1}\text{ if }\alpha\in\Delta_{-},\\\
c(\alpha)^{1}\text{ if }\alpha\in\Phi_{+}\backslash\Delta_{-},\end{cases}$
(24) $\displaystyle\mathcal{R}_{m}(\alpha^{1})$
$\displaystyle=\begin{cases}(-\alpha)^{1}\text{ if }\alpha\in-\Delta_{+},\\\
c(\alpha)^{1}\text{ if }\alpha\in-\Delta_{-}.\end{cases}$ (25)

######  Proposition 6.2 ([16]).

Assume $W$ is irreducible, with Coxeter number $h$. Then each orbit $\omega$
for the action of $\mathcal{R}_{m}$ on $\Phi^{(m)}_{\geq-1}$ satisfy:

  * •

either $\\#\omega=\frac{mh+2}{2}$ and $\\#(-\Delta)\cap\omega=1$,

  * •

or $\\#\omega=mh+2$ and $\\#(-\Delta)\cap\omega=2$.

Moreover, in the latter case the two elements $-\delta_{i}$ and $-\delta_{j}$
of $(-\Delta)\cap\omega$ are related by $w_{\circ}(\delta_{i})=\delta_{j}$,
where $w_{\circ}$ is the longest element of $W$.

Note that the previous proposition says that negative simple roots have the
same proportion in each orbit. This is similar to a classical result by
Steinberg [33] for the action of the bipartite Coxeter element on $T$ by
conjugation. It states that each orbit is:

  * •

either an $h/2$-element set containing one simple reflection,

  * •

or an $h$-element set containing two simple reflections.

###  6.2 The compatibility relation

The generalized cluster complex can be defined as the flag simplicial complex
associated to a binary relation on $\Phi^{(m)}_{\geq-1}$, called
compatibility.

######  Definition 6.3 ([16]).

The compatibility relation $\mathrel{\|}$ is the unique symmetric and
irreflexive binary relation on $\Phi^{(m)}_{\geq-1}$, characterized by the
conditions:

  * •

for all $\alpha,\beta\in\Phi^{(m)}_{\geq-1}$, we have
$\alpha\mathrel{\|}\beta$ iff
$\mathcal{R}_{m}(\alpha)\mathrel{\|}\mathcal{R}_{m}(\beta)$,

  * •

if $\alpha\in-\Delta$ and $\beta\in\Phi^{(m)}_{\geq-1}$, we have
$\alpha\mathrel{\|}\beta$ iff $\alpha$ does not appear in the expansion of
$\beta$ (forgetting its color) as a linear combination of simple roots.

In particular, the elements of $-\Delta$ are mutually compatible by the second
condition. Note that these two conditions suffice to decide if
$\alpha\mathrel{\|}\beta$ holds, for any $\alpha,\beta\in\Phi^{(m)}_{\geq-1}$.
Indeed, we can proceed as follows:

  * •

By Proposition 6.2, there exists $i$ such that $\mathcal{R}_{m}^{i}(\alpha)$
is in $-\Delta$,

  * •

By the first condition above, $\alpha\mathrel{\|}\beta$ holds iff
$\mathcal{R}_{m}^{i}(\alpha)\mathrel{\|}\mathcal{R}_{m}^{i}(\beta)$ holds,

  * •

by the second condition above, we can decide if
$\mathcal{R}_{m}^{i}(\alpha)\mathrel{\|}\mathcal{R}_{m}^{i}(\beta)$ by
expanding $\mathcal{R}_{m}^{i}(\beta)$ in terms of simple roots.

In particular, it follows that there exists at most one such relation
$\mathrel{\|}$. Fomin and Reading [16] proved that such a relation does exist.

######  Definition 6.4 ([16]).

The generalized cluster complex $\Upsilon(W,m)$ is the flag simplicial complex
generated by the compatibility relation $\|$ of Definition 6.3: its vertex set
is $\Phi^{(m)}_{\geq-1}$, and its faces are sets of pairwise compatible
vertices. The positive part of $\Upsilon(W,m)$, denoted $\Upsilon^{+}(W,m)$,
is its full subcomplex having $\Phi^{(m)}_{+}$ as vertex set. In the case
$m=1$, we write $\Upsilon(W):=\Upsilon(W,1)$ and
$\Upsilon^{+}(W):=\Upsilon^{+}(W,1)$.

Before stating more properties of $\Upsilon(W,m)$ and $\Upsilon^{+}(W,m)$, let
us give some reformulations of the definition.

###  6.3 Reflection ordering

An alternative characterization of the complex $\Upsilon(W,m)$ can be given
via a reflection ordering. It is due to Brady and Watt [10] in the case $m=1$,
and Tzanaki [36] in the general case.

######  Proposition 6.5 ([33]).

There exists an indexing $\Phi_{\geq-1}=(\alpha_{i})_{1\leq i\leq nh/2+n}$
with the following properties:

  * •

$c(\alpha_{i})=\alpha_{i+n}$ for $1\leq i\leq nh/2$.

  * •

$-\Delta_{-}=\\{\alpha_{1},\dots,\alpha_{r}\\}$ and
$\Delta_{+}=\\{\alpha_{r+1},\dots,\alpha_{n}\\}$ (where $r$ is the cardinality
of $\Delta_{-}$),

  * •

$\Delta_{-}=\\{\alpha_{nh/2+1},\dots,\alpha_{nh/2+r}\\}$ and
$-\Delta_{+}=\\{\alpha_{nh/2+r+1},\dots,\alpha_{nh/2+n}\\}$.

Note that the first two conditions above uniquely define the sequence. It
remains to check the last condition, and the fact that each element in
$\Phi_{\geq-1}$ appear exactly once. In fact, Steinberg’s construction in [33]
gives an indexing of the whole root system $\Phi=(\alpha_{i})_{1\leq i\leq
nh}$ from which we can extract the above indexing of $\Phi_{\geq-1}$.

######  Remark 6.6.

The map $\mathcal{R}$ is essentially a rotation through this indexing:

$\mathcal{R}(\alpha_{i})=\begin{cases}\alpha_{i+n}&\text{ if }1\leq i\leq
nh/2,\\\ w_{\circ}(\alpha_{i-nh/2})&\text{ otherwise}\end{cases}$

where $w_{\circ}$ is the longest element in $W$.

The indexing of the previous proposition defines a total order $\prec$ on
$\Phi_{\geq-1}$ by the condition $\alpha_{i}\prec\alpha_{j}\Longleftrightarrow
i<j$. We refer to it as the reflection ordering.

######  Proposition 6.7 (Brady and Watt [10, Section 8]).

Let $(\rho_{i})_{1\leq i\leq n}$ be a tuple of $n$ distinct elements of
$\Phi_{\geq-1}$, ordered so that
$\rho_{1}\succ\rho_{2}\succ\dots\succ\rho_{n}$. Then it is a facet of
$\Upsilon(W)$ iff $c=R(\rho_{1})R(\rho_{2})\cdots R(\rho_{n})$.

The extension to general $m$ is as follows. Define an indexing
$\Phi^{(m)}_{\geq-1}=\\{\beta_{i}\\}_{1\leq i\leq mnh/2+n}$ (and accordingly,
a total order $\prec$ on $\Phi^{(m)}_{\geq-1}$ where we omit the dependence in
$m$ in the notation) as follows. As a sequence, it is obtained by the
concatenation of the three following sequences:

  * •

$\alpha^{1}_{1},\dots,\alpha^{1}_{r}$ (elements of $-\Delta_{-}$ with color
$1$),

  * •

$\alpha^{m}_{r+1},\dots,\alpha^{m}_{nh/2+r},\alpha^{m-1}_{r+1},\dots,\alpha^{m-1}_{nh/2+r},\dots,\alpha^{1}_{r+1},\dots,\alpha^{1}_{nh/2+r}$
(colored positive roots),

  * •

$\alpha^{1}_{nh/2+r+1},\dots,\alpha^{1}_{nh/2+n}$ (elements of $-\Delta_{+}$
with color $1$).

Note that the rotation $\mathcal{R}_{m}$ has a priori no simple description
using this order, unlike in the case $m=1$.

######  Proposition 6.8 (Tzanaki [36]).

Let $(\rho_{i})_{1\leq i\leq n}$ be a tuple of $n$ distinct elements of
$\Phi^{(m)}_{\geq-1}$, ordered so that
$\rho_{1}\succ\rho_{2}\succ\dots\succ\rho_{n}$. Then it is a facet of
$\Upsilon(W,m)$ iff $c=R(\rho_{1})\cdots R(\rho_{n})$.

In what follows, we generally assume that each face $f\in\Upsilon(W,m)$ is
indexed in decreasing order, and write
$f=\\{\rho_{1}\succ\rho_{2}\succ\cdots\succ\rho_{k}\\}$.

This completes the definition and characterization of the generalized cluster
complex. Note that our exposition is not exhaustive: another construction of
$\Upsilon(W,m)$ is via subword complexes, see Stump, Thomas and Williams [34].

###  6.4 Other properties

Fomin and Reading proved various properties about their generalized cluster
complex, some of them will be useful in the present work. First,
$\Upsilon(W,m)$ and $\Upsilon^{+}(W,m)$ are purely $n-1$-dimensional. Their
number of facets are respectively the Fuß-Catalan number
$\operatorname{Cat}^{(m)}(W)$ and the positive Fuß-Catalan number
$\operatorname{Cat}_{+}^{(m)}(W)$. Another property that naturally follows
from the definition is:

######  Proposition 6.9.

Let $\rho\in\Delta$, $s=R(\rho)$, and denote the irreducible factor of
$W_{(s)}$ as $W_{1}$, $W_{2}$, etc. Then the link of the vertex $-\rho$ in
$\Upsilon(W,m)$ is the join
$\Upsilon(W_{1},m)\star\Upsilon(W_{2},m)\star\cdots$.

By the above result, it is natural to extend the definition of $\Upsilon(W,m)$
in the reducible case, by declaring that it is the join of the simplicial
complexes $\Upsilon(W^{\prime},m)$ where $W^{\prime}$ runs through the
irreducible factors of $W$.

The results of Brady and Watt and their extension by Tzanaki make clear that
the cluster complex is related with minimal factorizations of the Coxeter
element. In particular, to each face we can associate a noncrossing partition
in a natural way:

######  Definition 6.10.

Let $f=\\{\rho_{1}\succ\dots\succ\rho_{k}\\}\in\Upsilon(W,m)$. We define

$\prod f:=R(\rho_{1})\cdots R(\rho_{k})\in\operatorname{NC}(W).$

To see that this is well-defined, first note that from Proposition 6.8 we have
$R(\rho_{1})\cdots R(\rho_{k})\allowbreak=c$ if $k=n$ (i.e., $f$ is a facet).
In general, since a face is a subset of a facet we get that $R(\rho_{1})\cdots
R(\rho_{k})$ is a subword of a minimal reflection factorization of $c$. It
follows that $\prod f\in\operatorname{NC}(W)$.

######  Proposition 6.11.

For $w\in\operatorname{NC}(W)$, we have:

$\\#\Big{\\{}f\in\Upsilon^{+}(W,m)\;:\;\prod
f=w\Big{\\}}=\operatorname{Cat}^{(m)}_{+}(W_{\operatorname{Fix}(w)}).$

###### Proof.

Essentially, it is possible to identify those $f\in\Upsilon^{+}(W,m)$ such
that $\prod f=w$ with facets of $\Upsilon^{+}(W_{\operatorname{Fix}(w)},m)$.
However, some care is needed to do that. Although $w$ is a Coxeter element of
$W_{\operatorname{Fix}(w)}$ by Proposition 2.6, it might not be bipartite
Coxeter element. We thus need to use the generalized cluster complex
associated to any standard Coxeter element, following the definition of Stump,
Thomas, Williams [34].

An alternative path is to first get the result for $m=1$. Indeed, [9,
Proposition 6.3] give a characterization of the compatibility relation on
(uncolored) positive roots which is clearly stable by restriction to parabolic
subgroups, allowing the identification mentioned above. Now, we have:

$\\#\Big{\\{}f\in\Upsilon^{+}(W,m)\;:\;\prod f=w\Big{\\}}=\sum_{w=w_{1}\cdots
w_{m}}\prod_{i=1}^{m}\\#\Big{\\{}f\in\Upsilon^{+}(W)\;:\;\prod
f=w_{i}\Big{\\}},$ (26)

where we sum over length-additive factorizations with $m$ factors. Indeed,
this can be proved bijectively: to $f\in\Upsilon^{+}(W,m)$ we associate
$f_{1},\dots,f_{m}\in\Upsilon^{+}(W)$ where $f_{i}$ contains roots in $f$ of
color $i$. Using the case $m=1$ of the proposition, we get

$\\#\Big{\\{}f\in\Upsilon^{+}(W,m)\;:\;\prod f=w\Big{\\}}=\sum_{w=w_{1}\cdots
w_{m}}\prod_{i=1}^{m}\operatorname{Cat}_{+}(W_{\operatorname{Fix}(w_{i})}).$

The result follows from the identity:

$\operatorname{Cat}^{(m)}_{+}(W_{\operatorname{Fix}(w)})=\sum_{w=w_{1}\cdots
w_{m}}\prod_{i=1}^{m}\operatorname{Cat}_{+}(W_{\operatorname{Fix}(w_{i})}),$

where we sum over length-additive factorizations with $m$ factors. This
identity can be proved as follows: since it is invariant under conjugating
$w$, we can assume $w$ is a bipartite Coxeter element of
$W_{\operatorname{Fix}(w)}$, and the result comes from double counting of
facets in $\Upsilon(W_{\operatorname{Fix}(w)},m)$ as above. ∎

###  6.5 The map to orbit of flats

The Kreweras complement is an anti-automorphism of $\operatorname{NC}(W)$
defined by $K(w):=cw^{-1}$. We use here a variant adapted to the bipartite
Coxeter element:

######  Lemma 6.12.

The map $w\mapsto c_{+}wc_{-}$ is an involutive anti-automorphism of
$\operatorname{NC}(W)$.

See for example [9, Proposition 3.5].

######  Definition 6.13.

For $f\in\Upsilon(W,m)$, we define

$\underline{f}:=c_{+}\big{(}\prod f\big{)}c_{-},$

which is in $\operatorname{NC}(W)$ by the previous lemma. The natural map to
orbits of flats is:

$\displaystyle\Upsilon(W,m)$ $\displaystyle\to L(W)/W$ $\displaystyle f$
$\displaystyle\mapsto\text{ the orbit of }\operatorname{Fix}(\underline{f}).$

Accordingly, we define for any $X\in\Theta$:

$\displaystyle\Upsilon(W,X,m)$
$\displaystyle:=\big{\\{}f\in\Upsilon(W,m)\;:\;\operatorname{Fix}(\underline{f})\sim
X\big{\\}},$ $\displaystyle\Upsilon^{+}(W,X,m)$
$\displaystyle:=\big{\\{}f\in\Upsilon^{+}(W,m)\;:\;\operatorname{Fix}(\underline{f})\sim
X\big{\\}},$

and their cardinalities

$\displaystyle\gamma(W,X,m)$ $\displaystyle:=\\#\Upsilon(W,X,m),$
$\displaystyle\gamma^{+}(W,X,m)$ $\displaystyle:=\\#\Upsilon^{+}(W,X,m).$

We could actually take the usual Kreweras complement $K(w)$ instead of its
bipartite variant. Indeed, $cw^{-1}$ and $c_{+}wc_{-}$ are conjugated (it
easily follows from the fact that each element is conjugated to its inverse,
see [18, Corollary 3.2.14]). The bipartite Kreweras complement is somewhat
more adapted to the present situation.

In the next sections, we give the explicit formulas for the quantities
$\gamma$ and $\gamma^{+}$. Another important property is that the natural map
in the previous definition is invariant under the rotation $\mathcal{R}_{m}$
(see Proposition 11.1).

As a final remark for this section, observe that $\gamma(W,X,m)$ and
$\gamma^{+}(W,X,m)$ are polynomial in $m$. Indeed, the quantity in Proposition
6.11 is polynomial (from the definition of Fuß-Catalan numbers), and by
summing over a finite set of $w$ we get $\gamma(W,X,m)$ or
$\gamma^{+}(W,X,m)$.

##  7 Combinatorial reciprocities

In this section, we prove combinatorial reciprocities between the quantities
$\kappa$, $\kappa^{+}$ on one side and $\gamma^{+}$, $\gamma$ on the other
side. We deduce the formulas for $\gamma^{+}$, $\gamma$ in terms of
characteristic polynomials (or Orlik-Solomon exponents), knowing those for
$\kappa$ and $\kappa^{+}$ (see Corollary 7.3).

######  Theorem 7.1.

We have

$\displaystyle(-1)^{k}\kappa(W,X,-m)$ $\displaystyle=\gamma^{+}(W,X,m),$ (27)
$\displaystyle(-1)^{k}\kappa^{+}(W,X,-m)$ $\displaystyle=\gamma(W,X,m).$ (28)

Before proving this, we need to state inclusion-exclusion formulas that relate
$\kappa(W,X,m)$ to $\kappa^{+}(W,X,m)$ on one side, and $\gamma(W,X,m)$ to
$\gamma^{+}(W,X,m)$ on the other side. To do this, it is convenient to extend
the definitions of these quantities. Using the bijection from Lemma 2.4, we
write $\kappa(W,\mathcal{X},m)$ in place of $\kappa(W,X,m)$ and similarly for
$\kappa^{+}$, $\gamma$ and $\gamma^{+}$. Eventually, we extend this definition
in an additive way: if $\mathcal{Y}\subset W$ is a disjoint union of parabolic
conjugacy classes, say $\mathcal{Y}=\uplus_{i=1}^{j}\mathcal{X}_{i}$, we
write:

$\kappa(W,\mathcal{Y},m)=\sum_{i=1}^{j}\kappa(W,\mathcal{X}_{i},m)$

and similarly for $\kappa^{+}$, $\gamma$, and $\gamma^{+}$. In particular, for
each standard parabolic subgroup $W_{I}\subset W$ and $\mathcal{X}$ a
parabolic conjugacy class of $W$, the intersection $\mathcal{X}\cap W_{I}$ is
such a union.

Now, we can state:

######  Proposition 7.2.

We have:

$\displaystyle\kappa(W,\mathcal{X},m)$ $\displaystyle=\sum_{I\subset
S}\kappa^{+}(W_{I},\mathcal{X}\cap W_{I},m),$ (29)
$\displaystyle\kappa^{+}(W,\mathcal{X},m)$ $\displaystyle=\sum_{I\subset
S}(-1)^{n-\\#I}\kappa(W_{I},\mathcal{X}\cap W_{I},m),$ (30)

and

$\displaystyle\gamma(W,\mathcal{X},m)$ $\displaystyle=\sum_{I\subset
S}\gamma^{+}(W_{I},\mathcal{X}\cap W_{I},m),$ (31)
$\displaystyle\gamma^{+}(W,\mathcal{X},m)$ $\displaystyle=\sum_{I\subset
S}(-1)^{n-\\#I}\gamma(W_{I},\mathcal{X}\cap W_{I},m).$ (32)

###### Proof.

The number $\kappa^{+}(W_{I},\mathcal{X}\cap W_{I},m)$ counts chains
$w_{1}\leq\dots\leq w_{m}$ in $\operatorname{NC}(W_{I})$ where
$w_{1}\in\mathcal{X}\cap W_{I}$ and $w_{m}$ has full support in $W_{I}$. By
summing over $I$, we get the first equation. The second can be deduced,
essentially via the same proof as in the classical inclusion-exclusion
principle.

Let’s fix $I\subset S$. Then the map

$f\mapsto f^{+}:=f\cap\Phi^{(m)}_{+}$

is a bijection from faces $f\in\Upsilon(W,m)$ such that
$f\cap(-\Delta)=\\{-\delta_{i}\;:\;i\in S\backslash I\\}$ to
$\Upsilon^{+}(W_{I},m)$. To check the relation between $\underline{f}$ and
$\underline{f^{+}}$, note that with $f$ as above we have:

$\prod f=\big{(}\prod_{i\in S^{+}\backslash I}s_{i}\big{)}\big{(}\prod
f^{+}\big{)}\big{(}\prod_{i\in S^{-}\backslash I}s_{i}\big{)},$

so that

$\underline{f}=\big{(}\prod_{i\in S^{+}\cap I}s_{i}\big{)}\big{(}\prod
f^{+}\big{)}\big{(}\prod_{i\in S^{-}\cap I}s_{i}\big{)}=\underline{f^{+}}.$

The last equality comes from the fact that $\underline{f^{+}}$ is defined with
respect to the bipartite Coxeter element of $W_{I}$. It follows that
$\underline{f}\in\mathcal{X}$ is equivalent to
$\underline{f^{+}}\in\mathcal{X}\cap W_{I}$. By summing over $I\subset S$, we
get (31). The inverse relation follows as before. ∎

###### Proof of Theorem 7.1.

Recall that $\kappa(W,\mathcal{X},m)$ counts chains $w_{1}\leq\dots\leq w_{m}$
in $\operatorname{NC}(W)$ where $w_{1}\in\mathcal{X}$. If $w_{1}$ is fixed,
the other elements $w_{2},\dots,w_{m}$ can be mapped to
$w_{1}^{-1}w_{2},\dots,\allowbreak w_{1}^{-1}w_{m}$ and are thus in bijection
with $m-1$-element chains in $\operatorname{NC}(W)$ where the top element is
below $w_{1}^{-1}c$. The number of such tuples $(w_{2},\dots,w_{m})$ is thus
given by the Fuß-Catalan number
$\operatorname{Cat}^{(m-1)}(W_{\operatorname{Fix}(w_{1}^{-1}c)})$. We get:

$\kappa(W,\mathcal{X},m)=\sum_{w\in\mathcal{X}}\operatorname{Cat}^{(m-1)}(W_{\operatorname{Fix}(w^{-1}c)}).$
(33)

Note that we have a combinatorial reciprocity for Fuß-Catalan numbers:

$\displaystyle\operatorname{Cat}^{(-m)}(W)=(-1)^{n}\operatorname{Cat}^{(m-1)}_{+}(W).$
(34)

This relation follows from the two formulas in terms of the exponents given in
the introduction, and the fact that the (increasingly sorted) exponents
satisfy $e_{i}=h-e_{n+1-i}$. Using this reciprocity, Equation (33) gives:

$(-1)^{k}\kappa(W,\mathcal{X},-m)=\sum_{w\in\mathcal{X}}(-1)^{k}\operatorname{Cat}^{(-m-1)}(W_{\operatorname{Fix}(w^{-1}c)})=\sum_{w\in\mathcal{X}}\operatorname{Cat}_{+}^{(m)}(W_{\operatorname{Fix}(w^{-1}c)}).$

Since $w^{-1}c$ is conjugated to $c_{+}wc_{-}$, we also have:

$(-1)^{k}\kappa(W,\mathcal{X},-m)=\sum_{w\in\mathcal{X}}\operatorname{Cat}_{+}^{(m)}(W_{\operatorname{Fix}(c_{+}wc_{-})}).$

By Proposition 6.11, each term
$\operatorname{Cat}_{+}^{(m)}(W_{\operatorname{Fix}(c_{+}wc_{-})})$ is the
number of positive faces in $f\in\Upsilon^{+}(W,m)$ such that $\prod
f=c_{+}wc_{-}$. This is also the number of positive faces
$f\in\Upsilon^{+}(W,m)$ such that $\underline{f}=w$. So the sum is
$\gamma^{+}(W,\mathcal{X},m)$ by definition, and we have proved the first
equation.

Then, by substitution $m\leftarrow-m$ in (30) we get:

$\kappa^{+}(W,\mathcal{X},-m)=\sum_{I\subset
S}(-1)^{n-\\#I}\kappa(W_{I},\mathcal{X}\cap W_{I},-m)$

Using (27) that we have just proved, we get:

$\kappa(W_{I},\mathcal{X}\cap
W_{I},-m)=(-1)^{\\#I-(n-k)}\gamma^{+}(W_{I},\mathcal{X}\cap W_{I},m).$

To check the sign, note that in Theorem 7.1 the integer $k=\dim(X)$ is
$n-\ell(w)$ for some $w\in\mathcal{X}$. Here, the elements of $\mathcal{X}\cap
W_{I}$ have reflection length $n-k$ in $W_{I}$ (it is easily seen to be the
same as their reflection length in $W$). In $W_{I}$, which has rank $\\#I$,
this sign is thus given by the difference $\\#I-(n-k)$. The previous two
equations thus give:

$\kappa^{+}(W,\mathcal{X},-m)=(-1)^{k}\sum_{I\subset
S}\gamma^{+}(W_{I},\mathcal{X}\cap W_{I},m).$

From (31), the right-hand side of the previous equation is
$(-1)^{k}\gamma(W,\mathcal{X},m)$. We have thus proved the second equation. ∎

######  Corollary 7.3.

We have

$\displaystyle\gamma(W,X,m)=(-1)^{\dim(X)}\frac{\chi_{X}(-mh-1)}{[N(W_{X}):W_{X}]},$
(35)
$\displaystyle\gamma^{+}(W,X,m)=(-1)^{\dim(X)}\frac{\chi_{X}(-mh+1)}{[N(W_{X}):W_{X}]}.$
(36)

###### Proof.

Using the combinatorial reciprocities in Theorem 7.1, this follows from the
corresponding formulas for $\kappa$ and $\kappa^{+}$ in Equations (11) and
(15). ∎

##  8 Bijections between faces of the generalized cluster complex and chains
of noncrossing partitions

We give another proof of the formulas in Corollary 7.3, via a bijection which
is of independent interest. The combinatorial reciprocities in the previous
section gave connections $\kappa\leftrightarrow\gamma^{+}$ and
$\kappa^{+}\leftrightarrow\gamma$ (schematically), but here the bijection will
give connections $\kappa\leftrightarrow\gamma$ and
$\kappa^{+}\leftrightarrow\gamma^{+}$.

######  Proposition 8.1.

We have:

$\displaystyle\gamma(W,X,m)$ $\displaystyle=\sum_{Y\in\Theta}\operatorname{\bf
N}_{X,Y}\kappa(W,Y,m),$ (37) $\displaystyle\gamma^{+}(W,X,m)$
$\displaystyle=\sum_{Y\in\Theta}\operatorname{\bf N}_{X,Y}\kappa^{+}(W,Y,m).$
(38)

Note that these two identities are related to each other: one is the
consequence of the other, using the combinatorial reciprocity in (27) and (28)
and the inverse of the matrix $\operatorname{\bf N}$. Here we prove both
identities: the bijection proving the first one, suitably restricted, also
proves the second one.

One of our main tool is the following:

######  Lemma 8.2.

For any $u,w\in\operatorname{NC}(W)$ such that $u\leq w$, the elements
$v\in\operatorname{NC}(W)$ such that $u\sqsubset v\ll w$ are in bijection with
faces $f\in\Upsilon^{+}(W)$ such that $\prod f=u^{-1}w$.

###### Proof.

This follows from the results in [9] (though in this reference we only deal
with the case where $u$ is the minimal element, this slight generalization is
proved similarly). More explicitly, the construction is as follows.

Start from $f=\\{\alpha_{1},\dots,\alpha_{k}\\}$ as above, and let
$t_{i}=R(\alpha_{i})$. By Lemmas 8.7 and 8.8 from [9], we can reindex the
elements of $f$ (switching pairs of orthogonal reflections) so that
$u\sqsubset ut_{1}\sqsubset\hbox{}\dots\hbox{}\sqsubset ut_{1}\cdots t_{j}\ll
ut_{1}\cdots t_{j+1}\ll\hbox{}\dots\hbox{}\ll ut_{1}\cdots t_{k}=w$. Then the
bijection sends $f$ to $ut_{1}\cdots t_{j}$.

In the other direction, write $u^{-1}v=t_{1}\cdots t_{k}$ where the factors
are the simple reflections of $W_{\operatorname{Fix}(u^{-1}v)}$, and similarly
$v^{-1}w=u_{1}\cdots u_{j}$ where the factors are the simple reflections of
$W_{\operatorname{Fix}(v^{-1}w)}$. Then $R(f)$ contains the reflections

  * •

$t_{k}\cdots t_{i+1}t_{i}t_{i+1}\cdots t_{k}$ for $1\leq i\leq k$,

  * •

$u_{1}\cdots u_{i-1}u_{i}u_{i-1}\cdots u_{1}$ for $1\leq i\leq j$.

See [9, Section 8] for details. ∎

###### Proof of Proposition 8.1.

We give a bijection between faces $f\in\Upsilon(W,X,m)$ and chains of
noncrossing partitions of the form $w_{0}\sqsubset w_{1}\leq\dots\leq w_{m}$
such that $\operatorname{Fix}(w_{0})\sim X$.

Let us start from a chain in $\operatorname{NC}(W)$ as above. Using
Proposition 2.8, a chain $w_{0}\sqsubset w_{1}\leq\dots\leq w_{m}\leq c$ can
be completed in a unique way as a chain:

$w_{0}\sqsubset w_{1}\ll w^{\prime}_{1}\sqsubset w_{2}\ll
w^{\prime}_{2}\sqsubset\dots w_{m-1}\ll w^{\prime}_{m-1}\sqsubset w_{m}\ll
w^{\prime}_{m}\sqsubset c.$

Let us denote $w^{\prime}_{0}:=w_{0}$ for convenience. Each element $w_{i}$
(with $1\leq i\leq m$) is such that $w^{\prime}_{i-1}\sqsubset w_{i}\ll
w^{\prime}_{i}$. Via Lemma 8.2, it corresponds bijectively to a face
$f_{i}\in\Upsilon^{+}(W)$ such that

$\prod f_{i}=(w^{\prime}_{i-1})^{-1}w^{\prime}_{i}$ (39)

Define $f\in\Upsilon(W,m)$ as follows:

  * •

$f$ contains the colored positive roots $\alpha^{i}$ for $\alpha\in f_{i}$,

  * •

$f$ contains the negative roots $-\delta_{i}$ for $i\in
S\backslash\operatorname{supp}(w^{\prime}_{m})$.

To check that this is indeed in $\Upsilon(W,m)$, first note that from (39) we
get

$\prod f^{+}=(w^{\prime}_{0})^{-1}w^{\prime}_{m}$ (40)

with the notation $f^{+}=f\cap\Phi^{(m)}_{+}$ as before. Indeed, the order is
such that (reading this product from left to right) we first read roots with
color $1$, then roots with color $2$, etc. By Proposition 6.8, this shows that
$f^{+}\in\Upsilon^{+}(W,m)$. Then, each element $-\delta_{i}\in f$ is such
that $s_{i}\notin\operatorname{supp}(w_{m})$. Consequently,
$s_{i}\notin\operatorname{supp}(t)$ for each $t\in T$ such that $t\leq w_{m}$.
Thus this $-\delta_{i}$ is compatible with the roots in $f^{+}$, and it
follows $f\in\Upsilon(W,m)$.

To compute $\underline{f}$, let’s keep the notation as in the proof of
Proposition 7.2 so that $f\cap(-\Delta)=\\{-\delta_{i}\;:\;i\in S\backslash
I\\}$ for some $I\subset S$. From (40), we get

$\prod f=\big{(}\prod_{i\in S^{+}\backslash
I}\big{)}(w^{\prime}_{0})^{-1}w^{\prime}_{m}\big{(}\prod_{i\in S^{-}\backslash
I}\big{)}$

so that

$\underline{f}=\big{(}\prod_{i\in S^{+}\cap
I}\big{)}(w^{\prime}_{0})^{-1}w^{\prime}_{m}\big{(}\prod_{i\in S^{-}\cap
I}\big{)}.$

By definition, we have

$w^{\prime}_{m}=\big{(}\prod_{i\in S^{+}\cap I}\big{)}\big{(}\prod_{i\in
S^{-}\cap I}\big{)}.$

Indeed, the condition $w^{\prime}_{m}\sqsubset c$ implies that a reduced
factorization of $w^{\prime}_{m}$ can be extracted from the canonical one of
$c=c_{+}c_{-}$, and the definition of $f$ implies that the elements that
appear are those indexed by $I$. From the previous two equations, we get:

$\underline{f}=\big{(}\prod_{i\in S^{+}\cap
I}\big{)}(w^{\prime}_{0})^{-1}\big{(}\prod_{i\in S^{+}\cap I}\big{)}.$

We thus have $f\in\Upsilon(W,X,m)$, since
$\operatorname{Fix}(\underline{f})\sim\operatorname{Fix}((w^{\prime}_{0})^{-1})\sim\operatorname{Fix}(w^{\prime}_{0})\sim
X$.

Describing the inverse bijection is straightforward. Starting from
$f\in\Upsilon(W,m)$, define:

  * •

$w^{\prime}_{m}\sqsubset c$ is the product of the reflections $R(\alpha)$ for
$\alpha\in(-\Delta)\backslash f$,

  * •

$(w^{\prime}_{i-1})^{-1}w^{\prime}_{i}$ is the product of $R(\alpha^{i})$ over
positive roots $\alpha^{i}$ (of color $i$) in $f$.

From $w^{\prime}_{i-1}$, $w^{\prime}_{i}$, and the positive roots of color $i$
in $f$, we use the inverse bijection from Lemma 8.2 to get $w_{i}$ such that
$w^{\prime}_{i-1}\sqsubset w_{i}\ll w^{\prime}_{i}$. We omit details about
checking that the two maps are indeed inverse bijections.

We thus have proved (37). Eventually, observe that when we restrict this
bijection to chains such that $w_{m}\ll c$, we get $w^{\prime}_{m}=c$ (keeping
the same notation), and the corresponding faces
$f\in\Upsilon(W,\mathcal{X},m)$ contains no vertex in $-\Delta$. So the
bijection proves (38) as well. ∎

In the case where $X$ is the minimal flat, our bijection specializes into a
bijection between facets of $\Upsilon(W,m)$ and chains $w_{1}\leq\dots\leq
w_{m}$. Such a bijection was first given via representation theory by Buan,
Reiten, and Thomas [11]. A more combinatorial one and various related
bijections were given by Stump, Thomas, and Williams [34].

To end this section, we use the previous bijection to get:

###### Proof of Corollary 7.3.

From Lemma 5.3 and
$\operatorname{\boldsymbol{\Psi}}_{-1}=(-1)^{n}\boldsymbol{\epsilon}$, we
have:

$(-1)^{n}\operatorname{\bf
D}\operatorname{\boldsymbol{\Psi}}_{-t}=\operatorname{\bf
N}\operatorname{\boldsymbol{\Psi}}_{t},\qquad$

By taking the coefficients via Proposition 3.6 and evaluating at $t=mh+1$ and
$t=mh-1$, this rescpectively gives:

$\displaystyle(-1)^{\dim(X)}\frac{\chi_{X}(-mh-1)}{[N(W_{X}):W_{X}]}$
$\displaystyle=\sum_{Y\in\Theta}\operatorname{\bf
N}_{X,Y}\frac{\chi_{Y}(mh+1)}{[N(W_{Y}):W_{Y}]},$ (41)
$\displaystyle(-1)^{\dim(X)}\frac{\chi_{X}(-mh+1)}{[N(W_{X}):W_{X}]}$
$\displaystyle=\sum_{Y\in\Theta}\operatorname{\bf
N}_{X,Y}\frac{\chi_{Y}(mh-1)}{[N(W_{Y}):W_{Y}]}.$ (42)

Using the formulas for $\kappa$ and $\kappa^{+}$ obtained in (11) and (15), we
see that the right-hand sides of the previous two equations give,
respectively, the right-hand sides in (37) and (38).

By identifying the left-hand sides of the previous two equations with those of
(37) and (38), we immeditely obtain the formulas for $\gamma$ and $\gamma^{+}$
in Corollary 7.3. ∎

##  9 Cluster parking spaces

Via Proposition 3.6, the formulas we obtained in Corollary 7.3 imply:

$\displaystyle\sum_{X\in\Theta}(-1)^{\dim(X)}\gamma(W,X,m)\operatorname{\boldsymbol{\Phi}}_{X}$
$\displaystyle=\operatorname{\boldsymbol{\Psi}}_{-mh-1}=(-1)^{n}\boldsymbol{\epsilon}\otimes\operatorname{\boldsymbol{\Psi}}_{mh+1},$
(43)
$\displaystyle\sum_{X\in\Theta}(-1)^{\dim(X)}\gamma^{+}(W,X,m)\operatorname{\boldsymbol{\Phi}}_{X}$
$\displaystyle=\operatorname{\boldsymbol{\Psi}}_{-mh+1}=(-1)^{n}\boldsymbol{\epsilon}\otimes\operatorname{\boldsymbol{\Psi}}_{mh-1}.$
(44)

This suggests a representation theoretic interpretation, which is the goal of
this section. The following definition extends one given in [13] (in the case
of type A, with the “linear” Coxeter element), in the context of a poset of
parking functions.

######  Definition 9.1.

The set of cluster parking functions is:

$\operatorname{CPF}(W,m):=\biguplus_{f\in\Upsilon(W,m)}\\{f\\}\times
W/W_{\operatorname{Fix}(\underline{f})}.$

An element is thus a pair $(f,wW_{\operatorname{Fix}(\underline{f})})$ where
$f\in\Upsilon(W,m)$ and $w\in W$. Denote
$\operatorname{CPF}^{+}(W,m)\allowbreak\subset\operatorname{CPF}(W,m)$ the
subset of such pairs where $f\in\Upsilon^{+}(W,m)$. Moreover:

  * •

$W$ acts on $\operatorname{CPF}(W,m)$ and $\operatorname{CPF}^{+}(W,m)$ via
the natural left action on the quotient
$W/W_{\operatorname{Fix}(\underline{f})}$ (as in the case of parking
functions, see Equation (8)),

  * •

a partial order on $\operatorname{CPF}(W,m)$ (respectively, on
$\operatorname{CPF}^{+}(W,m)$) is given by

$(f^{\prime},w^{\prime}W_{\operatorname{Fix}(\underline{f^{\prime}})})\leq(f,wW_{\operatorname{Fix}(\underline{f})})\quad\Longleftrightarrow\quad
f^{\prime}\subset f\text{ and
}w^{\prime}W_{\operatorname{Fix}(\underline{f^{\prime}})}=wW_{\operatorname{Fix}(\underline{f^{\prime}})}.$

To see that the order is well-defined, note that $f^{\prime}\subset f$ implies
$\underline{f}\leq\underline{f^{\prime}}$, then
$\operatorname{Fix}(\underline{f^{\prime}})\subset\operatorname{Fix}(\underline{f})$
and $W_{\operatorname{Fix}(\underline{f})}\subset
W_{\operatorname{Fix}(\underline{f^{\prime}})}$. So the coset
$wW_{\operatorname{Fix}(\underline{f})}$ unambiguously defines a coset
$wW_{\operatorname{Fix}(\underline{f^{\prime}})}$ via the natural quotient
map.

Also, note that $\operatorname{CPF}(W,m)$ and $\operatorname{CPF}^{+}(W,m)$
are ranked posets where the rank function is
$(f,wW_{\operatorname{Fix}(\underline{f})})\mapsto\\#f$.

There is a similarity with the set
$\operatorname{PF}(W):=\operatorname{PF}(W,1)$ of parking functions. The
precise formulation is the following: we can see $\operatorname{CPF}(W,m)$ as
the (setwise) fiber product of $\Upsilon(W,m)$ and $\operatorname{PF}(W)$ over
$\operatorname{NC}(W)$, along the maps:

$f\mapsto\underline{f}\qquad\text{ and
}\qquad(w_{1},w_{2}W_{\operatorname{Fix}(w_{1})})\mapsto w_{1}.$

And the same holds with $\Upsilon^{+}(W,m)$ and $\operatorname{PF}^{+}(W)$ in
place of $\Upsilon(W,m)$ and $\operatorname{PF}(W)$.

######  Lemma 9.2.

Let $E\subset\operatorname{NC}(W)$, and denote $\wedge E$ its meet in the
lattice $\operatorname{NC}(W)$. Then we have:

$W_{\operatorname{Fix}(\wedge E)}=\bigcap_{w\in E}W_{\operatorname{Fix}(w)}.$

###### Proof.

An explicit construction of $\wedge E$ is given by Brady and Watt [10], and it
follows that:

$\big{\\{}t\in T\;:\;t\leq\wedge E\big{\\}}=\bigcap_{w\in E}\big{\\{}t\in
T\;:\;t\leq w\big{\\}}.$

Moreover, each set $\big{\\{}t\in T\;:\;t\leq w\big{\\}}$ is the set of
reflections in $W_{\operatorname{Fix}(w)}$, and this permits to complete the
proof. ∎

Note that $\operatorname{NC}(W)$ is embedded (as a subposet, but not a
sublattice) in the lattice of parabolic subgroups of $W$, via the map
$w\mapsto W_{\operatorname{Fix}(w)}$. In the latter lattice, a meet is clearly
given by taking an intersection of subgroups. So the previous lemma says that
the meet of $E$ as a subset of $\operatorname{NC}(W)$ coincides with its meet
in the lattice of parabolic subgroups. (The analog statement about the join
does not hold.)

######  Proposition 9.3.

The poset $\operatorname{CPF}(W,m)$ is the face poset of a simplicial complex,
that we still denote $\operatorname{CPF}(W,m)$. The same holds for
$\operatorname{CPF}^{+}(W,m)$.

###### Proof.

Observe that in the definition of the order relation, the coset
$w^{\prime}W_{\operatorname{Fix}(\underline{f^{\prime}})}$ is uniquely defined
by $f^{\prime}$ and $w$. It follows that the elements below
$(f,wW_{\operatorname{Fix}(\underline{f})})$ can be identified with subsets of
$f$, and form a boolean lattice of rank $\\#f$.

Now, consider
$(f,wW_{\operatorname{Fix}(\underline{f})})\in\operatorname{CPF}(W,m)$, and
write $f=\\{\alpha_{1},\dots,\alpha_{k}\\}$. The rank $1$ elements below
$(f,wW_{\operatorname{Fix}(\underline{f})})$ are
$(\\{\alpha_{i}\\},wW_{\operatorname{Fix}(\underline{\\{\alpha_{i}\\}})})$ for
$1\leq i\leq k$. We easily recover $f$ as $\cup_{i=1}^{k}\\{\alpha_{i}\\}$.
Note that $\prod f$ is the join of the elements $R(\alpha_{i})$ for $1\leq
i\leq k$. Using Lemma 6.12, we see that $\underline{f}$ is the meet of the
elements $\underline{\\{\alpha_{i}\\}}$ for $1\leq i\leq k$. Using the
previous lemma, we get:

$wW_{\operatorname{Fix}(\underline{f})}=\bigcap_{i=1}^{k}wW_{\operatorname{Fix}(\underline{\\{\alpha_{i}\\}})}.$

What we have showed is that each element of $\operatorname{CPF}(W,m)$ can be
recovered from the rank 1 elements below it. This completes the proof: the
simplicial complex is realized by identifying each element with the set of
rank 1 element below it. The same arguments show the result for
$\operatorname{CPF}^{+}(W,m)$. ∎

Let $\Delta$ be a finite simplicial complex, endowed with an action of $W$
preserving dimension and inclusion of faces. The equivariant Euler
characteristic of $\Delta$ is:

$\chi(\Delta)=\sum_{i=-1}^{\dim(\Delta)}(-1)^{i}C_{i}(\Delta)$

where $C_{i}(\Delta)$, the $i$th chain space, is the free abelian group over
$i$-dimensional faces of $\Delta$. By the Hopf trace formula (see [37, Theorem
2.3.9]), $\chi(\Delta)$ is also equal to the alternating sum of the (reduced
and simplicial) homology groups of $\Delta$ over $\mathbb{Z}$. This definition
gives $\chi(\Delta)$ as an element of the representation ring of $W$, but we
can as well see it as an element of its character ring. Also note that
“equivariant Euler characteristic” is not really a standard terminology. The
Euler characteristic is the evaluation at $1$ of this character, and the
evaluation at other group elements are called Lefschetz numbers.

######  Theorem 9.4.

We have:

$\chi(\operatorname{CPF}(W,m))=(-1)^{n}\boldsymbol{\epsilon}\otimes\operatorname{\boldsymbol{\Psi}}_{mh+1},\quad\text{
and
}\quad\chi(\operatorname{CPF}^{+}(W,m))=(-1)^{n}\boldsymbol{\epsilon}\otimes\operatorname{\boldsymbol{\Psi}}_{mh-1}.$

###### Proof.

By definition of cluster parking functions, each $f\in\Upsilon(W,X,m)$
contributes to a term $\operatorname{\boldsymbol{\Phi}}_{X}$ in the $k$th
chain space $C_{k}(\operatorname{CPF}(W,m))$ (with $k=\dim(X)$). We can thus
identify $\chi(\operatorname{CPF}(W,m)$ and $\chi^{+}(\operatorname{CPF}(W,m)$
with the left-hand sides of (43) and (44), respectively. The right-hand sides
give the expected formulas. ∎

It would be very interesting to investigate whether these simplicial complexes
are shellable. If this holds, the equivariant Euler characteristic is (up to
the sign $(-1)^{n}$) the top homology group:

$\displaystyle\chi(\operatorname{CPF}(W,m))$
$\displaystyle=(-1)^{n}\tilde{H}_{n-2}(\operatorname{CPF}(W,m)),$ (45)
$\displaystyle\chi(\operatorname{CPF}^{+}(W,m))$
$\displaystyle=(-1)^{n}\tilde{H}_{n-2}(\operatorname{CPF}^{+}(W,m)).$ (46)

Assume that this property holds. Knowing the combinatorial interpretation of
the characters $\operatorname{\boldsymbol{\Psi}}_{mh+1}$ and
$\operatorname{\boldsymbol{\Psi}}_{mh-1}$ from Section 3, our results raise
the following problem: find an explicit basis
$(Z_{p})_{p\in\operatorname{PF}(W,m)}$ of
$\tilde{H}_{n-2}(\operatorname{CPF}(W,m))$ such that $w\cdot
Z_{p}=(-1)^{\ell(w)}Z_{w\cdot p}$, and also an explicit basis
$(Z^{\prime}_{p})_{p\in\operatorname{PF}^{\prime}(W,m)}$ of
$\tilde{H}_{n-2}(\operatorname{CPF}^{+}(W,m))$ with the same property.

######  Remark 9.5.

By a combinatorial version of the Hopf-Lefschetz theorem due to Baclawski and
Björner [7], the character $\chi(\operatorname{CPF}(W,m))$ evaluated at $w\in
W$ is the Euler characteristic of the fixed-point subcomplex
$\operatorname{CPF}(W,m)^{w}$. It could be interesting to make explicit what
are those fixed-point subcomplex.

##  10 A refinement of the $f$\- to $h$\- transformation

Through this section, let $(f_{i})_{-1\leq i\leq n-1}$ denote the $f$-vector
of $\Upsilon(W,m)$, i.e., $f_{i}$ is the number of $i$-dimensional faces in
the complex. Similarly, $(f^{+}_{i})_{-1\leq i\leq n-1}$ denote the $f$-vector
of $\Upsilon^{+}(W,m)$. In terms of $\gamma$ and $\gamma^{+}$, we thus have:

$\displaystyle f_{k-1}=\sum_{X\in\Theta,\;\dim(X)=k}\gamma(W,X,m)\quad\text{
and }\quad f^{+}_{k-1}=\sum_{X\in\Theta,\;\dim(X)=k}\gamma^{+}(W,X,m).$ (47)

First, let us explain how our formulas for $\gamma$ completely explain some
numerology concerning $f_{k}$ obtained in [16, Section 8]. By examining the
tables in [25, Appendix C], we note that the Orlik-Solomon exponents
$b_{i}^{X}$ of $X\in\Theta$ (with $\dim(X)=k)$ often look like
$e_{1},\dots,e_{k}$, namely the $k$ smallest exponents of $W$ (in fact, this
happens exactly for the so-called [38, §3.1.5] coincidental types
$A_{n}=S_{n+1},B_{n},I_{2}(m),H_{3}$, see [23, §3.3]). Now, consider the
formula for $f_{k}$ in (47) as a polynomial in $m$. If some integer $b$ is an
Orlik-Solomon exponent for all $X\in\Theta$ with $\dim(X)=k$, each term
contains a factor $(mh+1-b)$ so that the sum $f_{k}$ also has this factor.
Also, we can observe that there is an equivalence between the two statements:

  * •

The integer $b$ is an Orlik-Solomon exponent for all $X\in\Theta$ with
$\dim(X)=k$.

  * •

The integer $b$ is an Orlik-Solomon exponent for all $X\in\Theta$ with
$\dim(X)\geq k$.

(It is easily checked on a case-by-case basis.) Note that it might a priori
happen that $f_{k}$ contains a factor $mh+1-b$ even though some of the terms
in the sum of (47) don’t contain it. This actually never happen. The upshot of
this discussion is:

######  Proposition 10.1.

An exponent $e_{i}$ of $W$ has level $j$ in the sense of Fomin and Reading
[16] if and only if $e_{i}$ is an Orlik-Solomon exponent of all $X\in\Theta$
with $\dim(X)\geq j$.

Now, consider the $h$-vector $(h_{i})_{0\leq i\leq n}$ of $\Upsilon(W,m)$ and
$(h^{+}_{i})_{0\leq i\leq n}$ of $\Upsilon^{+}(W,m)$, defined in terms of the
$f$-vector via the polynomial relation:

$\displaystyle\sum_{i=0}^{n}f_{i-1}z^{n-i}=\sum_{i=0}^{n}h_{i}(z+1)^{n-i},\qquad\sum_{i=0}^{n}f^{+}_{i-1}z^{n-i}=\sum_{i=0}^{n}h^{+}_{i}(z+1)^{n-i}.$
(48)

These $h$-vectors have been obtained by Athanasiadis and Tzanaki [4, 5] and
coincide with the Fuß-Narayana numbers of Armstrong [1, Chapter 5]. These
results can be stated as follows:

######  Proposition 10.2 (Athanasiadis and Tzanaki [4, 5]).

For $0\leq k\leq n$, we have:

$\displaystyle h_{n-k}$
$\displaystyle=\\#\Big{\\{}(w_{1},\dots,w_{m})\in\operatorname{NC}(W)^{m}\;:\;\ell(w_{1})=k\text{
and }w_{1}\leq\dots\leq w_{m}\Big{\\}},$ $\displaystyle h^{+}_{n-k}$
$\displaystyle=\\#\Big{\\{}(w_{1},\dots,wa_{m})\in\operatorname{NC}(W)^{m}\;:\;\ell(w_{1})=k,\;w_{1}\leq\dots\leq
w_{m},\text{ and }\operatorname{supp}(w_{m})=S\Big{\\}}.$

Note that the right-hand sides above are respectively the sums

$\displaystyle\sum_{X\in\Theta,\;\dim(X)=k}\kappa(W,X,m),\quad\text{ and
}\quad\sum_{X\in\Theta,\;\dim(X)=k}\kappa^{+}(W,X,m),$

similar to the sums in Equation (47). So, our Equations (37) and (38) can be
seen as a refinement of the $f$\- to $h$\- vector transformation, which is
obtained by summing over $X\in\Theta$ of a given dimension $k$.

In order to make that more explicit, let us show how the bijection from
Section 8 gives the formulas above for $h_{k}$ and $h^{+}_{k}$.

###### Proof of Proposition 10.2.

Via the bijection from Section 8, faces of $\Upsilon(W,m)$ of dimension $k-1$
are in bijection with chains $w_{0}\sqsubset w_{1}\leq\dots\leq w_{m}$ in
$\operatorname{NC}(W)$ such that $\ell(w_{0})=n-k$. So, we get:

$\sum_{k=0}^{n}f_{k-1}z^{n-k}=\sum_{w_{0}\sqsubset w_{1}\leq\dots\leq
w_{m}}z^{\ell(w_{0})}.$

Since the order ideal containing elements below $w_{1}$ for $\sqsubset$ is a
boolean lattice, the binomial theorem gives:

$\sum_{k=0}^{n}f_{k-1}z^{n-k}=\sum_{w_{1}\leq\dots\leq
w_{m}}(1+z)^{\ell(w_{1})}.$

By comparing with (48), we immediately get the combinatorial formula for
$h_{k}$. The one for $h^{+}_{k}$ is obtained similarly. ∎

It is well-known that a shelling of a simplicial complex can be used to find
its $h$-vector. We refer to [37]. Explicitly, suppose we have a shelling of
$\Upsilon(W,m)$, i.e., an indexing $F_{1},F_{2},\dots$ of its facets such that
for each $j$ the (geometric) intersection

$\displaystyle\big{(}\bigcup_{i=1}^{j-1}F_{i}\big{)}\cap F_{j}$ (49)

is purely $1$-codimensional in $F_{j}$. Then $h_{k}$ is the number of indices
$j$ such that the intersection in (49) is the union of $n-k$ facets of
$F_{j}$. It would be very interesting to show that our Equation (37) can also
be obtained via such a shelling of $\Upsilon(W,m)$, and similarly Equation
(38) from a shelling of $\Upsilon^{+}(W,m)$.

##  11 A recursion for the refined enumeration of faces

We give a recursion satisfied by the numbers $\gamma(W,X,m)$, in the spirit of
Fomin and Reading’s recursion for the $f$-vector of $\Upsilon(W,m)$. Our main
tool is also the rotation $\mathcal{R}_{m}$, and the method relies on the
following key result.

######  Proposition 11.1.

For each face $f\in\Upsilon(W,m)$, $\underline{f}$ and
$\underline{\mathcal{R}_{m}(f)}$ are conjugated in $W$. In other words,
$\Upsilon(W,X,m)$ is stable under the action of $\mathcal{R}_{m}$.

###### Proof.

Denote $f=\\{\rho_{1}\succ\dots\succ\rho_{k}\\}$, so that $\prod
f=R(\rho_{1})\cdots R(\rho_{k})\in\operatorname{NC}(W,c)$ and
$\underline{f}=c_{+}(\prod f)c_{-}$. There is a factorization $\prod
f=w_{+}w_{1}\cdots w_{m}w_{-}$ obtained as follows: $w_{+}$ (respectively,
$w_{i}$, $w_{-}$) is obtained from $R(\rho_{1})\cdots R(\rho_{k})$ by keeping
the factors $R(\rho_{j})$ such that $\rho_{j}\in-\Delta_{+}$ (respectively,
such that $\rho_{j}$ is a positive root with color $i$, such that
$\rho_{j}\in-\Delta_{-}$). This is possible because the definition of these
factors agrees with the total order $\prec$ on $\Phi^{(m)}_{\geq-1}$: the
elements in $-\Delta_{+}$ come first in $\rho_{1},\rho_{2},\dots$, etc. Using
similar notations, the noncrossing partition associated to the face
$f^{\prime}=\mathcal{R}_{m}(f)$ is denoted $\prod
f^{\prime}=w^{\prime}_{+}w^{\prime}_{1}\cdots w^{\prime}_{m}w^{\prime}_{-}$.

To find what are the factors of $\prod f^{\prime}$, we need to refine the
factorization of $\prod f$. Note that from Proposition 6.5, we get

$\\{\alpha_{nh/2-n+s+1},\dots,\alpha_{nh/2}\\}=c^{-1}(-\Delta_{+}).$

We write

$w_{m}=w_{m,3}w_{m,2}w_{m,1}$

where $w_{m,3}$ contains the factors $R(\alpha^{m})$ where
$\alpha\in\Delta_{-}$, $w_{m,2}$ contains the factors $R(\alpha^{m})$ where
$\alpha\in c^{-1}(-\Delta_{+})$, and $w_{m,1}$ contains the other factors
$R(\alpha^{m})$. As above, this is possible because this definition agrees
with the ordering on $\Phi^{(m)}_{\geq-1}$.

By examining the action of $\mathcal{R}_{m}$, we can check:

$\displaystyle w^{\prime}_{+}$ $\displaystyle=cw_{m,2}c^{-1},$ $\displaystyle
w^{\prime}_{1}$ $\displaystyle=(cw_{m,1}c^{-1})(cw_{-}c^{-1})w_{+},$
$\displaystyle w^{\prime}_{i}$ $\displaystyle=w_{i-1}\text{ for }2\leq i\leq
m,$ $\displaystyle w^{\prime}_{-}$ $\displaystyle=w_{m,3}.$

For example, $w^{\prime}_{-}=w_{m,3}$ comes from the case
$\mathcal{R}_{m}(\alpha^{m})=-\alpha$ if $\alpha\in\Delta_{-}$. Gathering the
factors gives:

$\displaystyle\prod f^{\prime}$
$\displaystyle=(cw_{m,2}c^{-1})(cw_{m,1}c^{-1}\cdot cw_{-}c^{-1}\cdot
w_{+})w_{1}\cdots w_{m-1}(w_{m,3})$
$\displaystyle=(cw_{m,2}w_{m,1}w_{-}c^{-1}\cdot w_{+})w_{1}\cdots
w_{m-1}w_{m,3}$ $\displaystyle=(cw_{m,2}w_{m,1}w_{-}c^{-1})\big{(}\prod
f\big{)}(w_{m,2}w_{m,1}w_{-})^{-1}.$

Using $c=c_{+}c_{-}$ and the definition of $\underline{f}$, it follows:

$\underline{f^{\prime}}=c_{-}(w_{m,2}w_{m,1}w_{-})c_{-}\underline{f}c_{-}(w_{m,2}w_{m,1}w_{-})^{-1}c_{-}.$

So $\underline{f^{\prime}}$ and $\underline{f}$ are conjugated. ∎

######  Proposition 11.2.

If $W=W_{1}\times W_{2}$ and $X=X_{1}\times X_{2}$, we have:

$\displaystyle\gamma(W,X,m)=\gamma(W_{1},X_{1},m)\cdot\gamma(W_{2},X_{2},m).$
(50)

###### Proof.

This follows from $\Upsilon(W,X,m)$ being the join of
$\Upsilon(W_{1},X_{1},m)$ and $\Upsilon(W_{2},X_{2},m)$. ∎

To state the next proposition, we extend the definition of $\gamma(W,X,m)$ in
the same way as we did for $\kappa$ and $\kappa^{+}$ in Section 7. We use
parabolic conjugacy classes rather than flats, and if
$\mathcal{Y}=\uplus_{i=1}^{j}\mathcal{X}_{i}$ is a disjoint union of parabolic
conjugacy classes, we write:

$\displaystyle\gamma(W,\mathcal{Y},m)=\sum_{i=1}^{j}\gamma(W,\mathcal{X}_{i},m).$
(51)

######  Proposition 11.3.

Assume that $W$ is irreducible and let $h$ be its Coxeter number. We have:

$\gamma(W,\mathcal{X},m)=\frac{mh+2}{2k}\sum_{s\in
S}\gamma(W_{(s)},\mathcal{X}\cap W_{(s)},m).$ (52)

###### Proof.

Following the idea in [16] used to count facets of $\Upsilon(W,m)$, we
consider pointed faces and let:

$\Upsilon^{\bullet}(W,X,m):=\big{\\{}(f,\rho)\;:\;f\in\Upsilon(W,X,m),\;\rho\in
f\big{\\}}.$

The result comes from double counting. First, we clearly have

$\\#\Upsilon^{\bullet}(W,\mathcal{X},m)=k\cdot\\#\Upsilon(W,\mathcal{X},m)=k\cdot\gamma(W,\mathcal{X},m),$

as each face $f\in\Upsilon(W,\mathcal{X},m)$ contains $k$ vertices.

Second, we consider the action of $\mathcal{R}_{m}$ on
$\Upsilon^{\bullet}(W,\mathcal{X},m)$ by
$\mathcal{R}_{m}\big{(}(f,\rho)\big{)}=\big{(}\mathcal{R}_{m}(f),\mathcal{R}_{m}(\rho)\big{)}$.
Let $\omega\subset\Phi_{\geq-1}^{(m)}$ be an orbit for the action of
$\mathcal{R}_{m}$. We will show that

$\displaystyle\\#\big{\\{}(f,\rho)\in\Upsilon^{\bullet}(W,\mathcal{X},m)\;:\;\rho\in\omega\big{\\}}=\frac{mh+2}{2}\sum_{s\in
R((-\Delta)\cap\omega)}\gamma(W_{(s)},\mathcal{X}\cap W_{(s)},m).$ (53)

By summing over the orbits $\omega$, it will follow that
$\\#\Upsilon^{\bullet}(W,\mathcal{X},m)$ is $k$ times the right hand side of
(52) (since the union of the sets $R((-\Delta)\cap\omega)$ is $R(-\Delta)=S$).

To prove (53), first observe that the cardinality in the left-hand side of
(53) actually doesn’t depend on $\rho$. Indeed, the rotation $\mathcal{R}_{m}$
immediately give a bijection between the set associated to $\rho$ and that
associated to $\mathcal{R}_{m}(\rho)$.

Consider the first case of Proposition 6.2. So, $\\#\omega=\frac{mh+2}{2}$ and
$\omega$ contains one element of $-\Delta$, that we denote $-\delta_{i}$. By
the previous observation, we get

$\displaystyle\\#\big{\\{}(f,\rho)\in\Upsilon^{\bullet}(W,\mathcal{X},m)\;:\;\rho\in\omega\big{\\}}=\frac{mh+2}{2}\\#\big{\\{}f\in\Upsilon(W,\mathcal{X},m)\;:\;-\delta_{i}\in
f\big{\\}}.$ (54)

The cardinality in the right-hand side is also
$\gamma(W_{(s_{i})},\mathcal{X}\cap W_{(s_{i})},m)$. We thus get the term
indexed by $s_{i}$ in (52) (up to the factor $k$).

Now, consider the second case of Proposition 6.2. So, $\\#\omega=mh+2$ and
$\omega$ contains two elements of $-\Delta$, that we denote $-\delta_{i}$ and
$-\delta_{j}$. In this case, we get:

$\displaystyle\\#\big{\\{}(f,\rho)\in\Upsilon^{\bullet}(W,\mathcal{X},m)\;:\;\rho\in\omega\big{\\}}=(mh+2)\cdot\\#\big{\\{}f\in\Upsilon(W,\mathcal{X},m)\;:\;-\delta_{i}\in
f\big{\\}}.$ (55)

Here, we get the two terms indexed by $s_{i}$ and $s_{j}$ in (52) (up to the
factor $k$). Indeed these two terms are equal (since $s_{i}$ and $s_{j}$ are
conjugated), and combine to give $(mh+2)$ times one of the two terms. ∎

To finish this section, note that Equations (50), (51), and (52) give an
inductive procedure to compute $\gamma(W,X,m)$ as follows:

  * •

if $\mathcal{X}$ is the conjugacy class of the Coxeter element, we have
$\gamma(W,\mathcal{X},m)=1$ (initial case),

  * •

if $W$ is reducible, use (50),

  * •

if $W$ is irreducible, use (52), then use (51) on each summand (in particular
$\gamma(W_{(s)},\mathcal{X}\cap W_{(s)},m)=0$ if $\mathcal{X}\cap
W_{(s)}=\varnothing$),

  * •

repeat the previous two steps, until each term can be treated by the initial
case.

Also, it is interesting to make explicit what is the intersection
$\mathcal{X}\cap W_{(s)}$ in the various cases of the finite type
classification. The common situation is that $\mathcal{X}\cap W_{(s)}$ is
itself a single conjugacy class of $W_{(s)}$. This comes from the fact that
there is often a unique conjugacy class $\mathcal{X}$ of $W_{(s)}$ of a given
Coxeter type (see tables in [24, Appendix C]), as all elements in
$\mathcal{X}\cap W_{(s)}$ have the same Coxeter type. The exceptions are given
explicitly as follows.

First, let $W$ be of type $D_{n}$, where $n>4$ is odd. Assume that
$\mathcal{X}$ contains elements of type $A_{n-2}$, and $W_{(s)}$ is the
standard parabolic subgroup of type $D_{n-1}$. There is a unique order 2
automorphism of the diagram of type $D_{n-1}$ (unless $n=5$, we leave details
to the reader), and it defines an outer automorphism of $W_{(s)}$. Then
$\mathcal{X}\cap W_{(s)}$ is the union of two conjugacy classes, which are
image of each other under this outer automorphism.

In the case of $E_{8}$, there is a unique conjugacy class of type $A_{5}$, but
the intersection with the parabolic subgroup of type $E_{7}$ is the union of
two conjugacy classes. These are given the five-element subsets of the Dynkin
diagram of $E_{7}$ as follows:

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} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{19.91684pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@lineto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{59.75052pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{59.75052pt}{0.0pt}\pgfsys@lineto{79.66737pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{79.66737pt}{0.0pt}\pgfsys@lineto{99.58421pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@lineto{39.83368pt}{19.91684pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@moveto{23.9002pt}{0.0pt}\pgfsys@curveto{23.9002pt}{2.19998pt}{22.11682pt}{3.98337pt}{19.91684pt}{3.98337pt}\pgfsys@curveto{17.71686pt}{3.98337pt}{15.93347pt}{2.19998pt}{15.93347pt}{0.0pt}\pgfsys@curveto{15.93347pt}{-2.19998pt}{17.71686pt}{-3.98337pt}{19.91684pt}{-3.98337pt}\pgfsys@curveto{22.11682pt}{-3.98337pt}{23.9002pt}{-2.19998pt}{23.9002pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{19.91684pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@moveto{43.81705pt}{0.0pt}\pgfsys@curveto{43.81705pt}{2.19998pt}{42.03366pt}{3.98337pt}{39.83368pt}{3.98337pt}\pgfsys@curveto{37.6337pt}{3.98337pt}{35.85031pt}{2.19998pt}{35.85031pt}{0.0pt}\pgfsys@curveto{35.85031pt}{-2.19998pt}{37.6337pt}{-3.98337pt}{39.83368pt}{-3.98337pt}\pgfsys@curveto{42.03366pt}{-3.98337pt}{43.81705pt}{-2.19998pt}{43.81705pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@moveto{103.56758pt}{0.0pt}\pgfsys@curveto{103.56758pt}{2.19998pt}{101.7842pt}{3.98337pt}{99.58421pt}{3.98337pt}\pgfsys@curveto{97.38423pt}{3.98337pt}{95.60085pt}{2.19998pt}{95.60085pt}{0.0pt}\pgfsys@curveto{95.60085pt}{-2.19998pt}{97.38423pt}{-3.98337pt}{99.58421pt}{-3.98337pt}\pgfsys@curveto{101.7842pt}{-3.98337pt}{103.56758pt}{-2.19998pt}{103.56758pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{99.58421pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{79.66737pt}{0.0pt}\pgfsys@moveto{83.65074pt}{0.0pt}\pgfsys@curveto{83.65074pt}{2.19998pt}{81.86736pt}{3.98337pt}{79.66737pt}{3.98337pt}\pgfsys@curveto{77.46739pt}{3.98337pt}{75.684pt}{2.19998pt}{75.684pt}{0.0pt}\pgfsys@curveto{75.684pt}{-2.19998pt}{77.46739pt}{-3.98337pt}{79.66737pt}{-3.98337pt}\pgfsys@curveto{81.86736pt}{-3.98337pt}{83.65074pt}{-2.19998pt}{83.65074pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{79.66737pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{59.75052pt}{0.0pt}\pgfsys@moveto{63.73389pt}{0.0pt}\pgfsys@curveto{63.73389pt}{2.19998pt}{61.9505pt}{3.98337pt}{59.75052pt}{3.98337pt}\pgfsys@curveto{57.55054pt}{3.98337pt}{55.76715pt}{2.19998pt}{55.76715pt}{0.0pt}\pgfsys@curveto{55.76715pt}{-2.19998pt}{57.55054pt}{-3.98337pt}{59.75052pt}{-3.98337pt}\pgfsys@curveto{61.9505pt}{-3.98337pt}{63.73389pt}{-2.19998pt}{63.73389pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{59.75052pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;.$

The same phenomenon appear when $\mathcal{X}$ has type $A_{1}\times A_{3}$,
and then the two conjugacy classes in $E_{7}$ are given by:

$\leavevmode\hbox to105.96pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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