# q, t-Fuß-Catalan numbers for complex reflection groups

Christian Stump†

Faculty of Mathematics, Nordbergstraße 15, 1090 Vienna, Austria

**Abstract.** In type A, the $q, t$-Fuß-Catalan numbers $Cat_n^{(m)}(q, t)$ can be defined as a bigraded Hilbert series of a module associated to the symmetric group $S_n$. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in $q$ and $t$. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.

**Résumé.** Dans le cas du type A, les $q, t$-nombres de Fuß-Catalan $Cat_n^{(m)}(q, t)$ peuvent être définis comme la série de Hilbert bigraduée d'un certain module associé au groupe symétrique $S_n$. Nous généralisons cette construction aux groupes de réflexion complexes (finis) et nous formulons de jolies propriétés (conjecturales) algébriques et combina-toires de ces polynômes en $q$ et $t$. Enfin, nous décrivons une idée sur la manière dont ces polynômes pourraient être liés à certaines séries de Hilbert de modules apparaissant dans le contexte des algèbres de Cherednik rationnelles. Ceci est un travail en cours.

**Keywords:** $q, t$-Catalan numbers, reflection group, Shi arrangement, coinvariant ring, rational Cherednik algebras

## 1 Introduction

Within the last 15 years the $q, t$-Fuß-Catalan numbers of type A, $Cat_n^{(m)}(q, t)$, arose in more and more contexts in mathematics, namely in symmetric functions theory, algebraic and enumerative combinatorics, representation theory and algebraic geometry. They first appeared in a paper by A. Garsia and M. Haiman in the context of modified Macdonald polynomials, [11]. Later, in his work on the $n!-$ and on the $(n+1)^{n-1}$-conjecture, M. Haiman showed that $Cat_n^{(m)}(q, t)$ is equal to the Hilbert series of the alternating component of the diagonal coinvariant ring, [15]. J. Haglund, [14], and N. Loehr, [18], found a very interesting (partially conjectured) combinatorial interpretation of $Cat_n^{(m)}(q, t)$. The $q, t$-Catalan numbers have many interesting algebraic and combinatorial properties. To mention some: they are symmetric functions in $q$ and $t$ with positive integer coefficients and give a $q, t$-extension of the famous Fuß-Catalan numbers

$$ Cat_n^{(m)} := \frac{1}{mn+1} \binom{(m+1)n}{n}. $$

<footnote>†Supported by FWF Austrian Science Fund grant P17563-N13.</footnote>
Specializing $t = 1$ in $Cat_n^{(m)}(q, t)$ gives some $q$-Fuß-Catalan numbers introduced by J. Fürlinger and J. Hofbauer in [9] and specializing $t = q^{-1}$ gives another $q$-extension of the Fuß-Catalan numbers introduced in [19] by P.A. MacMahon.

We generalize the definition of Fuß-Catalan numbers by means of a Hilbert series from the symmetric group to arbitrary finite complex reflection groups. Furthermore, we present conjectured generalizations of the properties mentioned above concerning specializations of $q$ and $t$, and combinatorial interpretations, see Conjectures 3.10, 3.11 and 4.17. Finally, we present an idea which would relate them to some graded Hilbert series constructed by I. Gordon [12] and by Y. Berest, P. Etingof and V. Ginzburg [4] in the context of rational Cherednik algebras, see Conjecture 5.3 and the following corollary.

This extended abstract is organized as follows: in Section 2, we define alternating polynomials of type A and generalize this definition to complex reflection groups. In Section 3, we first define $q, t$-Fuß-Catalan numbers of type A as a bigraded Hilbert series and present some properties. Then we generalize this definition to $q, t$-Fuß-Catalan numbers for complex reflection groups, $Cat_n^{(m)}(W, q, t)$, and present two conjectures concerning the specializations $q = t = 1$ and $t = q^{-1}$. In Section 4, we combinatorially define $q$-Fuß-Catalan numbers in terms of the extended Shi arrangement and conjecture that they appear in the specialization $t = 1$ of $Cat_n^{(m)}(W, q, t)$. In Section 5, we present another conjecture which would connect $Cat_n^{(m)}(W, q, t)$ to a graded Hilbert series in the context of rational Cherednik algebras.

# 2 Alternating polynomials

## 2.1 Alternating polynomials associated to the symmetric group

The symmetric group $S_n$, which is the reflection group of type $A_{n-1}$, acts on the polynomial ring

$$ \mathbb{C}[\mathbf{x}, \mathbf{y}] := \mathbb{C}[x_1, y_1, \dots, x_n, y_n] $$

by permuting the coordinates in $\mathbf{x}$ and $\mathbf{y}$ simultaneously amongst themselves. This is the *diagonal action*

$$ \sigma(x_i) := x_{\sigma(i)}, \sigma(y_i) := y_{\sigma(i)} \quad \text{for } \sigma \in S_n. $$

A polynomial $p \in \mathbb{C}[\mathbf{x}, \mathbf{y}]$ is called *alternating* if

$$ \sigma(p) = \operatorname{sgn}(\sigma)p \quad \text{for all } \sigma \in S_n, $$

where $\operatorname{sgn}(\sigma)$ is the usual sign of the permutation $\sigma$. We denote the space of all alternating polynomials by $\mathbb{C}[\mathbf{x}, \mathbf{y}]^\epsilon$. As a vector space, $\mathbb{C}[\mathbf{x}, \mathbf{y}]^\epsilon$ has a well-known basis: For $G = \{(\alpha_1, \beta_1), \dots, (\alpha_n, \beta_n)\} \subseteq \mathbb{N} \times \mathbb{N}$ define a bivariate analogue of the Vandermonde determinant $\Delta_G$ by

$$ \Delta_G = \det \begin{pmatrix} x_1^{\alpha_1} y_1^{\beta_1} & \dots & x_1^{\alpha_n} y_1^{\beta_n} \\ \vdots & \ddots & \vdots \\ x_n^{\alpha_1} y_n^{\beta_1} & \dots & x_n^{\alpha_n} y_n^{\beta_n} \end{pmatrix}. $$

The set

$$ \mathcal{B} := \{ \Delta_G : G \subseteq \mathbb{N} \times \mathbb{N}, |G| = n \} $$

forms a vector space basis of $\mathbb{C}[\mathbf{x}, \mathbf{y}]^\epsilon$ and in particular the ideal generated by $\mathcal{B}$ is the same as the ideal generated by all alternating polynomials.
## 2.2 Alternating polynomials associated to any complex reflection group

The concept for polynomials to be alternating can be generalized to any (finite) complex reflection group in the following way: let $V$ be an $n$-dimensional complex vector space and let $W \subseteq \text{GL}(V)$ be a (finite) complex reflection group acting on $V$. For definitions and further information on complex reflection groups see e.g. [6].

The contragredient action of $W$ on $V^* = \text{Hom}(V, \mathbb{C})$ is given by

$$ \omega(\rho) := \rho \circ \omega^{-1}. $$

This induces an action of $W$ on the symmetric algebra $S(V^*)$ which is equal to $\mathbb{C}[\mathbf{x}]$. “Doubling up” this action diagonally defines a *diagonal action* of $W$ on $\mathbb{C}[\mathbf{x}, \mathbf{y}]$.

**Definition 2.1** Let $W$ be a complex reflection group acting on a complex vector space of dimension $n$. We call a polynomial $p \in \mathbb{C}[\mathbf{x}, \mathbf{y}]$ alternating if

$$ \det(\omega)\omega(p) = p \quad \text{for all } \omega \in W. $$

For $W$ being the complex reflection group of type $A_{n-1}$ - which is the symmetric group $S_n$ - this definition reduces to the definition of alternating polynomials given above. We denote the space of all alternating polynomials by $\mathbb{C}[\mathbf{x}, \mathbf{y}]^{\epsilon, W}$.

**Remark 2.2** If $W$ is a real reflection group or, equivalently, if $W$ is a (finite) Coxeter group then

$$ \det(\omega) = (-1)^{l(\omega)}, $$

where $l$ is the length function in the Coxeter group $W$.

The reason why we call a polynomial $p$ alternating if $\det(\omega)\omega(p) = p$ and not if $\omega(p) = \det(\omega)p$ is the following: Define the *sign idempotent* $e_\epsilon$ by

$$ e_{\epsilon} := \frac{1}{|W|} \sum_{\omega \in W} \det(\omega)\omega $$

and the *sign representation* $\epsilon$ by $\omega(z) := \det(\omega)z$ for all $\omega \in W$ and $z \in \mathbb{C}$. Then

$$ \mathbb{C}[\mathbf{x}, \mathbf{y}]^{\epsilon, W} = (\mathbb{C}[\mathbf{x}, \mathbf{y}] \otimes \epsilon)^W = e_{\epsilon} \mathbb{C}[\mathbf{x}, \mathbf{y}]. $$

As for the symmetric group, $\mathbb{C}[\mathbf{x}, \mathbf{y}]^{\epsilon, W}$ has a vector space basis given by

$$ \mathcal{B}_W := \{ e_\epsilon(m) : m \text{ monomial in } \mathbf{x}, \mathbf{y} \text{ with } e_\epsilon(m) \neq 0 \}. $$

**Remark 2.3** • For $W$ of type $A$, $\mathcal{B}_W$ reduces to $\mathcal{B}$ defined above,

• for $W$ of type $B$, $\mathcal{B}_W$ reduces to $\{\Delta_G : G \subseteq \mathbb{N} \times \mathbb{N}, |G| = n, \alpha_i + \beta_i \equiv 1 \mod 2\}$,

• for $W$ of type $D$, $\mathcal{B}_W$ reduces to $\{\Delta_G : G \subseteq \mathbb{N} \times \mathbb{N}, |G| = n, \alpha_i + \beta_i \equiv \alpha_j + \beta_j \mod 2\}$.
# 3 $q, t$-Fuß-Catalan numbers

Before we define $q, t$-Fuß-Catalan numbers in general, we review the definition and the properties about the well-studied case $W = S_n$ which they seem to generalize. To refer to the parameter $m \in \mathbb{N}$, we use the term *Fuß-Catalan* which has commonly been used in the literature for higher Catalan numbers of general type, see e.g. [2] and [8]. In the literature concerning only the case $W = S_n$, the name *generalized $q, t$-Catalan numbers* was more usual.

## 3.1 $q, t$-Fuß-Catalan numbers associated to the symmetric group

Let $S_n$ act on $\mathbb{C}[\mathbf{x}, \mathbf{y}]$ as described in Section 2.1 and let $I \subseteq \mathbb{C}[\mathbf{x}, \mathbf{y}]$ be the ideal generated by all alternating polynomials. Define the $S_n$-module $M^{(m)}$ to be the minimal generating space of $I^m$,

$$M^{(m)} := I^m / \langle \mathbf{x}, \mathbf{y} \rangle I^m.$$

It carries a natural bigrading by degree in $\mathbf{x}$ and degree in $\mathbf{y}$, $M^{(m)} = \bigoplus_{i,j \ge 0} M_{ij}^{(m)}$.

**Remark 3.1** The name minimal generating space comes from the fact that - as a vector space - $M^{(m)}$ is isomorphic to the complex vector space with basis in one-to-one correspondence to any homogeneous minimal generating set of $I^m$.

The following definition is due to M. Haiman:

**Definition 3.2** The $q, t$-Fuß-Catalan numbers of type $A_{n-1}$ are defined as the bigraded Hilbert series of the $S_n$-module $M^{(m)}$,

$$\mathrm{Cat}_n^{(m)}(q, t) := \mathcal{H}(M^{(m)}; q, t) = \sum_{i,j \ge 0} \dim (M_{ij}^{(m)}) q^i t^j.$$

**Remark 3.3** $\mathrm{Cat}_n^{(m)}(q, t)$ was originally defined in [11] as the complicated rational function

$$\sum_{\mu \vdash n} \frac{q^{(m+1)n(\mu')} t^{(m+1)n(\mu)} (1-q)(1-t) \prod_{\mu \in D(\mu)} B_{\mu}(q,t)}{\prod_{c \in D(\mu)} (q^{a(c)} - t^{l(c)+1})(t^{l(c)} - q^{a(c)+1})}$$

in the context of modified Macdonald polynomials. M. Haiman later showed that this rational function is in fact equal to the bigraded Hilbert series in the definition above, [16].

In [10], A. Garsia and J. Haglund proved a simple combinatorial interpretation of $\mathrm{Cat}_n(q, t) := \mathrm{Cat}_n^{(1)}(q, t)$ which was conjectured by J. Haglund in [14], where he introduced the *bounce statistic* on *Catalan paths* or, equivalently, on the set $D_n$ of all partitions that fit inside the partition $(n-1, \dots, 2, 1)$:

$$\mathrm{Cat}_n(q, t) = \sum_{\lambda \in \mathcal{D}_n} q^{\mathrm{area}(\lambda)} t^{\mathrm{bounce}(\lambda)}.$$

Together with N. Loehr, they extended the definitions of *area* and *bounce* to the set $D_n^{(m)}$ of $m$-Catalan paths, these are partitions that fit inside the partition $((n-1)m, \dots, 2m, m)$, and conjectured a combinatorial interpretation of $\mathrm{Cat}_n^{(m)}(q, t)$ in terms of these statistics, [18]:

$$\mathrm{Cat}_n^{(m)}(q, t) = \sum_{\lambda \in \mathcal{D}_n^{(m)}} q^{\mathrm{area}(\lambda)} t^{\mathrm{bounce}(\lambda)}.$$
**Remark 3.4** The specialization $t = 1$ was proved by A. Garsia and M. Haiman in [11].

**Corollary 3.5** The specialization $q = t = 1$ reduces $\mathrm{Cat}_n^{(m)}(q, t)$ to the $\mathrm{Cat}_n^{(m)}$.

Also in [11], it was shown that the specialization $t = q^{-1}$ yields the following $q$-extension of $\mathrm{Cat}_n^{(m)}$:

**Theorem 3.6 (Garsia, Haiman)**

$$q^{m(\frac{n}{2})} \mathrm{Cat}_n^{(m)}(q, q^{-1}) = \frac{1}{[mn+1]_q} \begin{bmatrix} (m+1)n \\ n \end{bmatrix}_q,$$

$$\text{where } [k]_q := 1 + q + \dots + q^{k-1}, [k]_q! := [1]_q[2]_q \cdots [k]_q \text{ and } \begin{bmatrix} k \\ l \end{bmatrix}_q := [k]_q!/[l]_q![k-l]_q!.$$

**Remark 3.7** For $m = 1$, this reduces to the $q$-Catalan numbers defined by P.A. MacMahon in [19].

## 3.2 $q, t$-Fuß-Catalan numbers for complex reflection groups

Recall that a complex reflection group $W$ acts on the polynomial ring $\mathbb{C}[\mathbf{x}, \mathbf{y}]$ diagonally as described in Section 2.2. Let $I \trianglelefteq \mathbb{C}[\mathbf{x}, \mathbf{y}]$ be the ideal generated by all alternating polynomials and define the $W$-module $M^{(m)} := I^m/\langle\mathbf{x}, \mathbf{y}\rangle I^m$.

**Definition 3.8** The $q, t$-Fuß-Catalan numbers associated to $W$ are defined as

$$\mathrm{Cat}^{(m)}(W, q, t) := \mathcal{H}(M^{(m)}; q, t) = \sum_{i,j \ge 0} \dim (M_{ij}) q^i t^j.$$

**Remark 3.9** Using the computer algebra systems Singular and Macaulay 2, we computed the dimension of $M^{(m)}$ as well as $\mathrm{Cat}^{(m)}(W, q, t)$ for the classical types at least up to rank 4 and small $m$ and following exceptional types

$$I_2(k) \text{ for } k \in \{5, 6, 10, 12\}, \quad H_3, \quad G(k, 1, 1) \text{ for } k \le 10, \quad G(4, 2, 2).$$

All following conjectures are based on these computations.

In [20], V. Reiner defined Fuß-Catalan numbers for classical reflection groups, and in [5] D. Bessis generalized this definition to well-generated complex reflection group $W$:

$$\mathrm{Cat}^{(m)}(W) := \prod_{i=1}^{l} \frac{d_i + mh}{d_i},$$

where $l$ is the rank of $W$, $d_1 \le \dots \le d_l$ are the degrees of the fundamental invariants and $h = d_l$ is the Coxeter number. For definitions and further information see e.g. [2, Section 2.7] and [5].

For more or less general classes of reflection groups, this number counts a bunch of interesting combinatorial objects, see e.g. [2] and [3]. For $W = S_n$, it reduces to $\mathrm{Cat}_n^{(m)}$ and for any real reflection group, it reduces for $m = 1$ to the well-known Catalan numbers $\mathrm{Cat}(W)$ for real reflection groups, which are shown in Fig. 1.
<table><caption>Fig. 1: Cat($W$) for the irreducible real reflection groups.</caption><thead><tr><th>A<sub>n-1</sub></th><th>B<sub>n</sub></th><th>D<sub>n</sub></th><th>I<sub>2</sub>(k)</th><th>H<sub>3</sub></th><th>H<sub>4</sub></th><th>F<sub>4</sub></th><th>E<sub>6</sub></th><th>E<sub>7</sub></th><th>E<sub>8</sub></th></tr></thead><tbody><tr><td>(<sup>2n</sup>n)</td><td>(<sup>2n</sup>n)</td><td>(<sup>2n</sup>n) - (<sup>2(n-1)</sup>n-1)</td><td>k + 2</td><td>32</td><td>280</td><td>105</td><td>833</td><td>4160</td><td>25080</td></tr></tbody></table>

Our first conjecture concerns $\mathrm{Cat}^{(m)}(W, 1, 1)$, the dimension of $M$:

**Conjecture 3.10** Let $W$ be a well-generated complex reflection group. Then

$$ \mathrm{Cat}^{(m)}(W, 1, 1) = \mathrm{Cat}^{(m)}(W). $$

In [6], D. Bessis and V. Reiner defined a $q$-extension of $\mathrm{Cat}^{(m)}(W)$ by

$$ \prod_{i=1}^{l} \frac{[d_i + mh]_q}{[d_i]_q}. $$

The following conjecture, which is obviously stronger than Conjecture 3.10, would generalize Theorem 3.6 and would thereby give a new answer to a question of C. Kriloff and V. Reiner in [1, Problem 2.2]:

**Conjecture 3.11** Let $W$ be a well-generated complex reflection group. Then

$$ q^{mN} \mathrm{Cat}^{(m)}(W, q, q^{-1}) = \prod_{i=1}^{l} \frac{[d_i + mh]_q}{[d_i]_q}, $$

where $N := \sum (d_i - 1)$.

**Remark 3.12** For a real reflection group $W$, the $(d_i - 1)$'s appearing in the conjecture are the exponents associated to $W$ and $N$ is equal to the number of positive roots.

**Open Problem 3.13** Are there statistics $qstat$ and $tstat$ on objects counted by $\mathrm{Cat}^{(m)}(W)$ which generalize area and bounce on Catalan paths $\mathcal{D}_n^{(m)}$ such that

$$ \mathrm{Cat}^{(m)}(W, q, t) = \sum_{\lambda} q^{qstat(\lambda)} t^{tstat(\lambda)}. $$

In the next section, we will present some conjectures concerning this open problem.

# 4 A generalization of the area statistic to crystallographic reflection groups

## 4.1 The area statistic for Catalan paths of type A

Specializing $t = 1$ reduces $\mathrm{Cat}_n^{(m)}(q, t)$ to the well-known Carlitz $q$-Fuß-Catalan numbers defined by

$$ \mathrm{Cat}_n^{(m)}(q) := \mathrm{Cat}_n^{(m)}(q, 1) = \sum_{\lambda \in \mathcal{D}_n^{(m)}} q^{\mathrm{area}(\lambda)}. $$

They satisfy the following recurrence which can be deduced from a generating function identity proved by C. Krattenthaler in [17, Theorem 9]:
<figure><img src="image_1.png" /><figcaption>Fig. 2: All type B Catalan paths of length 2.</figcaption></figure>

**Theorem 4.1**

$$ \mathrm{Cat}_{n+1}^{(m)}(q) = \sum_{k_1+\dots+k_{m+1}=n} q^{n(\mathbf{k})} \mathrm{Cat}_{k_1}^{(m)}(q) \dots \mathrm{Cat}_{k_{m+1}}^{(m)}(q), \quad \mathrm{Cat}_0^{(m)}(q) = 1, $$

$$ \text{where } n(\mathbf{k}) = n(k_1, \dots, k_{m+1}) := \sum (m+1-i)k_i. $$

## 4.2 An area statistic for Catalan paths of type B

For $m=1$, we define an area-statistic on type B Catalan paths and establish an analogous recurrence.

**Definition 4.2** A type B Catalan path of length $n$ is a lattice paths of $2n$ steps, either north or east, that starts at (0, 0) and stays above the diagonal $x = y$. For such a path $\lambda$, we define area($\lambda$) to be the number of boxes in the region confined by the path, the diagonal $x = y$ and the anti-diagonal $x = 2n - y$, not counting the halfboxes at the diagonal $x = y$ but counting the halfboxes at the anti-diagonal $x = 2n - y$.

**Example 4.3** In Fig. 2, all Catalan paths of type $B_2$ are shown, the boxes which contribute to the area are shaded.

In analogy to type A, we define q-Catalan numbers for type B in the following way:

**Definition 4.4**

$$ \mathrm{Cat}_{B_n}(q) := \sum q^{\mathrm{area}(\lambda)}, $$

where the sum ranges over all type B Catalan paths $\lambda$ of length $n$.

**Example 4.5** As shown in Example 4.3, we have $\mathrm{Cat}_{B_2}(q) = 1 + 2q + q^2 + q^3 + q^4$.

The definition is based on the following conjecture:

**Conjecture 4.6**

$$ \mathrm{Cat}^{(1)}(W_{B_n}, q, 1) = \mathrm{Cat}_{B_n}(q). $$

$\mathrm{Cat}_{B_n}(q)$ satisfy the following recurrence involving Catalan numbers of type A:

**Theorem 4.7**

$$ \mathrm{Cat}_{B_n}(q) = \mathrm{Cat}_n(q) + \sum_{k=0}^{n-1} q^{2k+1} \mathrm{Cat}_{B_k}(q) \mathrm{Cat}_{n-k}(q), \quad \mathrm{Cat}_{B_0}(q) = 1. $$
**Proof:** Let $\lambda$ be a type $B$ Catalan path of length $n$. Then either $\lambda$ has as many east as north steps, which means $\lambda$ is equal to a type $A$ Catalan path of length $n$, or there exists a last point $(k, k+1)$ where the path touches the diagonal $x+1=y$ and stays strictly above afterwards. Now, we have an initial type $A$ like Catalan path of length $k+1$ (where the last step is a north step instead of an east step). After this north step, a type $B$ Catalan path of length $n-k-1$ starts. This gives the following recurrence which is equivalent to the statement:

$$ \mathrm{Cat}_{B_n}(q) = \mathrm{Cat}_n(q) + \sum_{k=0}^{n-1} q \mathrm{Cat}_{k+1}(q) q^{2(n-k-1)} \mathrm{Cat}_{B_{n-k-1}}(q). $$

□

**Corollary 4.8** The $\mathrm{Cat}_{B_n}(q)$ satisfy the following generating function identity:

$$ \sum_{n \ge 0} \frac{x^n q^{-n(n-1)} (1 - qx)}{(-x; q^{-1})_{2n+1}} \mathrm{Cat}_{B_n}(q) = 1, $$

$$ \text{where } (a; q)_k := (1-a)(1-qa)\dots(1-q^{k-1}a). $$

We will see in the next section that both, $m$-Catalan paths of type $A$ and Catalan paths of type $B$ are special cases of a more general construction and that it is not possible to construct Catalan paths of type $B$ for higher $m$'s as lattice paths (at least not in the manner of defining an area generating function equal to the specialization $t=1$ of $q, t$-Fuß-Catalan numbers). This is likely to be the reason why we were - so far - not able to find a recurrence in type $B$ for higher $m$'s.

## 4.3 The extended Shi arrangement and the coheight statistic

Fix $\Phi$ to be a crystallographic root system and let $W = W_{\Phi}$ be the associated reflection group. The root poset of $\Phi$ is given by the partial order on the set of positive roots $\Phi^+$ defined by covering relation

$$ \alpha \prec \beta :\Leftrightarrow \beta - \alpha \text{ is a simple root.} $$

An order ideal $I \subseteq \Phi^+$ is a subset $I \subseteq \Phi^+$ such that $\alpha \le \beta \in I$ implies $\alpha \in I$.

**Theorem 4.9 (V. Reiner [20])** Let $\Phi$ be a crystallographic root system. Then

$$ \#\{I \subseteq \Phi^+\} = \mathrm{Cat}(W). $$

In [3], C.A. Athanasiadis generalized this theorem to $\mathrm{Cat}^{(m)}(W)$ as follows: Let $\mathcal{I}$ be an increasing chain of order ideals $I_1 \subseteq \dots \subseteq I_m \subseteq \Phi^+$. We call $\mathcal{I}$ a filtered chain of length $m$ if for $i, j \ge 1$,

$$ (I_i + I_j) \cap \Phi^+ \subseteq I_{i+j} \quad \text{with } i + j \le m, $$

$$ (J_i + J_j) \cap \Phi^+ \subseteq J_{i+j}, $$

$$ \text{where } J_i = \Phi^+ \setminus I_i \text{ and } J_i = J_m \text{ for } i > m. $$

Let $V$ be the vector space spanned by $\Phi$, with inner product $(\cdot, \cdot)$. The extended Shi arrangement $\mathrm{Shi}^{(m)}(\Phi)$
<figure><img src="image_2.png" /><figcaption>Fig. 3: The extended Shi arrangement of type $A_2$ and $m = 2$.</figcaption></figure>

is given by the collection of hyperplanes in $V$ defined by the affine equations $(\alpha, x) = k$ for $\alpha \in \Phi$ and $-m < k \le m$. Thus $\mathrm{Shi}^{(m)}(\Phi)$ is a deformation of the Coxeter arrangement $\mathcal{A}_{\Phi}$. A positive region of $\mathrm{Shi}^{(m)}(\Phi)$ is a connected component of $V \setminus \mathrm{Shi}^{(m)}(\Phi)$ which lies in the fundamental chamber of $\mathcal{A}_{\Phi}$.

C.A. Athanasiadis defined the following map $\Psi$ between positive regions of $\mathrm{Shi}^{(m)}(\Phi)$ and filtered chains in $\Phi^+$ and showed that $\Psi$ is a bijection: let $R$ be a positive region and let $x \in R$. Then $\Psi(R)$ is defined to be the filtered chain $I_1 \subseteq \dots \subseteq I_m \subseteq \Phi^+$ such that

$$ (\alpha, x) < i \quad , \quad \text{if } \alpha \in I_i $$

$$ (\alpha, x) > i \quad , \quad \text{if } \alpha \in J_i = \Phi^+ \setminus I_i. $$

**Theorem 4.10 (C.A. Athanasiadis [3])** Both the number of filtered chains in $\Phi^+$ and the number of positive regions of $\mathrm{Shi}^{(m)}(\Phi)$ is equal to $\mathrm{Cat}^{(m)}(W)$.

**Definition 4.11** Let $R^0$ be the fundamental region given by $0 < (x, \alpha) < 1$ for all $\alpha \in \Phi^+$. For any region $R$, define the height of $R$, denoted $\mathrm{h}(R)$, to be the number of hyperplanes in $\mathrm{Shi}^{(m)}(\Phi)$ that separate $R$ from $R^0$ and the coheight by $\mathrm{coh}(R) := mN - \mathrm{h}(R)$. Furthermore, we (combinatorially) define q-Fuß-Catalan numbers associated to $W$ by

$$ \mathrm{Cat}^{(m)}(W, q) := \sum_{R} q^{\mathrm{coh}(R)}, $$

where the sum ranges over all positive regions of $\mathrm{Shi}^{(m)}(\Phi)$.

**Example 4.12** Let $W$ be the reflection group of type $A_2$ and $m = 2$. In Fig. 3, the extended Shi arrangement of the given type is shown. The positive roots are denoted by $\alpha_1, \alpha_2$ and $\alpha_3 = \alpha_1 + \alpha_2$, the fundamental chamber is shaded and the positive regions are labelled by their coheights. This gives

$$ \mathrm{Cat}^{(2)}(W_{A_2}, q) = 1 + 2q + 3q^2 + 2q^3 + 2q^4 + q^5 + q^6. $$
**Remark 4.13** The poset of all regions, defined by the covering relation $R \prec R'$ if $h(R) = h(R') - 1$ and $R, R'$ share a common face, is isomorphic to the poset $NN^{(k)}(W)$ described in [2, Definition 5.1.20].

**Proposition 4.14** Let $I_1 \subseteq \dots \subseteq I_m \subseteq \Phi^+$ be a filtered chain and let $R$ be the associated region. Then the bijection given above implies that

$$ \mathrm{coh}(R) = \sum \#I_i. $$

The next theorem shows that the coheight on regions in the fundamental chamber reduces for type A to the area on m-Catalan paths of type A and for type B with $m = 1$ to the area on Catalan paths of type B.

**Theorem 4.15** For all integers $m \ge 1$, we have

$$ \mathrm{Cat}^{(m)}(W_{A_{n-1}}, q) = \mathrm{Cat}_{n}^{(m)}(q) \quad , \quad \mathrm{Cat}^{(1)}(W_{B_n}, q) = \mathrm{Cat}_{B_n}(q). $$

**Remark 4.16** Counting lattice paths consisting of north and east steps having a boundary can always be seen as counting order ideals in very special kinds of posets. The posets occurring for $\mathrm{Cat}^{(m)}(W, q)$ with $W$ of type $D_n$ with $n \ge 4$ and of type $B_n$ with $n, m \ge 2$ fail to have this property.

The definition of coheight is motivated by the following conjecture which would partially answer Open Problem 3.13:

**Conjecture 4.17** Let $W$ be a crystallographic reflection group. Then

$$ \mathrm{Cat}^{(m)}(W, q, 1) = \mathrm{Cat}^{(m)}(W, q). $$

## 4.4 Non-crystallographic reflection groups

So far, no definition for root posets for non-crystallographic reflection groups is known. In [2], D. Armstrong suggests, how these root posets should look like in types $I_2(m)$ and $H_3$. Our computations confirm these ideas: Let $W$ be a reflection group of one of the following types: $I_2(5)$, $I_2(10)$, $I_2(12)$, $H_3$. Then

$$ \mathrm{Cat}^{(m)}(W, q, 1) = \sum_{I \le \Phi^+} q^{\#I}, $$

where $\Phi^+$ is Armstrong's suggested root poset of type $W$. The situation in the cyclic group of order $k$, $W = G(k, 1, 1)$ is the following: The $q, t$-Fuß-Catalan numbers associated to $G(k, 1, 1)$ are all equal. As we already know the classical case $k = 2$, we get

$$ \mathrm{Cat}^{(m)}(W, q, t) = \mathrm{Cat}_{2}^{(m)}(q, t) = q + t. $$

We also computed $\mathrm{Cat}^{(m)}(W, q, t)$ for $W$ the non-well-generated reflection group of type $G(4, 2, 2)$ and the result was, up to $m = 3$,

$$ \mathrm{Cat}^{(m)}(W, q, t) = \mathrm{Cat}^{(m)}(W_{G_2}, q, t). $$
# 5 Connections to rational Cherednik algebras

Let W be a real reflection group or, equivalently, let W be a (finite) Coxeter group acting on the polynomial ring $\mathbb{C}[\mathbf{x}, \mathbf{y}]$ and let $M^{(m)}$ be the W-module defined in Section 3.2. It is easy to see that $M^{(m)}$ is, except for the sign twist, equal to the alternating component of the module

$$ \mathbf{R}[\mathbf{x}, \mathbf{y}] := (I^{m-1}/I^{m-1}J) \otimes \epsilon^{m-1}, $$

where J is the ideal generated by all invariant polynomials without constant term,

$$ M^{(m)} \cong (\mathbf{R}[\mathbf{x}, \mathbf{y}] \otimes \epsilon)^W = \mathbf{e}_{\epsilon}\mathbf{R}[\mathbf{x}, \mathbf{y}]. $$

**Remark 5.1** $\mathbf{R}[\mathbf{x}, \mathbf{y}]$ reduces for $m = 1$ to the coinvariant ring $\mathbb{C}[\mathbf{x}, \mathbf{y}]/J$ and was, in type A, introduced in [11].

The conjectured connection to rational Cherednik algebras is the following: in [7], C.F. Dunkl and E. Opdam constructed a certain W-module L depending on a non-negative integer m. This module carries a natural tensor product filtration and in [4, Theorem 1.6] Y. Berest, P. Etingof and V. Ginzburg showed that the Hilbert series of the trivial component of its associated graded module gr(L) is given by

$$ \mathcal{H}(\mathbf{e}(\mathrm{gr}(L)); q) = q^{-mN} \prod_{i=1}^{l} \frac{[d_i + mh]_q}{[d_i]_q}. $$

Using [13, Lemma 6.7 (2)] together with equalities (7.6) and (7.8) in [4], we obtain the following result which partially generalizes [12, Theorem 5].

**Theorem 5.2** Let W be a real reflection group and let $\mathbf{R}[\mathbf{x}, \mathbf{y}]$ be graded by degree in x minus degree in y. Then there exists a natural surjection of graded W-modules,

$$ \mathbf{R}[\mathbf{x}, \mathbf{y}] \otimes \epsilon \to \mathrm{gr}(L). $$

**Conjecture 5.3** The kernel of this surjection does not contain a copy of the trivial representation.

This conjecture would imply the following corollary:

**Corollary 5.4** Let $M^{(m)}$ be graded by degree in x minus degree in y. Then

$$ M^{(m)} \cong \mathbf{e}(\mathrm{gr}(L)) $$

as graded W-modules.

**Remark 5.5** Together with the above discussion, Corollary 5.4 would imply Conjecture 3.11.

# References

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