# 0-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra

Jia Huang*

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

**Abstract.** We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q, t)-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics.

**Résumé.** Nous définissons une action de l'algèbre de Hecke-0 de type A sur l'anneau Stanley-Reisner de l'algèbre de Boole. En étudiant cette action, on obtient une famille de fonctions symétriques non commutatives multivariées, qui se spécialisent pour les non commutatives fonctions de Hall-Littlewood symétriques et leur (q, t)-analogues introduits par Bergeron et Zabrocki. Nous obtenons également des identités de fonction quasisymmetric multivariées, qui se spécialisent à la suite de Garsia et Gessel sur la fonction génératrice de la distribution conjointe de cinq statistiques de permutation.

**Keywords:** 0-Hecke algebra, Stanley-Reisner ring, Boolean algebra, noncommutative Hall-Littlewood symmetric function, multivariate quasisymmetric function.

## 1 Introduction

Let $\mathbb{F}$ be any field. The symmetric group $\mathfrak{S}_n$ naturally acts on the polynomial ring $\mathbb{F}[X] := \mathbb{F}[x_1, \dots, x_n]$ by permuting the variables $x_1, \dots, x_n$. The invariant algebra $\mathbb{F}[X]^{\mathfrak{S}_n}$, which consists of all the polynomials fixed by this $\mathfrak{S}_n$-action, is a polynomial algebra generated by the elementary symmetric functions $e_1, \dots, e_n$. The coinvariant algebra $\mathbb{F}[X]/(\mathbb{F}[X]^{\mathfrak{S}_n})$, with $(\mathbb{F}[X]^{\mathfrak{S}_n}) = (e_1, \dots, e_n)$, is a vector space of dimension $n!$ over $\mathbb{F}$, and when $\mathbb{F}$ has characteristic larger than $n$ the coinvariant algebra carries the regular representation of $\mathfrak{S}_n$. A well known basis for $\mathbb{F}[X]/(\mathbb{F}[X]^{\mathfrak{S}_n})$ consists of the descent monomials. Garsia [7] obtained this basis by transferring a natural basis from the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ of the Boolean algebra $\mathcal{B}_n$ to the polynomial ring $\mathbb{F}[X]$. Here the Boolean algebra $\mathcal{B}_n$ is the set of all subsets of $[n] := \{1, 2, \dots, n\}$ partially ordered by inclusion, and the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ is the quotient of the polynomial algebra $\mathbb{F}[y_A : A \subseteq [n]]$ by the ideal $(y_A y_B : A \text{ and } B \text{ are incomparable in } \mathcal{B}_n)$.

The 0-Hecke algebra $\mathcal{H}_n(0)$ (of type A) is a deformation of the group algebra of $\mathfrak{S}_n$. It acts on $\mathbb{F}[X]$ by the Demazure operators, also known as the isobaric divided difference operators, having the same

<footnote>*Email: jiahuangustc@gmail.com. Supported by NSF grant DMS-1001933.</footnote>
invariant algebra as the $\mathfrak{S}_n$-action on $\mathbb{F}[X]$. In our earlier work [13], we showed that the coinvariant algebra $\mathbb{F}[X]/(\mathbb{F}[X]_+^{\mathfrak{S}_n})$ is also isomorphic to the regular representation of $H_n(0)$, for any field $\mathbb{F}$, by constructing another basis for $\mathbb{F}[X]/(\mathbb{F}[X]_+^{\mathfrak{S}_n})$ which consists of certain polynomials whose leading terms are the descent monomials. This and the previously mentioned connection between the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ and the polynomial ring $\mathbb{F}[X]$ motivate us to define an $H_n(0)$-action on $\mathbb{F}[\mathcal{B}_n]$.

It turns out that our $H_n(0)$-action on $\mathbb{F}[\mathcal{B}_n]$ has similar properties to the $H_n(0)$-action on $\mathbb{F}[X]$. It preserves an $\mathbb{N}^{n+1}$-multigrading of $\mathbb{F}[\mathcal{B}_n]$ and has invariant algebra equal to a polynomial algebra $\mathbb{F}[\Theta]$, where $\Theta$ is the set of *rank polynomials* $\theta_i$ (the usual analogue of $e_i$ in $\mathbb{F}[\mathcal{B}_n]$). We show that the $H_n(0)$-action is $\Theta$-linear and thus descends to the coinvariant algebra $\mathbb{F}[\mathcal{B}_n]/(\Theta)$. Using the descent monomials in $\mathbb{F}[\mathcal{B}_n]$ it is not hard to see that $\mathbb{F}[\mathcal{B}_n]/(\Theta)$ carries the regular representation of $H_n(0)$.

It is well known that every finite dimensional (complex) $\mathfrak{S}_n$-representation is a direct sum of simple (i.e. irreducible) $\mathfrak{S}_n$-modules, and the simple $\mathfrak{S}_n$-modules are indexed by partitions $\lambda$ of $n$, which correspond to the Schur functions $s_\lambda$ via the *Frobenius characteristic map*. Hotta-Springer [11] and Garsia-Procesi [9] discovered that the cohomology ring of the *Springer fiber* indexed by a partition $\mu$ of $n$ is isomorphic to certain quotient ring $R_\mu$ of $\mathbb{F}[X]$, which admits a graded $\mathfrak{S}_n$-module structure corresponding to the *modified Hall-Littlewood symmetric function* $\tilde{H}_\mu(x; t)$ via the *Frobenius characteristic map*. The coinvariant algebra of $\mathfrak{S}_n$ is nothing but $R_{1n}$.

In our previous work [13] we established a partial analogue of the above result by showing that the $H_n(0)$-action on $\mathbb{F}[X]$ descends to $R_\mu$ if and only if $\mu = (1^k, n-k)$ is a hook, and if so then $R_\mu$ has graded quasisymmetric characteristic equal to $\tilde{H}_\mu(x; t)$ and graded noncommutative characteristic $\tilde{H}_\mu(x; t)$. Here $\tilde{H}_\alpha(x; t)$ is the noncommutative modified Hall-Littlewood symmetric function introduced by Bergeron and Zabrocki [3] for any composition $\alpha$ of $n$. Using an analogue of the nabla operator Bergeron and Zabrocki [3] also introduced a $(q, t)$-analogue $\tilde{H}_\alpha(x; q, t)$ for any composition $\alpha$. Now we provide in Theorem 1.1 below a complete representation theoretic interpretation for $\tilde{H}_\alpha(x; t)$ and $\tilde{H}_\alpha(x; q, t)$ by the $H_n(0)$-action on $\mathbb{F}[\mathcal{B}_n]$.

To state our result, we first recall the two characteristic maps for representations of $H_n(0)$ introduced by Krob and Thibon [14], which we call the *quasisymmetric characteristic* and the *noncommutative characteristic*. The simple $H_n(0)$-modules are indexed by compositions $\alpha$ of $n$ and correspond to the *fundamental quasisymmetric functions* $F_\alpha$ via the quasisymmetric characteristic; the projective indecomposable $H_n(0)$-modules are also indexed by compositions $\alpha$ of $n$ and correspond to the *noncommutative ribbon Schur functions* $s_\alpha$ via the noncommutative characteristic. See §2 for details.

**Theorem 1.1** Let $\alpha$ be a composition of $n$. Then there exists a homogeneous $H_n(0)$-invariant ideal $I_\alpha$ of the multigraded algebra $\mathbb{F}[\mathcal{B}_n]$ such that the quotient algebra $\mathbb{F}[\mathcal{B}_n]/I_\alpha$ becomes a projective $H_n(0)$-module with multigraded noncommutative characteristic equal to

$$ \tilde{\mathbf{H}}_{\alpha}(\mathbf{x}; t_1, \dots, t_{n-1}) := \sum_{\beta \preceq \alpha} \underline{t}^{D(\beta)} s_{\beta} \quad \text{inside} \quad \mathbf{NSym}[t_1, \dots, t_{n-1}]. $$

One has $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; t, t^2, \dots, t^{n-1}) = \tilde{\mathbf{H}}_\alpha(\mathbf{x}; t)$, and obtains $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; q, t)$ from $\tilde{\mathbf{H}}_{1n}(\mathbf{x}; t_1, \dots, t_{n-1})$ by taking $t_i = t^i$ for all $i \in D(\alpha)$, and $t_i = q^{n-i}$ for all $i \in [n-1] \setminus D(\alpha)$.

Here $D(\alpha)$ is the set of partial sums of $\alpha$, the notation $\beta \preceq \alpha$ means $\alpha$ and $\beta$ are compositions of $n$ with $D(\beta) \subseteq D(\alpha)$, and $\underline{t}^S$ denotes the product $\prod_{i \in S} t_i$ over all elements $i$ in a multiset $S$, including the repeated ones. Taking $\alpha = (1^n)$ shows that $\mathbb{F}[\mathcal{B}_n]/(\Theta)$ carries the regular representation of $H_n(0)$.
Specializations of $\tilde{\mathbf{H}}_{1^n}(\mathbf{x}; t_1, \dots, t_{n-1})$ include not only $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; q, t)$, but also a more general family of noncommutative symmetric functions depending on parameters associated with paths in binary trees introduced recently by Lascoux, Novelli, and Thibon [15].

Next we study the quasisymmetric characteristic of $\mathbb{F}[\mathcal{B}_n]$. We combine the usual $\mathbb{N}^{n+1}$-multigrading of $\mathbb{F}[\mathcal{B}_n]$ (recorded by $\underline{t} := t_0, \dots, t_n$) with the length filtration of $H_n(0)$ (recorded by $q$) and obtain an $\mathbb{N} \times \mathbb{N}^{n+1}$-multigraded quasisymmetric characteristic for $\mathbb{F}[\mathcal{B}_n]$.

**Theorem 1.2** The $\mathbb{N} \times \mathbb{N}^{n+1}$-multigraded quasisymmetric characteristic of $\mathbb{F}[\mathcal{B}_n]$ is

$$ 
\begin{align*}
\mathrm{Ch}_{q, \underline{t}}(\mathbb{F}[\mathcal{B}_n]) &= \sum_{k \ge 0} \sum_{\alpha \in \mathrm{Com}(n, k+1)} \underline{t}^{D(\alpha)} \sum_{w \in \mathfrak{S}^{\alpha}} q^{\mathrm{inv}(w)} F_{D(w^{-1})} \\
&= \sum_{w \in \mathfrak{S}_n} \frac{q^{\mathrm{inv}(w)} \underline{t}^{D(w)} F_{D(w^{-1})}}{\prod_{0 \le i \le n} (1 - t_i)} \\
&= \sum_{k \ge 0} \sum_{\mathbf{p} \in [k+1]^n} t_{p'_1} \cdots t_{p'_k} q^{\mathrm{inv}(\mathbf{p})} F_{D(\mathbf{p})}.
\end{align*}
$$

Here we identify $F_I$ with $F_\alpha$ if $D(\alpha) = I \subseteq [n-1]$. The set $\mathrm{Com}(n, k)$ consists of all weak compositions of $n$ with length $k$, i.e. all the sequences $\alpha = (\alpha_1, \dots, \alpha_k)$ of $k$ nonnegative integers with $|\alpha| := \sum_{i=1}^k \alpha_i = n$. The descent multiset of the weak composition $\alpha$ is the multiset

$$ D(\alpha) := \{\alpha_1, \alpha_1 + \alpha_2, \dots, \alpha_1 + \dots + \alpha_{k-1}\}. $$

We also define $\mathfrak{S}^\alpha := \{w \in \mathfrak{S}_n : D(w) \subseteq D(\alpha)\}$. The set $[k+1]^n$ consists of all words of length $n$ on the alphabet $[k+1]$. Given $\mathbf{p} = (p_1, \dots, p_n) \in [k+1]^n$, we write $p'_i := \#\{j : p_j \le i\}$, $\mathrm{inv}(\mathbf{p}) := \#\{(i, j) : 1 \le i < j \le n : p_i > p_j\}$, and $D(\mathbf{p}) := \{i : p_i > p_{i+1}\}$.

Let $\mathbf{ps}_{q; \underline{t}}(F_\alpha) := F_\alpha(1, q, q^2, \dots, q^{\ell-1}, 0, 0, \dots)$. Applying the linear transformation $\sum_{\ell \ge 0} u_1^\ell \mathbf{ps}_{q_1; \ell+1}$ and the specialization $t_i = q_2^i u_2$ for all $i = 0, 1, \dots, n$ to Theorem 1.2, we recover a result of Garsia and Gessel [8, Theorem 2.2] on the generating function of the joint distribution of five permutation statistics:

$$ \frac{\sum_{w \in \mathfrak{S}_n} d_0^{\mathrm{inv}(w)} q_1^{\mathrm{maj}(w^{-1})} u_1^{\mathrm{des}(w^{-1})} q_2^{\mathrm{maj}(w)} u_2^{\mathrm{des}(w)}}{(u_1; q_1)_n (u_2; q_2)_n} = \sum_{\ell, k \ge 0} u_1^\ell u_2^k \sum_{(\lambda, \mu) \in \mathcal{B}(\ell, k)} q_0^{\mathrm{inv}(\mu)} q_1^{\lambda|} q_2^{\mu|} \quad (1) $$

Here $(u; q)_n := \prod_{0 \le i \le n} (1 - q^i u)$, the set $B(\ell, k)$ consists of pairs of weak compositions $\lambda = (\lambda_1, \dots, \lambda_n)$ and $\mu = (\mu_1, \dots, \mu_n)$ satisfying the conditions $\ell \ge \lambda_1 \ge \dots \ge \lambda_n$, $\max\{\mu_i : 1 \le i \le n\} \le k$, and $\lambda_i = \lambda_{i+1} \Rightarrow \mu_i \ge \mu_{i+1}$ (such pairs $(\lambda, \mu)$ are sometimes called bipartite partitions), and $\mathrm{inv}(\mu)$ is the number of inversion pairs in $\mu$. Some further specializations of Theorem 1.2 imply identities of Carlitz-MacMahon [6, 17] and Adin-Brenti-Roichman [1].

The structure of this paper is as follows. Section 2 reviews the representation theory of the 0-Hecke algebra. Section 3 studies the Stanley-Reisner ring of the Boolean algebra. Section 4 defines a 0-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra. The noncommutative and quasisymmetric characteristics are discussed in Section 5 and Section 6. Finally we give some remarks and questions for future research in Section 7, including a generalization to an action of the Hecke algebra of any finite Coxeter group on the Stanley-Reisner ring of the Coxeter complex.
## 2 Representation theory of the 0-Hecke algebra

We review the representation theory of the 0-Hecke algebra in this section. The (type A) Hecke algebra $H_n(q)$ is the associative $\mathbb{F}(q)$-algebra generated by $T_1, \dots, T_{n-1}$ with relations

$$ \left\{ \begin{array}{ll} (T_i + 1)(T_i - q) = 0, & 1 \le i \le n-1, \\ T_i T_j = T_j T_i, & 1 \le i, j \le n-1, |i-j| > 1, \\ T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1}, & 1 \le i \le n-2. \end{array} \right. \qquad (2) $$

It has an $\mathbb{F}(q)$-basis $\{T_w : w \in \mathfrak{S}_n\}$ where $T_w := T_{i_1} \cdots T_{i_k}$ if $w = s_{i_1} \cdots s_{i_k}$ is a reduced expression.

Specializing $q=1$ gives the group algebra of $\mathfrak{S}_n$, with $s_i = T_i|_{q=1}$ and $w = T_w|_{q=1}$. Let $w \in \mathfrak{S}_n$. The length of $w$ equals $\mathrm{inv}(w) := \#\{(i, j) : 1 \le i < j \le n, w(i) > w(j)\}$, and the descent set of $w$ is $D(w) = \{i : 1 \le i \le n-1, w(i) > w(i+1)\}$. We write $\mathrm{des}(w) := |D(w)|$ and $\mathrm{maj}(w) := \sum_{i \in D(w)} i$.

Let $\alpha$ be a (weak) composition of $n$, and let $\alpha^c$ be the composition of $n$ with $D(\alpha^c) = [n-1] \setminus D(\alpha)$. The parabolic subgroup $\mathfrak{S}_\alpha$ is the subgroup of $\mathfrak{S}_n$ generated by $\{s_i : i \in D(\alpha^c)\}$. The set of all minimal $\mathfrak{S}_\alpha$-coset representatives is $\mathfrak{S}^\alpha := \{w \in \mathfrak{S}_n : D(w) \subseteq D(\alpha)\}$. The descent class of $\alpha$ consists of the permutations in $\mathfrak{S}_n$ with descent set equal to $D(\alpha)$, and turns out to be an interval under the (left) weak order of $\mathfrak{S}_n$, denoted by $[w_0(\alpha), w_1(\alpha)]$. One sees that $w_0(\alpha)$ is the longest element of the parabolic subgroup $\mathfrak{S}_{\alpha^c}$, and $w_1(\alpha)$ is the longest element in $\mathfrak{S}^\alpha$ (c.f. Björner and Wachs [5, Theorem 6.2]).

Another interesting specialization of $H_n(q)$ is the 0-Hecke algebra $H_n(0)$, with generators $\pi_i = T_i|_{q=0}$ for $i = 1, \dots, n-1$, and an $\mathbb{F}$-basis $\{\pi_w = T_w|_{q=0} : w \in \mathfrak{S}_n\}$. Let $\pi_i := \pi_i + 1$. Then $\pi_1, \dots, \pi_{n-1}$ form another generating set for $H_n(0)$, with the same relations as (2) except $\pi_i^2 = \pi_i$, $1 \le i \le n-1$. The element $\pi_w := \pi_{i_1} \cdots \pi_{i_k}$ is well defined for any $w \in \mathfrak{S}_n$ with a reduced expression $w = s_{i_1} \cdots s_{i_k}$, and $\{\pi_w : w \in \mathfrak{S}_n\}$ is another $\mathbb{F}$-bases for $H_n(0)$. One can check that $\pi_w$ equals the sum of $\pi_u$ over all $u$ less than or equal to $w$ in the Bruhat order of $\mathfrak{S}_n$. In particular, $\pi_{w_0(\alpha)}$ is the sum of $\pi_u$ for all $u \in \mathfrak{S}_{\alpha^c}$.

Norton [18] decomposed the 0-Hecke algebra $H_n(0)$ into a direct sum of projective indecomposable submodules $\mathbf{P}_\alpha := H_n(0) \cdot \pi_{w_0(\alpha)}\pi_{w_0(\alpha^c)}$ for all $\alpha \models n$ (i.e. compositions of $n$). Each $\mathbf{P}_\alpha$ has an $\mathbb{F}$-basis $\{\pi_w \pi_{w_0(\alpha^c)} : w \in [w_0(\alpha), w_1(\alpha)]\}$. Its radical $\mathrm{rad} \mathbf{P}_\alpha$ is the unique maximal $H_n(0)$-submodule spanned by $\{\pi_w \pi_{w_0(\alpha^c)} : w \in (w_0(\alpha), w_1(\alpha)]\}$. Although $\mathbf{P}_\alpha$ itself is not necessarily simple, its top $\mathbf{C}_\alpha := \mathbf{P}_\alpha / \mathrm{rad} \mathbf{P}_\alpha$ is a one-dimensional simple $H_n(0)$-module with the action of $H_n(0)$ given by

$$ \pi_i = \begin{cases} -1, & \text{if } i \in D(\alpha), \\ 0, & \text{if } i \notin D(\alpha). \end{cases} $$

It follows from general representation theory of algebras (see e.g. [2, §I.5]) that $\{\mathbf{P}_\alpha : \alpha \models n\}$ and $\{\mathbf{C}_\alpha : \alpha \models n\}$ are the complete lists of pairwise non-isomorphic projective indecomposable and simple $H_n(0)$-modules, respectively.

Krob and Thibon [14] introduced a correspondence between $H_n(0)$-representations and the dual Hopf algebras QSym and NSym, which we review next. The Hopf algebra QSym has a free $\mathbb{Z}$-basis of fundamental quasisymmetric functions $F_\alpha$, and the dual Hopf algebra NSym has a dual basis of non-commutative ribbon Schur functions $s_\alpha$, for all compositions $\alpha$.

Let $M = M_0 \supseteq M_1 \supseteq \dots \supseteq M_k \supseteq M_{k+1} = 0$ be a composition series of $H_n(0)$-modules with simple factors $M_i/M_{i+1} \cong \mathbf{C}_{\alpha^{(i)}}$ for $i = 0, 1, \dots, k$. Then the quasisymmetric characteristic of $M$ is

$$ \mathrm{Ch}(M) := F_{\alpha^{(0)}} + \dots + F_{\alpha^{(k)}}. $$
The noncommutative characteristic of a projective $H_n(0)$-module $M \cong \mathbf{P}_{\alpha^{(1)}} \oplus \cdots \oplus \mathbf{P}_{\alpha^{(k)}}$ is

$$ \mathrm{ch}(M) := \mathbf{s}_{\alpha^{(1)}} + \cdots + \mathbf{s}_{\alpha^{(k)}}. $$

It is not hard to extend these characteristic maps to $H_n(0)$-modules with gradings and filtrations.

## 3 Stanley-Reisner ring of the Boolean algebra

In this section we study the Stanley-Reisner ring of the Boolean algebra. The Boolean algebra $\mathcal{B}_n$ is the ranked poset of all subsets of $[n] := \{1, 2, \dots, n\}$ ordered by inclusion, with minimum element $\emptyset$ and maximum element $[n]$. The rank of a subset of $[n]$ is defined as its cardinality. The Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ of the Boolean algebra $\mathcal{B}_n$ is the quotient of the polynomial algebra $\mathbb{F}[y_A : A \subseteq [n]]$ by the ideal $(y_A y_B : A \text{ and } B \text{ are incomparable in } \mathcal{B}_n)$. It has an $\mathbb{F}$-basis $\{y_M\}$ indexed by the multichains $M$ in $\mathcal{B}_n$, and is multigraded by the rank multisets $r(M)$ of the multichains $M$.

The symmetric group $\mathfrak{S}_n$ acts on the Boolean algebra $\mathcal{B}_n$ by permuting the integers $1, \dots, n$. This induces an $\mathfrak{S}_n$-action on the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$, preserving its multigrading. The invariant algebra $\mathbb{F}[\mathcal{B}_n]^{\mathfrak{S}_n}$ consists of all elements in $\mathbb{F}[\mathcal{B}_n]$ invariant under this $\mathfrak{S}_n$-action. One can show that $\mathbb{F}[\mathcal{B}_n]^{\mathfrak{S}_n} = \mathbb{F}[\Theta]$, where $\Theta := \{\theta_0, \dots, \theta_n\}$. Garsia [7] showed that $\mathbb{F}[\mathcal{B}_n]$ is a free $\mathbb{F}[\Theta]$-module on the basis of descent monomials

$$ Y_w := \prod_{i \in D(w)} y_{\{w(1), \dots, w(i)\}}, \quad \forall w \in \mathfrak{S}_n. \qquad (3) $$

There is an analogy between the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ and the polynomial ring $\mathbb{F}[X]$ via the transfer map $\tau : \mathbb{F}[\mathcal{B}_n] \to \mathbb{F}[X]$ defined by

$$ \tau(y_M) := \prod_{1 \le i \le k} \prod_{j \in A_i} x_j $$

for all multichains $M = (A_1 \subseteq \cdots \subseteq A_k)$ in $\mathcal{B}_n$. It is not a ring homomorphism (e.g. $y_{\{1\}}y_{\{2\}} = 0$ but $x_1x_2 \ne 0$). Nevertheless, it induces an isomorphism $\tau : \mathbb{F}[\mathcal{B}_n]/(\theta_0) \cong \mathbb{F}[X]$ of $\mathfrak{S}_n$-modules. Moreover, it sends the rank polynomials $\theta_1, \dots, \theta_n$ to the elementary symmetric polynomials $e_1, \dots, e_n$, and sends the descent monomials $Y_w$ in $\mathbb{F}[\mathcal{B}_n^*]$ defined by (3) to the corresponding descent monomials in $\mathbb{F}[X]$ for all $w \in \mathfrak{S}_n$.

**Example 3.1** The Boolean algebra $\mathcal{B}_3$ consists of all subsets of $\{1, 2, 3\}$. Its Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_3]$ is a free $\mathbb{F}[\Theta]$-module with a basis of descent monomials $Y_1 = 1$, $Y_{s_1} = y_2$, $Y_{s_2s_1} = y_3$, $Y_{s_2} = y_{13}$, $Y_{s_1s_2} = y_{23}$, $Y_{s_1s_2s_1} = y_{23}y_3$, where $\Theta$ consists of the rank polynomials $\theta_0 = y_\emptyset$, $\theta_1 := y_1 + y_2 + y_3$, $\theta_2 := y_{12} + y_{13} + y_{23}$, $\theta_3 := y_{123}$. The transfer map $\tau$ sends $\theta_1, \theta_2, \theta_3$ to $e_1, e_2, e_3$, and sends the six descent monomials in $\mathbb{F}[\mathcal{B}_3]$ to the six descent monomials $1, x_2, x_3, x_1x_3, x_2x_3, x_2x_3^2$ in $\mathbb{F}[x_1, x_2, x_3]$.

The homogeneous components of $\mathbb{F}[\mathcal{B}_n]$ are indexed by multisets with elements in $\{0, \dots, n\}$, or equivalently by weak compositions $\alpha$ of $n$. The $\alpha$-homogeneous component $\mathbb{F}[\mathcal{B}_n]_\alpha$ has an $\mathbb{F}$-basis $\{y_M : r(M) = D(\alpha)\}$. Denote by $\mathrm{Com}(n, k)$ the set of all weak compositions of $n$ with length $k$. If $M = (A_1 \subseteq \cdots \subseteq A_k)$ is a multichain of length $k$ in $\mathcal{B}_n$ then we set $A_0 := \emptyset$ and $A_{k+1} := [n]$ by convention. Define $\alpha(M) := (\alpha_1, \dots, \alpha_{k+1})$, where $\alpha_i = |A_i| - |A_{i-1}|$ for all $i \in [k+1]$. Then
$\alpha(M) \in \text{Com}(n, k+1)$ and $D(\alpha(M)) = r(M)$, i.e. $\alpha(M)$ indexes the homogeneous component containing $y_M$. Define $\sigma(M)$ to be the minimal element in $\mathfrak{S}_n$ which sends the standard multichain $[\alpha_1] \subseteq [\alpha_1 + \alpha_2] \subseteq \cdots \subseteq [\alpha_1 + \cdots + \alpha_k]$ with rank multiset $D(\alpha(M))$ to $M$. Then $\sigma(M) \in \mathfrak{S}^{\alpha(M)}$.

The map $M \mapsto (\alpha(M), \sigma(M))$ is a bijection between multichains of length $k$ in $B_n$ and the pairs $(\alpha, \sigma)$ of $\alpha \in \text{Com}(n, k+1)$ and $\sigma \in \mathfrak{S}^\alpha$. A short way to write down this encoding of $M$ is to insert bars at the descent positions of $\sigma(M)$. For example, the length-4 multichain $\{2\} \subseteq \{2\} \subseteq \{1, 2, 4\} \subseteq [4]$ in $B_4$ is encoded by $2||14||3|$.

There is another way to encode the multichain $M$. Let $p_i(M) := \min\{j \in [k+1] : i \in A_j\}, 1 \le i \le n$. So $p_i(M)$ is the first position where $i$ appears in $M$. One checks that

$$ \left\{ \begin{array}{l} p_i(M) > p_{i+1}(M) \Leftrightarrow i \in D(\sigma(M)^{-1}), \\ p_i(M) = p_{i+1}(M) \Leftrightarrow i \notin D(\sigma(M)^{-1}), D(s_i\sigma(M)) \not\subseteq D(\alpha(M)), \\ p_i(M) < p_{i+1}(M) \Leftrightarrow i \notin D(\sigma(M)^{-1}), D(s_i\sigma(M)) \subseteq D(\alpha(M)). \end{array} \right. \qquad (4) $$

The map $M \mapsto p(M) := (p_1(M), \dots, p_n(M))$ is an bijection between the set of multichains with length $k$ in $B_n$ and the set $[k+1]^n$ of all words of length $n$ on the alphabet $[k+1]$, for any fixed integer $k \ge 0$.

Let $p(M) = \mathbf{p} = (p_1, \dots, p_n) \in [k+1]^n$. Then $\text{inv}(p(M)) = \text{inv}(\sigma(M))$. Let $\mathbf{p}' := (p'_1, \dots, p'_k)$ where $p'_i := |\{j : p_j(M) \le i\}| = |A_i|$. Then the rank multiset of $M$ consists of $p'_1, \dots, p'_k$. Define $D(\mathbf{p}) := \{i \in [n-1] : p_i > p_{i+1}\}$. For example, the multichain $3|14||2|5$ corresponds to $\mathbf{p} = (2, 4, 1, 2, 5) \in [5]^5$, and one has $\mathbf{p}' = (1, 3, 3, 4)$, $D(2, 5, 1, 2, 4) = \{2\}$.

These two encodings (with slightly different notation) were already used by Garsia and Gessel [8] in their work on generating functions of multivariate distributions of permutation statistics.

## 4 0-Hecke algebra action

We saw an analogy between $\mathbb{F}[B_n]$ and $\mathbb{F}[X]$ in the last section. The usual $H_n(0)$-action on the polynomial ring $\mathbb{F}[X]$ is via the Demazure operators

$$ \bar{\pi}_i(f) := \frac{x_{i+1}f - x_{i+1}s_i f}{x_i - x_{i+1}}, \quad \forall f \in \mathbb{F}[X], 1 \le i \le n-1. \qquad (5) $$

The above definition is equivalent to

$$ \bar{\pi}_i(x_i^a x_{i+1}^b m) = \begin{cases} (x_i^{a-1} x_{i+1}^{b+1} + x_i^{a-2} x_{i+1}^{b+2} + \dots + \underline{x_i^b x_{i+1}^a})m, & \text{if } a > b, \\ 0, & \text{if } a = b, \\ -(\underline{x_i^a x_{i+1}^b} + x_i^{a+1} x_{i+1}^{b-1} + \dots + x_i^{b-1} x_{i+1}^{a+1})m, & \text{if } a < b. \end{cases} \qquad (6) $$

Here $m$ is any monomial in $\mathbb{F}[X]$ containing neither $x_i$ nor $x_{i+1}$. Denote by $\bar{\pi}'_i$ the operator obtained from (6) by taking only the leading term (underlined) in the lexicographic order of the result. Then $\bar{\pi}'_1, \dots, \bar{\pi}'_{n-1}$ realize another $H_n(0)$-action on $\mathbb{F}[X]$. We call it the transferred $H_n(0)$-action because it can be obtained by applying the transfer map $\tau$ to our $H_n(0)$-action on $\mathbb{F}[B_n]$, which we now define.

Let $M = (A_1 \subseteq \cdots \subseteq A_k)$ be a multichain in $B_n$. Recall that $p_i(M) := \min\{j \in [k+1] : i \in A_j\}, 1 \le i \le n$. We define

$$ \bar{\pi}_i(y_M) := \begin{cases} -y_M, & p_i(M) > p_{i+1}(M), \\ 0, & p_i(M) = p_{i+1}(M), \\ s_i(y_M), & p_i(M) < p_{i+1}(M) \end{cases} \qquad (7) $$
for $i = 1, \dots, n-1$. Applying the transfer map $\tau$ one recovers $\bar{\pi}'_i$. For instance, when $n=4$ one has

$$ 
\begin{align*}
\bar{\pi}_1(y_{1|34||2|}) &= y_{2|34||1|}, & \bar{\pi}'_1(x_1^4 x_2 x_3^3 x_4^3) &= x_1 x_2^4 x_3^3 x_4^3, \\
\bar{\pi}_2(y_{1|34||2|}) &= -y_{1|34||2|}, & \bar{\pi}'_2(x_1^4 x_2 x_3^3 x_4^3) &= -x_1^4 x_2 x_3^3 x_4^3, \\
\bar{\pi}_3(y_{1|34||2|}) &= 0, & \bar{\pi}'_3(x_1^4 x_2 x_3^3 x_4^3) &= 0.
\end{align*}
$$

One can check that $\bar{\pi}_1, \dots, \bar{\pi}_{n-1}$ realize an $H_n(0)$-action on $\mathbb{F}[\mathcal{B}_n]$ preserving the multigrading of $\mathbb{F}[\mathcal{B}_n]$. If one set $t_i = t^i$ for $i = 1, \dots, n$, then there is an isomorphism $\mathbb{F}[\mathcal{B}_n]/(\emptyset) \cong \mathbb{F}[X]$ of graded $H_n(0)$-modules (which can be given explicitly, but not via the transfer map $\tau$).

It is not hard to show that $\mathbb{F}[\mathcal{B}_n]^{H_n(0)} = \mathbb{F}[\Theta]$, where $\mathbb{F}[\mathcal{B}_n]^{H_n(0)}$ is the invariant algebra of the $H_n(0)$-action on $\mathbb{F}[\mathcal{B}_n]$, defined as

$$ \mathbb{F}[\mathcal{B}_n]^{H_n(0)} := \{f \in \mathbb{F}[\mathcal{B}_n] : \pi_i f = f, i = 1, \dots, n-1\}. $$

**Proposition 4.1** *The $H_n(0)$-action on $\mathbb{F}[\mathcal{B}_n]$ is $\Theta$-linear.*

Therefore the coinvariant algebra $\mathbb{F}[\mathcal{B}_n]/(\Theta)$ is a multigraded $H_n(0)$-module, which is isomorphic to the regular representation of $H_n(0)$ by Theorem 1.1. This cannot be obtained simply by applying the transfer map $\tau$, since $\tau$ is not a map of $H_n(0)$-modules.

# 5 Noncommutative characteristic

In this section we use the $H_n(0)$-action on the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ of the Boolean algebra $\mathcal{B}_n$ to provide a noncommutative analogue of the following remarkable result.

**Theorem 5.1 (Hotta-Springer [11], Garsia-Procesi [9])** For any partition $\mu = (0 < \mu_1 \le \dots \le \mu_k)$ of $n$, there exists an $\mathfrak{S}_n$-invariant ideal $J_\mu$ of $\mathbb{C}[X]$ such that $\mathbb{C}[X]/J_\mu$ is isomorphic to the cohomology ring of the Springer fiber indexed by $\mu$ and has graded Frobenius characteristic equal to the modified Hall-Littlewood symmetric function

$$ \tilde{H}_{\mu}(X; t) = \sum_{\lambda} t^{n(\mu)} K_{\lambda\mu}(t^{-1}) s_{\lambda} \quad \text{inside} \quad \text{Sym}[t] $$

where $n(\mu) := \mu_{k-1} + 2\mu_{k-2} + \dots + (k-1)\mu_1$ and $K_{\lambda\mu}(t)$ is the Kostka-Foulkes polynomial.

**Example 5.2** Tanisaki [19] gives a construction for the ideal $J_\mu$. If $\mu = (1^k, n-k)$ is a hook then $J_{1^k, n-k}$ is generated by $e_1, \dots, e_k$ and all monomials $x_{i_1} \cdots x_{i_{k+1}}$ with $1 \le i_1 < \cdots < i_{k+1} \le n$.

Now consider a composition $\alpha = (\alpha_1, \dots, \alpha_\ell)$. The major index of $\alpha$ is maj($\alpha$) := $\sum_{i \in D(\alpha)} i$. Viewing a partition $\mu = (0 < \mu_1 \le \mu_2 \le \dots)$ as a composition one has maj($\mu$) = $n(\mu)$. Recall that $\overleftarrow{\alpha} := (\alpha_\ell, \dots, \alpha_1)$ and $\alpha^c$ is the composition of $n$ with $D(\alpha^c) = [n-1] \setminus D(\alpha)$. We define $\alpha' := \overleftarrow{\alpha}^c = (\overleftarrow{\alpha})^c$. One can identify $\alpha$ with a ribbon diagram, i.e. a connected skew Young diagram without 2 by 2 boxes, which has row lengths $\alpha_1, \dots, \alpha_\ell$, ordered from bottom to top. Note that a ribbon diagram is a Young diagram if and only if it is a hook. One can check that $\alpha'$ is the transpose of $\alpha$; see the example below.

<figure><img src="image_1.png" /></figure>

$\alpha = (2, 3, 1, 1)$

<figure><img src="image_2.png" /></figure>

$\alpha^c = (1, 2, 1, 3)$

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

$\alpha' = (3, 1, 2, 1)$
Bergeron and Zabrocki [3] introduced a noncommutative modified Hall-Littlewood symmetric function

$$ \tilde{\mathbf{H}}_{\alpha}(\mathbf{x}; t) := \sum_{\beta \preceq \alpha} t^{\mathrm{maj}(\beta)} \mathbf{s}_{\beta} \quad \text{inside} \quad \mathbf{NSym}[t] \qquad (8) $$

and a $(q, t)$-analogue

$$ \tilde{\mathbf{H}}_{\alpha}(\mathbf{x}; q, t) := \sum_{\beta \models n} t^{c(\alpha, \beta)} q^{c(\alpha', \beta')} \mathbf{s}_{\beta} \quad \text{inside} \quad \mathbf{NSym}[q, t] \qquad (9) $$

for every composition $\alpha$, where $\mathbf{s}_\beta$ is the noncommutative ribbon Schur function indexed by $\beta$, and $c(\alpha, \beta) := \sum_{i \in D(\alpha) \cap D(\beta)} i$. In our earlier work [13] we provided a partial representation theoretic interpretation for $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; t)$ when $\alpha = (1^k, n-k)$ is a hook, using the $H_n(0)$-action on the polynomial ring $\mathbb{F}[X]$ by the Demazure operators.

**Theorem 5.3 ([13)** The ideal $J_\mu$ of $\mathbb{F}[X]$ is $H_n(0)$-invariant if and only if $\mu = (1^{n-k}, k)$ is a hook, and if that holds then $\mathbb{F}[X]/J_\mu$ becomes a graded projective $H_n(0)$-module with

$$ \mathrm{ch}_t(\mathbb{F}[X]/J_\mu) = \tilde{\mathbf{H}}_\mu(\mathbf{x}; t), $$

$$ \mathrm{Ch}_t(\mathbb{F}[X]/J_\mu) = \tilde{\mathbf{H}}_\mu(\mathbf{x}; t). $$

Now we switch to the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ and define $I_\alpha$ to be its ideal generated by

$$ \Theta_\alpha := \{\theta_i : i \in D(\alpha) \cup \{n\}\} \quad \text{and} \quad \{y_A : A \subseteq [n], |A| \notin D(\alpha) \cup \{n\}\} $$

for any composition $\alpha$ of $n$. The following result is a restatement of Theorem 1.1.

**Theorem 5.4** Let $\alpha$ be a composition of $n$. Then $\mathbb{F}[\mathcal{B}_n]/I_\alpha$ is a projective $H_n(0)$-module with multi-graded noncommutative characteristic equal to

$$ \tilde{\mathbf{H}}_{\alpha}(\mathbf{x}; t_1, \dots, t_{n-1}) := \sum_{\beta \preceq \alpha} t^{D(\beta)} \mathbf{s}_{\beta} \quad \text{inside} \quad \mathbf{NSym}[t_1, \dots, t_{n-1}]. $$

One has $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; t, t^2, \dots, t^{n-1}) = \tilde{\mathbf{H}}_\alpha(\mathbf{x}; t)$, and one obtains $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; q, t)$ from $\tilde{\mathbf{H}}_{1^n}(\mathbf{x}; t_1, \dots, t_{n-1})$ by taking $t_i = t^i$ for all $i \in D(\alpha)$, and $t_i = q^{n-i}$ for all $i \in [n-1] \setminus D(\alpha)$.

**Proof:** There is an $\mathbb{F}$-basis for $\mathbb{F}[\mathcal{B}_\alpha]/(\Theta_\alpha)$ given by the descent monomials $Y_w$ defined in (3) for all $w \in \mathfrak{S}^\alpha$. The result follows from the $H_n(0)$-action on this basis and (4). $\square$

The proof of this theorem is actually simpler than the proof of our partial interpretation for $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; t)$ in [13]. This is because $\pi_i$ sends a descent monomial in $\mathbb{F}[\mathcal{B}_n]$ to either 0 or $\pm 1$ times a descent monomial, but sends a descent monomial in $\mathbb{F}[X]$ to a polynomial in general (whose leading term is still a descent monomial). We view the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ (or $\mathbb{F}[\mathcal{B}_n]/(\emptyset)$) as a $q = 0$ analogue of the polynomial ring $\mathbb{F}[X]$. For an odd (i.e. $q = -1$) analogue, see Lauda and Russell [16].

**Remark 5.5** If $\alpha = (1^k, n-k)$ is a hook, one can check that the ideal $I_{1^k, n-k}$ of $\mathbb{F}[\mathcal{B}_n]$ has generators $\theta_1, \dots, \theta_k$ and all $y_A$ with $A \subseteq [n]$ and $|A| \notin [k]$. By Example 5.2, the images of these generators under the transfer map $\tau$ are the Tanisaki generators for the ideal $J_{1^k, n-k}$ of $\mathbb{F}[X]$, but $\tau(I_{1^k, n-k}) \neq J_{1^k, n-k}$.
For any composition $\alpha \models n$, one can view $\tilde{\mathbf{H}}_\alpha(\mathbf{x}; t_1, \dots, t_{n-1})$ as a modified version of

$$ \mathbf{H}_{\alpha} = \mathbf{H}_{\alpha}(\mathbf{x}; t_1, \dots, t_{n-1}) := \sum_{\beta \preceq \alpha} \underline{t}^{D(\alpha) \setminus D(\beta)} \mathbf{s}_{\beta}. $$

Below are some properties satisfied by $\mathbf{H}_\alpha$, generalizing the properties of $\mathbf{H}_\alpha(\mathbf{x}; t)$ given in [3].

**Proposition 5.6** Let $\alpha$ and $\beta$ be two compositions.

(i) $\mathbf{H}_\alpha(0, \dots, 0) = \mathbf{s}_\alpha, \mathbf{H}_\alpha(1, \dots, 1) = \mathbf{h}_\alpha$.

(ii) $\cup_{n \ge 0} \{\mathbf{H}_\alpha : \alpha \models n\}$ is a basis for $\mathbf{NSym}[t_1, t_2, \dots]$.

(iii) $\langle \mathbf{H}_\alpha, \mathbf{H}_\beta \rangle = (-1)^{|\alpha|+|\beta|} \delta_{\alpha, \beta^c}$ for any pair of compositions $\alpha$ and $\beta$.

(iv)

$$ \mathbf{H}_{\alpha} \cdot \mathbf{H}_{\beta} = \sum_{\gamma \preceq \beta} \left( \prod_{i \in D(\beta) \setminus D(\gamma)} (t_i - t_{|\alpha|+i}) \right) (\mathbf{H}_{\alpha\gamma} + (1 - t_{|\alpha|}) \mathbf{H}_{\alpha\triangleright\gamma}). $$

(v) If $n = |\alpha|$ and $t|n := (t_1, \dots, t_{n-1}, 1, t_1, \dots, t_{n-1}, 1, \dots)$ then

$$ \mathbf{H}_{\alpha}(\mathbf{x}; t_1, \dots, t_{n-1}) \mathbf{H}_{\beta}(\mathbf{x}; t|n) = \mathbf{H}_{\alpha\beta}(t|n). $$

# 6 Quasisymmetric characteristic

Now we study the quasisymmetric characteristic of $\mathbb{F}[\mathcal{B}_n]$. The following lemma follows easily from (4).

**Lemma 6.1** Let $\alpha$ be a weak composition of $n$. Then the $\alpha$-homogeneous component $\mathbb{F}[\mathcal{B}_n]_\alpha$ of the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$ is an $H_n(0)$-submodule of $\mathbb{F}[\mathcal{B}_n]$ with homogeneous multigrading $\underline{t}^{D(\alpha)}$ and isomorphic to the cyclic module $H_n(0)\pi_{w_0(\alpha^c)}$.

Since $\mathbb{F}[\mathcal{B}_n]_\alpha$ is a cyclic multigraded $H_n(0)$-module, we get an $\mathbb{N} \times \mathbb{N}^{n+1}$-multigraded quasisymmetric characteristic

$$ \mathrm{Ch}_{q, \underline{t}}(\mathbb{F}[\mathcal{B}_n]_\alpha) = \sum_{w \in \mathfrak{S}^\alpha} q^{\mathrm{inv}(w)} \underline{t}^{D(\alpha)} F_{D(w^{-1})} \quad (10) $$

where $q$ keeps track of the length filtration and $\underline{t}$ keeps track of the multigrading of $\mathbb{F}[\mathcal{B}_n]_\alpha$. This defines an $\mathbb{N} \times \mathbb{N}^{n+1}$-multigraded quasisymmetric characteristic for the Stanley-Reisner ring $\mathbb{F}[\mathcal{B}_n]$, which is explicitly given in Theorem 1.2 and restated below.

**Theorem 6.2** The $\mathbb{N} \times \mathbb{N}^{n+1}$-multigraded quasisymmetric characteristic of $\mathbb{F}[\mathcal{B}_n]$ is

$$ \begin{aligned} \mathrm{Ch}_{q, \underline{t}}(\mathbb{F}[\mathcal{B}_n]) &= \sum_{k \ge 0} \sum_{\alpha \in \mathrm{Com}(n, k+1)} \underline{t}^{D(\alpha)} \sum_{w \in \mathfrak{S}^\alpha} q^{\mathrm{inv}(w)} F_{D(w^{-1})} \\ &= \sum_{w \in \mathfrak{S}_n} \frac{q^{\mathrm{inv}(w)} \underline{t}^{D(w)} F_{D(w^{-1})}}{\prod_{0 \le i \le n} (1 - t_i)} \\ &= \sum_{k \ge 0} \sum_{\mathbf{p} \in [k+1]^n} t_{p'_1} \cdots t_{p'_k} q^{\mathrm{inv}(\mathbf{p})} F_{D(\mathbf{p})}. \end{aligned} $$
**Proof:** Use the two encodings of the multichains in $\mathcal{B}_n$ as well as the free $\mathbb{F}[\Theta]$-basis $\{Y_w : w \in \mathfrak{S}_n\}$ of descent monomials for $\mathbb{F}[\mathcal{B}_n]$ discussed in Section 3. $\square$

Next we explain here how this theorem specializes to (1), a result of Garsia and Gessel [8, Theorem 2.2] on the multivariate generating function of the permutation statistics $\text{inv}(w)$, $\text{maj}(w)$, $\text{des}(w)$, $\text{maj}(w^{-1})$, and $\text{des}(w^{-1})$ for all $w \in \mathfrak{S}_n$. First recall that

$$F_{\alpha} = \sum_{\substack{i_1 \ge \cdots \ge i_n \ge 1 \\ i \in D(\alpha) \Rightarrow i_j > i_{j+1}}} x_{i_1} \cdots x_{i_n}, \quad \forall \alpha \models n.$$

Let $\mathbf{ps}_{q;\ell}(F_{\alpha}) := F_{\alpha}(1, q, q^2, \dots, q^{\ell-1}, 0, 0, \dots)$, and $(u; q)_n := (1-u)(1-QU)(1-q^2U)\cdots(1-q^nU)$. It is not hard to check (see Gessel and Reutenauer [10, Lemma 5.2]) that

$$\sum_{\ell \ge 0} u^{\ell} \mathbf{ps}_{q; \ell+1}(F_{\alpha}) = \frac{q^{\text{maj}(\alpha)} u^{\text{des}(\alpha)}}{(u; q)_n}.$$

Then applying the linear transformation $\sum_{\ell \ge 0} u_1^{\ell} \mathbf{ps}_{q_1; \ell+1}$ and the specialization $t_i = q_2^i u_2$ for all $i = 0, 1, \dots, n$ to Theorem 6.2 we recover (1).

A further specialization of Theorem 6.2 gives a well known result which is often attributed to Carlitz [6] but actually dates back to MacMahon [17, Volume 2, Chapter 4].

**Corollary 6.3 (Carlitz-MacMahon)** Let $[k+1]_q := 1+q+q^2+\cdots+q^k$. Then

$$\frac{\sum_{w \in \mathfrak{S}_n} q^{\text{maj}(w)} u^{\text{des}(w)}}{(u; q)_n} = \sum_{k \ge 0} ([k+1]_q)^n u^k.$$

Theorem 6.2 also implies the following result, which was obtained by Adin, Brenti, and Roichman [1] from the Hilbert series of the coinvariant algebra $\mathbb{F}[X]/(\mathbb{F}[X]^+)^{\mathfrak{S}_n})$.

**Corollary 6.4 (Adin-Brenti-Roichman)** Let $\text{Par}(n)$ be the set of weak partitions $\lambda = (\lambda_1, \dots, \lambda_n)$ with $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$, and let $m(\lambda) = (m_0(\lambda), m_1(\lambda), \dots)$, where $m_j(\lambda) := \#\{1 \le i \le n : \lambda_i = j\}$. Then

$$\sum_{\lambda \in \text{Par}(n)} \binom{n}{m(\lambda)} \prod_{i=1}^{n} q_i^{\lambda_i} = \frac{\sum_{w \in \mathfrak{S}_n} \prod_{i \in D(w)} q_1 \cdots q_i}{(1-q_1)(1-q_1q_2)\cdots(1-q_1 \cdots q_n)}.$$

# 7 Remarks and questions for future research

## 7.1 Hecke algebra action

It is well known that the symmetric group $\mathfrak{S}_n$ is the Coxeter group of type $A_{n-1}$. The Stanley-Reisner ring of $\mathcal{B}_n$ is essentially the Stanley-Reisner ring of the Coxeter complex of $\mathfrak{S}_n$. The Hecke algebra $H_W(q)$ can be defined for any finite Coxeter group $W$. We can generalize our action $H_n(0)$-action on $\mathbb{F}[\mathcal{B}_n]$ to an $H_W(q)$-action on the Stanley-Reisner ring $\mathbb{F}[q][\Delta(W)]$ of the Coxeter complex $\Delta(W)$ of any finite Coxeter group $W$. We show similar results for this $H_W(q)$-action.
## 7.2 Gluing the group algebra and the 0-Hecke algebra

The group algebra $\mathbb{F}W$ of a finite Coxeter group $W$ naturally admits both actions of $W$ and $H_W(0)$. Hivert and Thiéry [12] defined the *Hecke group algebra* of $W$ by gluing these two actions. In type $A$, one can also glue the usual actions of $\mathfrak{S}_n$ and $H_n(0)$ on the polynomial ring $\mathbb{F}[X]$, but the resulting algebra is different from the Hecke group algebra of $\mathfrak{S}_n$.

Now one has a $W$-action and an $H_W(0)$-action on the Stanley-Reisner ring $\mathbb{F}[\Delta(W)]$. What can we say about the algebra generated by the operators $s_i$ and $\pi_i$ on $\mathbb{F}[\Delta(W)]$? Is it the same as the Hecke group algebra of $W$? If not, what properties (dimension, bases, presentation, simple and projective indecomposable modules, etc.) does it have?

## 7.3 *Tits Building*

Let $\Delta(G)$ be the *Tits building* of the general linear group $G = GL(n, \mathbb{F}_q)$ and its usual BN-pair over a finite field $\mathbb{F}_q$; see e.g. Björner [4]. The Stanley-Reisner ring $\mathbb{F}[\Delta(G)]$ is a $q$-analogue of $\mathbb{F}[B_n]$. The nonzero monomials in $\mathbb{F}[\Delta(G)]$ are indexed by multiflags of subspaces of $\mathbb{F}_q^n$, and there are $q^{\text{inv}(w)}$ many multiflags corresponding to a given multichain $M$ in $B_n$, where $w = \sigma(M)$. Can one obtain the multivariate quasisymmetric function identities in Theorem 1.2 by defining a nice $H_n(0)$-action on $\mathbb{F}[\Delta(G)]$?

## Acknowledgements

The author is grateful to Victor Reiner for providing valuable suggestions. He also thanks Ben Braun and Jean-Yves Thibon for helpful conversations and email correspondence.

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