# Unital versions of the higher order peak algebras

Marcelo Aguiar¹†, Jean-Christophe Novelli² and Jean-Yves Thibon² ‡

¹ Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

² Institut Gaspard Monge, Université Paris-Est Marne-la-Vallée, 77454 Marne-la-Vallée cedex 2, France

**Abstract.** We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411–430], and show that they can be obtained as homomorphic images of certain subalgebras of the Mantaci-Reutenauer algebras of type *B*. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781–2824].

**Résumé.** Nous construisons des extensions unitaires des algèbres de pics d'ordre supérieur définies par Krob et le troisième auteur dans [Ann. Comb. 9 (2005), 411–430], et nous montrons qu'elles peuvent être obtenues comme images homomorphes de certaines sous-algèbres des algèbres de Mantaci-Reutenauer de type *B*. Ceci généralise un résultat dû à Bergeron, Nyman et au premier auteur [Trans. AMS 356 (2004), 2781–2824].

**Keywords:** Descent algebras, Noncommutative symmetric functions, Peak algebras

## 1 Introduction

A descent of a permutation $\sigma \in \mathfrak{S}_n$ is an index $i$ such that $\sigma(i) > \sigma(i+1)$. A descent is a peak if moreover $i > 1$ and $\sigma(i) > \sigma(i-1)$. The sums of permutations with a given descent set span a subalgebra of the group algebra, the descent algebra $\Sigma_n$. The peak algebra $\mathring{P}_n$ of $\mathfrak{S}_n$ is a subalgebra of its descent algebra, spanned by sums of permutations having the same peak set. This algebra has no unit. Descent algebras can be defined for all finite Coxeter groups [19]. In [2], it is shown that the peak algebra of $\mathfrak{S}_n$ can be naturally extended to a unital algebra, which is obtained as a homomorphic image of the descent algebra of the hyperoctahedral group $B_n$.

The direct sum of the peak algebras turns out to be a Hopf subalgebra of the direct sum of all descent algebras, which can itself be identified with Sym, the Hopf algebra of noncommutative symmetric functions [9]. As explained in [5], it turns out that a fair amount of results on the peak algebras can be deduced from the case $q = -1$ of a $q$-identity of [11]. Specializing $q$ to other roots of unity, Krob and the third author introduced and studied higher order peak algebras in [12]. Again, these are non-unital, and it is natural to ask whether the construction of [2] can be extended to this case.

<footnote>†Partially supported by NSF grant DMS-0600973.</footnote>

<footnote>‡Partially supported by ANR grant 06-BLAN-0380.</footnote>
We will show that this is indeed possible. We first construct the unital versions of the higher order peak algebras by a simple manipulation of generating series. We then show that they can be obtained as homomorphic images of the *Mantaci-Reutenauer algebras* of type *B*. Hence no Coxeter groups other than $B_n$ and $\mathfrak{S}_n$ are involved in the process; in fact, the construction is related to the notion of superization, as defined in [16], rather than to root systems or wreath products.

# 2 Notations and background

## 2.1 Noncommutative symmetric functions

We will assume familiarity with the notations of [9] and with the main results of [12]. We recall a few definitions for the convenience of the reader.

The Hopf algebra of noncommutative symmetric functions is denoted by $\mathbf{Sym}$, or by $\mathbf{Sym}(A)$ if we consider the realization in terms of an auxiliary alphabet $A$. Linear bases of $\mathbf{Sym}_n$ are labelled by compositions $I = (i_1, \dots, i_r)$ of $n$ (we write $I \vdash n$). The noncommutative complete and elementary functions are denoted by $S_n$ and $\Lambda_n$, and $S^I = S_{i_1} \cdots S_{i_r}$. The ribbon basis is denoted by $R_I$. The descent set of $I$ is $\mathrm{Des}(I) = \{i_1, i_1+i_2, \dots, i_1+\cdots+i_{r-1}\}$. The descent composition of a permutation $\sigma \in \mathfrak{S}_n$ is the composition $I = D(\sigma)$ of $n$ whose descent set is the descent set of $\sigma$.

Recall from [8] that for an infinite totally ordered alphabet $A$, $\mathbf{FQSym}(A)$ is the subalgebra of $\mathbb{C}\langle A\rangle$ spanned by the polynomials

$$ \mathbf{G}_{\sigma}(A) = \sum_{\mathrm{std}(w)=\sigma} w, \qquad (1) $$

that is, the sum of all words in $A^n$ whose standardization is the permutation $\sigma \in \mathfrak{S}_n$. The noncommutative ribbon Schur function $R_I \in \mathbf{Sym}$ is then

$$ R_I = \sum_{D(\sigma)=I} \mathbf{G}_{\sigma}. \qquad (2) $$

This defines a Hopf embedding $\mathbf{Sym} \to \mathbf{FQSym}$. The Hopf algebra $\mathbf{FQSym}$ is self-dual under the pairing $(\mathbf{G}_{\sigma}, \mathbf{G}_{\tau}) = \delta_{\sigma, \tau-1}$ (Kronecker symbol). Let $\mathbf{F}_{\sigma} := \mathbf{G}_{\sigma-1}$, so that $\{\mathbf{F}_{\sigma}\}$ is the dual basis of $\{\mathbf{G}_{\sigma}\}$. The internal product $*$ of $\mathbf{FQSym}$ is induced by composition $\circ$ in $\mathfrak{S}_n$ in the basis $\mathbf{F}$, that is,

$$ \mathbf{F}_{\sigma} * \mathbf{F}_{\tau} = \mathbf{F}_{\sigma \circ \tau} \quad \text{and} \quad \mathbf{G}_{\sigma} * \mathbf{G}_{\tau} = \mathbf{G}_{\tau \circ \sigma}. \qquad (3) $$

Each subspace $\mathbf{Sym}_n$ is stable under this operation, and anti-isomorphic to the descent algebra $\Sigma_n$ of $\mathfrak{S}_n$. For $f_i \in \mathbf{FQSym}$ and $g \in \mathbf{Sym}$, we have the splitting formula

$$ (f_1 \dots f_r) * g = \mu_r \cdot (f_1 \otimes \dots \otimes f_r) *_r \Delta^r g, \qquad (4) $$

where $\mu_r$ is $r$-fold multiplication, and $\Delta^r$ the iterated coproduct with values in the $r$-th tensor power.

## 2.2 The Mantaci-Reutenauer algebra of level 2

We denote by $\mathbf{MR}$ the free product $\mathbf{Sym} * \mathbf{Sym}$ of two copies of the Hopf algebra of noncommutative symmetric functions [14]. That is, $\mathbf{MR}$ is the free associative algebra on two sequences $(S_n)$ and $(S_{\tilde{n}})$ ($n \ge 1$). We regard the two copies of $\mathbf{Sym}$ as noncommutative symmetric functions on two auxiliary
alphabets: $S_n = S_n(A)$ and $S_{\bar{n}} = S_n(\bar{A})$. We denote by $F \mapsto \bar{F}$ the involutive automorphism which exchanges $S_n$ and $S_{\bar{n}}$. The bialgebra structure is defined by the requirement that the series

$$ \sigma_1 = \sum_{n \ge 0} S_n \quad \text{and} \quad \bar{\sigma}_1 = \sum_{n \ge 0} S_{\bar{n}} \qquad (5) $$

are grouplike. The internal product of $\mathbf{MR}$ can be computed from the splitting formula and the conditions that $\sigma_1$ is neutral, $\bar{\sigma}_1$ is central, and $\bar{\sigma}_1 * \bar{\sigma}_1 = \sigma_1$.

In [15], an embedding of $\mathbf{MR}$ in the Hopf algebra $\mathbf{BFQSym}$ of free quasi-symmetric functions of type $B$ (spanned by colored permutations) is described. Under this embedding, left $*$-multiplication by $\Lambda_n = G_{nn-1...2,1}$ corresponds to right multiplication by $nn-1...2, 1$ in the group algebra of $B_n$. This implies that left $*$-multiplication by $\lambda_1$ is an involutive anti-automorphism of $\mathbf{BFQSym}$, hence of $\mathbf{MR}$.

## 2.3 Noncommutative symmetric functions of type $B$

The hyperoctahedral analogue $\mathbf{BSym}$ of $\mathbf{Sym}$, defined in [6], is the right $\mathbf{Sym}$-module freely generated by another sequence $(\tilde{S}_n)$ ($n \ge 0$, $\tilde{S}_0 = 1$) of homogeneous elements, with $\tilde{\sigma}_1$ grouplike. This is a coalgebra, but not an algebra. It is endowed with an internal product, for which each homogeneous component $\mathbf{BSym}_n$ is anti-isomorphic to the descent algebra of $B_n$.

# 3 Solomon descent algebras of type $B$

## 3.1 Descents in $B_n$

The hyperoctahedral group $B_n$ is the group of signed permutations. A signed permutation can be denoted by $w = (\sigma, \epsilon)$ where $\sigma$ is an ordinary permutation and $\epsilon \in \{\pm 1\}^n$, such that $w(i) = \epsilon_i \sigma(i)$. If we set $w(0) = 0$, then, $i \in [0, n-1]$ is a descent of $w$ if $w(i) > w(i+1)$. Hence, the descent set of $w$ is a subset $D = \{i_0, i_0 + i_1, \dots, i_0 + i_1 + \dots + i_{r-1}\}$ of $[0, n-1]$. We then associate to $D$ a so-called type-$B$ composition (a composition whose first part can be zero) $(i_0 - 0, i_1, \dots, i_{r-1}, n - i_{r-1})$. The sum of all signed permutations whose descent set is contained in $D$ is mapped to $\tilde{S}^I := \tilde{S}_{i_0} S^{I'}$ by Chow's anti-isomorphism [6], where $I' = (i_1, \dots, i_r)$.

## 3.2 Noncommutative supersymmetric functions

An embedding of $\mathbf{BSym}$ as a sub-coalgebra and sub-$\mathbf{Sym}$-module of $\mathbf{MR}$ can be deduced from [14]. To describe it, let us define, for $F \in \mathbf{Sym}(A)$,

$$ F^{\sharp} = F(A|\bar{A}) = F(A - q\bar{A})|_{q=-1} \qquad (6) $$

(the supersymmetric version of $F$). The superization of $F \in \mathbf{Sym}(A)$ can also be given by

$$ F^{\sharp} = F * \sigma_1^{\sharp}. \qquad (7) $$

Indeed, $\sigma_1^{\sharp}$ is grouplike, and for $F = S^I$, the splitting formula gives

$$ (S_{i_1} \cdots S_{i_r}) * \sigma_1^{\sharp} = \mu_r [(S_{i_1} \otimes \cdots \otimes S_{i_r}) * (\sigma_1^{\sharp} \otimes \cdots \otimes \sigma_1^{\sharp})] = S^{I^{\sharp}}. \qquad (8) $$
We have

$$ \sigma_1^\sharp = \bar{\lambda}_1 \sigma_1 = \sum \Lambda_i S_j. \qquad (9) $$

The element $\bar{\sigma}_1$ is central for the internal product, and

$$ \bar{\sigma}_1 * F = \bar{F} = F * \bar{\sigma}_1. \qquad (10) $$

Hence,

$$ \bar{\sigma}_1 * \sigma_1^\sharp = \lambda_1 \bar{\sigma}_1 := \sigma_1^\flat. \qquad (11) $$

The basis element $\tilde{S}^I$ of $BSym$, where $I = (i_0, i_1, \dots, i_r)$ is a type $B$-composition, can be embedded as

$$ \tilde{S}^I = S_{i_0}(A) S^{i_1 i_2 \cdots i_r} (A | \bar{A}). \qquad (12) $$

We will identify $BSym$ with its image under this embedding.

### 3.3 A proof that $BSym$ is $*-\text{stable}$

We are now in a position to understand why $BSym$ is a $*-\text{subalgebra of MR}$. The argument will be extended below to the case of unital peak algebras. Let $F, G \in \text{Sym}$. We want to understand why $\sigma_1 F^\sharp * \sigma_1 G^\sharp$ is in $\text{BSym}$. Using the splitting formula, we rewrite this as

$$ \mu[(\sigma_1 \otimes F^\sharp) * \Delta\sigma_1 \Delta G^\sharp] = \sum_{(G)} (\sigma_1 G_{(1)}^\sharp)(F^\sharp * \sigma_1 G_{(2)}^\sharp). \qquad (13) $$

We now only have to show that each term $F^\sharp * \sigma_1 G_{(2)}^\sharp$ is in $\text{Sym}^\sharp$. We may assume that $F = S^I$, and for any $G \in \text{Sym}$,

$$ S^{I^\sharp} * \sigma_1 G^\sharp = \sum_{(G)} \mu_r [(S_{i_1}^\sharp \otimes \cdots \otimes S_{i_r}^\sharp) * (\sigma_1 G_{(1)}^\sharp \otimes \cdots \otimes \sigma_1 G_{(r)}^\sharp)] \qquad (14) $$

so that it is sufficient to prove the property for $F = S_n$. Now,

$$ \begin{aligned} \sigma_1^\sharp * \sigma_1 G^\sharp &= (\bar{\lambda}_1 \sigma_1) * \sigma_1 G^\sharp \\ &= \sum_{(G)} (\bar{\lambda}_1 * \sigma_1 G_{(1)}^\sharp)(\sigma_1 G_{(2)}^\sharp) \end{aligned} \qquad (15) \\ = \sum_{(G)} (\bar{\sigma}_1 * \lambda_1 * \sigma_1 G_{(1)}^\sharp) \cdot \sigma_1 \cdot G_{(2)}^\sharp $$

Now,

$$ \lambda_1 * \sigma_1 G_{(1)}^\sharp = (\lambda_1 * G_{(1)}^\sharp)(\lambda_1 * \sigma_1) = (\lambda_1 * G_{(1)}^\sharp)\lambda_1, \qquad (16) $$

since $\lambda_1$ is an anti-automorphism. We then get

$$ \begin{aligned} \sigma_1^\sharp * \sigma_1 G^\sharp &= \sum_{(G)} (\bar{\sigma}_1 * ((\lambda_1 * G_{(1)}^\sharp)\lambda_1) \cdot \sigma_1 \cdot G_{(2)}^\sharp) \\ &= \sum_{(G)} (\bar{\sigma}_1 * \lambda_1 * G_{(1)}^\sharp) \cdot (\bar{\sigma}_1 * \lambda_1) \sigma_1 \cdot G_{(2)}^\sharp \end{aligned} \qquad (17) \\ = \sum_{(G)} (\bar{\lambda}_1 * G_{(1)}^\sharp) \cdot \sigma_1^\sharp \cdot G_{(2)}^\sharp $$
Now, the result will follow if we can prove that $\bar{\lambda}_1 * G^\sharp$ is in $\mathbf{Sym}^\sharp$ for any $G \in \mathbf{Sym}$.
For $G = S^I$,

$$ \bar{\lambda}_1 * S^I = \lambda_1 * \bar{\sigma}_1 * S^I * \sigma_1^\sharp = \lambda_1 * S^I * \bar{\sigma}_1 * \sigma_1^\sharp = \lambda_1 * S^I * \sigma_1^b. \quad (18) $$

Since left $*$-multiplication by $\lambda_1$ in an anti-automorphism, we only need to prove that $\lambda_1 * S_n^b$ is of the form $G^\sharp$. And indeed,

$$ \begin{aligned} \lambda_1 * S_n^b &= \sum_{i+j=n} \lambda_1 * (\Lambda_i S_j) \\ &= \sum_{i+j=n} (\lambda_1 * S_j)(\lambda_1 * \Lambda_i) \\ &= \sum_{i+j=n} \Lambda_j S_i = S_n^\sharp. \end{aligned} \quad (19) $$

This concludes the proof that $\mathbf{BSym}$ is a $*$-subalgebra of $\mathbf{BFQSym}$.

# 4 Unital versions of the higher order peak algebras

As shown in [5], much of the theory of the peak algebra can be deduced from a formula of [11] for $R_I((1-q)A)$, in the special case $q = -1$. In [12], this formula was studied in the case where $q$ is an arbitrary root of unity, and higher order analogs of the peak algebra were obtained. In [2], it was shown that the classical peak algebra can be extended to a unital algebra, which is obtained as a homomorphic image of the descent algebra of type $B$. In this section, we construct unital extensions of the higher order peak algebras.

Let $q$ be a primitive $r$-th root of unity. All objects introduced below will depend on $q$ (and $r$), although this dependence will not be made explicit in the notation. We denote by $\theta_q$ the endomorphism of $\mathbf{Sym}$ defined by

$$ \tilde{f} = \theta_q(f) = f((1-q)A) = f(A) * \sigma_1((1-q)A). \quad (20) $$

We denote by $\tilde{P}$ the image of $\theta_q$ and by $\mathcal{P}$ the right $\tilde{P}$-module generated by the $S_n$ for $n \ge 0$. Note that $\tilde{P}$ is by definition a left $*$-ideal of $\mathbf{Sym}$.

**Theorem 4.1** *P* is a unital $*$-subalgebra of $\mathbf{Sym}$. Its Hilbert series is

$$ \sum_{n \ge 0} \dim \mathcal{P}_n t^n = \frac{1}{1 - t - t^2 - \dots - t^r}. \quad (21) $$

*Proof* – Since the internal product of homogeneous elements of different degrees is zero, it is enough to show that, for any $f, g \in \mathbf{Sym}$, $\sigma_1 \tilde{f} * \sigma_1 \tilde{g}$ is in $\mathcal{P}$. Thanks to the splitting formula,

$$ \begin{aligned} \sigma_1 \tilde{f} * \sigma_1 \tilde{g} &= \mu[(\sigma_1 \otimes \tilde{f}) * \sum_{(g)} \sigma_1 \tilde{g}_{(1)} \otimes \sigma_1 \tilde{g}_{(2)}] \\ &= \sum_{(g)} (\sigma_1 \tilde{g}_{(1)})(\tilde{f} * \sigma_1 \tilde{g}_{(2)}). \end{aligned} \quad (22) $$
Thus, it is enough to check that $\tilde{f} * \sigma_1 \tilde{h}$ is in $\mathcal{P}$ for any $f, h \in \text{Sym}$. Now,

$$ \tilde{f} * \sigma_1 \tilde{h} = f * \sigma_1((1-q)A) * \sigma_1 \tilde{h}, \qquad (23) $$

and since $\mathcal{P}$ is a $\text{Sym}$ left $*$-ideal, we only have to show that $\sigma_1((1-q)A) * \sigma_1 \tilde{h}$ is in $\mathcal{P}$. One more splitting yields

$$ \begin{aligned} \sigma_1((1-q)A) * \sigma_1 \tilde{h} &= (\lambda_{-q} \sigma_1) * \sigma_1 \tilde{h} \\ &= \mu[(\lambda_{-q} \otimes \sigma_1) * \sum_{(h)} \sigma_1 \tilde{h}_{(1)} \otimes \sigma_1 \tilde{h}_{(2)}] \\ &= \sum_{(h)} (\lambda_{-q} * \sigma_1 \tilde{h}_{(1)})(\sigma_1 \tilde{h}_{(2)}) \\ &= \sum_{(h)} (\lambda_{-q} * \tilde{h}_{(1)}) \lambda_{-q} \sigma_1 \tilde{h}_{(2)} \end{aligned} \qquad (24) $$

(since left $*$-multiplication by $\lambda_{-q}$ is an anti-automorphism, namely the composition of the antipode and $q^{\text{degree}}$). The first parentheses $(\lambda_{-q} * \tilde{h}_{(1)})$ are in $\mathcal{P}$ since it is a left $*$-ideal. The middle term is $\sigma_1((1-q)A)$, and the last one is in $\mathcal{P}$ by definition.

Recall from [12, Prop. 3.5] that the Hilbert series of $\mathcal{P}$ is

$$ \sum_{n \ge 0} \dim \mathcal{P}_n t^n = \frac{1 - t^r}{1 - t - t^2 - \dots - t^r}. \qquad (25) $$

From [12, Lemma 3.13 and Eq. (3.9)], it follows that $S_n \notin \mathcal{P}$ if and only if $n \equiv 0 \mod r$, so that the Hilbert series of $\mathcal{P}$ is

$$ \sum_{n \ge 0} \dim \mathcal{P}_n t^n = \frac{1}{1 - t - t^2 - \dots - t^r}. \qquad (26) $$

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

# 5 Back to the Mantaci-Reutenauer algebra

The above proofs are in fact special cases of a master calculation in the Mantaci-Reutenauer algebra, which we carry out in this section.

Let $q$ be an arbitrary complex number or an indeterminate, and define, for any $F \in \text{MR}$,

$$ F^{\sharp} = F * \sigma_1(A - q\bar{A}) = F * \sigma_1^{\sharp}. \qquad (27) $$

Since $\sigma_1^{\sharp}$ is grouplike, it follows from the splitting formula that

$$ F \mapsto F^{\sharp} \qquad (28) $$

is an automorphism of $\text{MR}$ for the Hopf structure. In addition, it is clear from the definition that it is also an endomorphism of left $*$-modules. We refer to it as the $\sharp$ transform.
We now define

$$ \mathring{Q} = \mathbf{MR}^{\sharp}, \qquad (29) $$

the image of the $\sharp$ transform. Since the latter is an endomorphism of Hopf algebras and of left $*$-modules, $\mathring{Q}$ is both a Hopf subalgebra of $\mathbf{MR}$ and a left $*$-ideal. When $q$ is a root of unity, its image under the specialization $\bar{A} = A$ is the non-unital peak algebra $\mathring{P}$ of Section 4 (and for generic $q$, it is $\mathbf{Sym}$).

Let $\mathcal{Q}$ be the right $\mathring{Q}$-module generated by the $S_n$, for all $n \ge 0$. Clearly, the identification $\bar{A} = A$ maps $\mathcal{Q}$ onto $\mathcal{P}$, the unital peak algebra of Section 4.

**Theorem 5.1** $\mathcal{Q}$ is a $*$-subalgebra of $\mathbf{MR}$, containing $\mathring{Q}$ as a left ideal.

*Proof* – Let $F, G \in \mathbf{MR}$. As above, we want to show that $\sigma_1 F^{\sharp} * \sigma_1 G^{\sharp}$ is in $\mathcal{Q}$. Using the splitting formula, we rewrite this as

$$ \mu[(\sigma_1 \otimes F^{\sharp}) * \Delta\sigma_1 \Delta G^{\sharp}] = \sum_{(G)} (\sigma_1 G_{(1)}^{\sharp})(F^{\sharp} * \sigma_1 G_{(2)}^{\sharp}) \qquad (30) $$

and we only have to show that each term $F^{\sharp} * \sigma_1 G_{(2)}^{\sharp}$ is in $\mathring{Q}$. We may assume that $F = S^I$, where $I$ is now a bicolored composition, and for any $G \in \mathbf{MR}$,

$$ S^{I^{\sharp}} * \sigma_1 G^{\sharp} = \sum_{(G)} \mu_r [(S_{i_1}^{\sharp} \otimes \cdots \otimes S_{i_r}^{\sharp}) * (\sigma_1 G_{(1)}^{\sharp} \otimes \cdots \otimes \sigma_1 G_{(r)}^{\sharp})] \qquad (31) $$

so that it is sufficient to prove the property for $F = S_n$ or $S_{\bar{n}}$. Now,

$$ \begin{aligned} \sigma_1^{\sharp} * \sigma_1 G^{\sharp} &= (\bar{\lambda}_{-q}\sigma_1) * \sigma_1 G^{\sharp} \\ &= \sum_{(G)} (\bar{\lambda}_{-q}1 * \sigma_1 G_{(1)}^{\sharp})(\sigma_1 G_{(2)}^{\sharp}) \end{aligned} \qquad (32) \\ = \sum_{(G)} (\bar{\lambda}_{-q} * G_{(1)}^{\sharp}) \cdot \sigma_1^{\sharp} \cdot G_{(2)}^{\sharp} $$

which is in $\mathring{Q}$, since it is a subalgebra and a left $*$-ideal, and similarly,

$$ \begin{aligned} \bar{\sigma}_1^{\sharp} * \sigma_1 G^{\sharp} &= (\lambda_{-q}\bar{\sigma}_1) * \sigma_1 G^{\sharp} \\ &= \sum_{(G)} (\lambda_{-q} * \sigma_1 G_{(1)}^{\sharp})(\bar{\sigma}_1 \bar{G}_{(2)}^{\sharp}) \end{aligned} \qquad (33) \\ = \sum_{(G)} (\lambda_{-q} * G_{(1)}^{\sharp}) \cdot \bar{\sigma}_1^{\sharp} \cdot \bar{G}_{(2)}^{\sharp} $$

is also in $\mathring{Q}$.

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

The various algebras introduced in this paper and their interrelationships are summarized in the following diagram.

<figure><img src="image_3.png" /></figure>
Note that in the special case $q = -1$, by the results of Section 3.3, $Q_n$ is the (Solomon) descent algebra of $B_n$, $Q$ is isomorphic to $BSym$, and $P$ is the unital peak algebra of [2].

# 6 Further developments

## 6.1 Inversion of the generic $\sharp$ transform

For generic $q$, the endomorphism (27) of $\mathbf{MR}$ is invertible; therefore

$$ \mathring{Q} \sim \mathbf{MR}. \qquad (35) $$

The inverse endomorphism of $\mathbf{MR}$ arises from the transformation of alphabets

$$ A \mapsto (q\bar{A} + A)/(1 - q^2), \qquad (36) $$

which is to be understood in the following sense:

$$ \sigma_1 \left( \frac{q\bar{A} + A}{1 - q^2} \right) := \prod_{k \ge 0} \sigma_{q^{2k+1}}(\bar{A}) \sigma_{q^{2k}}(A). \qquad (37) $$

Indeed,

$$ \begin{aligned} \sigma_1 \left( \frac{q\bar{A} + A}{1 - q^2} \right) * \sigma_1(A - q\bar{A}) &= \prod_{k \ge 0} \sigma_{q^{2k+1}}(\bar{A} - q\bar{A}) \sigma_{q^{2k}}(A - q\bar{A}) \\ &= \prod_{k \ge 0} \lambda_{-q^{2k+2}}(A) \sigma_{q^{2k+1}}(\bar{A}) \lambda_{-q^{2k+1}}(\bar{A}) \sigma_{q^{2k}}(A) \qquad (38) \\ &= \sigma_1(A). \end{aligned} $$

By normalizing the term of degree $n$ in (37), we obtain $B_n$-analogs of the $q$-Klyachko elements defined in [9]:

$$ K_n(q; A, \bar{A}) := \prod_{i=1}^{n} (1 - q^{2i}) S_n \left( \frac{q\bar{A} + A}{1 - q^2} \right) = \sum_{I \vdash n} q^{2 \operatorname{maj}(I)} R_I(q\bar{A} + A). \qquad (39) $$

This expression can be completely expanded on signed ribbons. From the expression of $R_I$ in $\mathbf{FQSym}$, we have

$$ R_I(\bar{A} + A) = \sum_{C(\sigma)=I} \mathbf{G}_{\sigma}(\bar{A} + A) \qquad (40) $$

where $\bar{A} + A$ is the ordinal sum. If we order $\bar{A}$ by

$$ \bar{a}_1 < \bar{a}_2 < \dots < \bar{a}_k < \dots \qquad (41) $$

then, arguing as in [16], we have

$$ \mathbf{G}_{\sigma}(\bar{A} + A) = \sum_{\text{std}(\tau, \epsilon) = \sigma} \mathbf{G}_{\tau, \epsilon} \qquad (42) $$
so that

$$R_I(\bar{A} + A) = \sum_{\rho(J)=I} R_J \qquad (43)$$

where for a signed composition $J = (J, \epsilon)$, the unsigned composition $\rho(J)$ is defined as the shape of $\text{std}(\sigma, \epsilon)$, where $\sigma$ is any permutation of shape $J$.

Replacing $\bar{A}$ by $q\bar{A}$, one obtains the expansion of the q-Klyachko elements of type B:

$$K_n(q; A, \bar{A}) = \sum_J q^{\text{bmaj}(J)} R_J \qquad (44)$$

where

$$\text{bmaj}(J) = 2 \operatorname{maj}(\rho(J)) + |\epsilon|, \qquad (45)$$

where $|\epsilon|$ is the number of minus signs in $\epsilon$.

For example,

$$K_2(q) = R_2 + q^2 R_{\bar{2}} + q^2 R_{11} + q^3 R_{1\bar{1}} + q R_{\bar{1}1} + q^4 R_{\bar{1}\bar{1}}. \qquad (46)$$

$$\begin{aligned} K_3(q) = & R_3 + q^3 R_{\bar{3}} + q^4 R_{21} + q^5 R_{2\bar{1}} + q^2 R_{\bar{2}1} + q^7 R_{\bar{2}\bar{1}} + q^2 R_{12} + q^4 R_{1\bar{2}} \\ & + q R_{\bar{1}2} + q^5 R_{\bar{1}\bar{2}} + q^6 R_{111} + q^7 R_{1\bar{1}\bar{1}} + q^3 R_{1\bar{1}1} + q^8 R_{1\bar{1}\bar{1}} \\ & + q^5 R_{\bar{1}11} + q^6 R_{\bar{1}\bar{1}\bar{1}} + q^4 R_{1\bar{1}1} + q^9 R_{\bar{1}\bar{1}\bar{1}}. \end{aligned} \qquad (47)$$

This major index of type B is the flag major index defined in [1].

Following [1] and considering the signed composition (where $\epsilon$ is encoded as boolean vector for readability)

$$J = (2, 1, 1, \bar{3}, \bar{1}, \bar{2}, 4, \bar{1}, 2, 2) = (2113124122, 00001111110000100000) \qquad (48)$$

we can take the smallest permutation of shape $(2, 1, 3, 1, 2, 4, 1, 2, 2)$, which is

$$\alpha = 1543269871101213161514181719 \qquad (49)$$

sign it according to $\epsilon$, which yields

$$15432\bar{6}\bar{6}987\bar{1}\bar{1}10121316\bar{1}514181719 \qquad (50)$$

whose standardized is

$$81110912543612131416715181719 \qquad (51)$$

and has shape $\rho(J) = (2, 1, 3, 1, 6, 3, 2)$. The major index of $\rho(J)$ is 55, the number of minus signs in $\epsilon$ is 7, so $\text{bmaj}(J) = 2 \times 55 + 7 = 117$.

The major index of type B can be read directly on signed compositions without reference to signed permutations as follows: one can get $\rho(J)$ by first adding the absolute values of two consecutive parts if the left one is signed and the second one is not, then remove the signs and proceed as before.

A different solution consists in reading the composition from right to left, then associate weight 0 (resp. 1) to the rightmost part if it is positive (resp. negative) and then proceed left by adding 2 to the weight if the two parts are of the same sign and 1 if not. Finally, add up the product of the absolute values of the parts with their weight.

For example, with the same J as above we have the following weights:

$$J = (2, 1, 1, \bar{3}, \bar{1}, \bar{2}, 4, \bar{1}, 2, 2) \qquad (52)$$
weights : 14 12 10 9 7 5 4 3 2 0

$$\text{so that we get } 2 \cdot 14 + 1 \cdot 12 + 1 \cdot 10 + 3 \cdot 9 + 1 \cdot 7 + 2 \cdot 5 + 4 \cdot 4 + 1 \cdot 3 + 2 \cdot 2 + 2 \cdot 0 = 117.$$

This technique generalizes immediately to colored compositions with a fixed number $c$ of colors 0, 1, ..., $c-1$: the weight of the rightmost cell is its color and the weight of a part is equal to the sum of the weight of the next part and the unique representative of the difference of the colors of those parts modulo $c$ belonging to the interval $[1, c]$.
## 6.2 Generators and Hilbert series

For $n \ge 0$, let

$$S_n^{\pm} = S_n(A) \pm S_n(\bar{A}), \qquad (53)$$

and denote by $\mathcal{H}_n$ the subalgebra of $\mathbf{MR}$ generated by the $S_k^{\pm}$ for $k \le n$. For $n \ge 0$, we have

$$(S_n^{\pm})^{\#} \equiv (1 \mp q^n)S_n^{\pm} \quad \mod \mathcal{H}_{n-1}, \qquad (54)$$

so that the $(S_n^{\pm})^{\#}$ such that $1 \mp q^n \ne 0$ form a set of free generators in $\mathbf{MR}^{\#}$.

**Conjecture 6.1** If $r$ is odd, a basis of $\mathbf{MR}^{\#}$ will be parametrized by colored compositions such that parts of color 0 are not $\equiv 0 \mod r$ and parts of color 1 are arbitrary. The Hilbert series is then

$$H_r(t) = \frac{1 - t^r}{1 - 2(t + t^2 + \dots + t^r)}. \qquad (55)$$

If $r$ is even, there is the extra condition that parts of color 1 are not $\equiv r/2 \mod r$. The Hilbert series is then

$$H_r(t) = \frac{1 - t^r}{1 - 2(t + t^2 + \dots + t^r) + t^{r/2}}. \qquad (56)$$

For example,

$$H_2(t) = 1 + t + 2t^2 + 4t^3 + 8t^4 + 16t^5 + 32t^6 + 64t^7 + 128t^8 + O(t^9) \qquad (57)$$

$$H_3(t) = 1 + 2t + 6t^2 + 17t^3 + 50t^4 + 146t^5 + 426t^6 + 1244t^7 + 3632t^8 + O(t^9) \qquad (58)$$

$$H_4(t) = 1 + 2t + 5t^2 + 14t^3 + 38t^4 + 104t^5 + 284t^6 + 776t^7 + 2120t^8 + O(t^9) \qquad (59)$$

If these conjectures are correct, the Hilbert series of the right $\mathbf{MR}^{\#}$-modules generated by the $S_n$ are respectively

$$\frac{1}{1 - 2(t + t^2 + \dots + t^r)}, \qquad (60)$$

or

$$\frac{1}{1 - 2(t + t^2 + \dots + t^r) + t^{r/2}}. \qquad (61)$$

according to whether $r$ is odd or even.

The cases $r=1$ and $r=2$ are easily proved as follows. Assume first that $q=1$. Set

$$f = 1 + (\sigma_1^+)^{\#} = (\sigma_1 + \lambda_{-1})(A - \bar{A}), \qquad (62)$$

$$g = (\sigma_1^-)^{\#} - 1 = (\sigma_1 - \lambda_{-1})(A - \bar{A}). \qquad (63)$$

Then, $f^2 = g^2 + 4$, so that

$$f = 2 \left( 1 + \frac{1}{4} g^2 \right)^{\frac{1}{2}} \qquad (64)$$

which proves that the $(S_n^+)^{\#}$ can be expressed in terms of the $(S_m^-)^{\#}$.

Similarly, for $q = -1$, one can express

$$f = \sum_{n \ge 1} (S_{2n}^{+})^{\#} + \sum_{n \ge 0} (S_{2n+1}^{-})^{\#} \qquad (65)$$
in terms of

$$g = \sum_{n \ge 1} (S_{2n}^{-})^{\sharp} + \sum_{n \ge 0} (S_{2n+1}^{+})^{\sharp} \qquad (66)$$

since, as is easily verified,

$$(f+2)^2 = g^2 + 4, \text{ i.e., } f = -2 + 2 \left(1 + \frac{1}{4}g^2\right)^{\frac{1}{2}}. \qquad (67)$$

Apparently, this approach does not work anymore for higher roots of unity.

# 7 Appendix: monomial expansion of the $(1-q)$-kernel

The results of [16, 7] allow us to write down a new expansion of $S_n((1-q)A)$, in terms of the monomial basis of [4]. The special case $q=1$ gives back a curious expression of Dynkin's idempotent, first obtained in [3].

Let $\sigma$ be a permutation. We then define its left-right minima set $LR(\sigma)$ as the values of $\sigma$ that have no smaller value to their left. We will denote by $lr(\sigma)$ the cardinality of $LR(\sigma)$. For example, with $\sigma = 46735182$, we have $LR(\sigma) = \{4, 3, 1\}$, and $lr(\sigma) = 3$.

Let us now decompose $S_n((1-q)A)$ on the monomial basis $M_\sigma$ (see [4]) of FQSym. Thanks to the Cauchy formula of FQSym [7], we have

$$S_n((1-q)A) = \sum_{\sigma} S^{\sigma}(1-q)M_{\sigma}(A), \qquad (68)$$

where $S$ is the dual basis of $M$. Given the transition matrix between $M$ and $G$, we see that

$$S^{\sigma} = \sum_{\tau \le \sigma^{-1}} F_{\tau}, \qquad (69)$$

where $\le$ is the right weak order, e.g., $S^{312} = F_{123} + F_{213} + F_{231}$. Thanks to [16], we know that $F_\sigma(1-q)$ is either $(-q)^k$ if $\text{Des}(\sigma) = \{1, \dots, k\}$ or 0 otherwise. Let us define hook permutations of hook $k$ the permutations $\sigma$ such that $\text{Des}(\sigma) = \{1, \dots, k\}$. Now, $S^\sigma(1-q)$ amounts to compute the list of hook permutations smaller than $\sigma$. Note that hook permutations are completely characterized by their left-right minima. Moreover, if $\tau$ is smaller than $\sigma$ in the right weak order, then $LR(\tau) \subset LR(\sigma)$.

Hence all hook permutations smaller than a given permutation $\sigma$ belong to the set of hook permutations with left-right minima in $LR(\sigma)$. Since by elementary transpositions decreasing the length, one can get from $\sigma$ to the hook permutation with the same left-right minima and then from this permutation to all the others, we have:

**Theorem 7.1** Let $n$ be an integer. Then

$$S_n((1-q)A) = \sum_{\sigma \in S_n} (1-q)^{\text{lr}(\sigma)} M_{\sigma}. \qquad (70)$$

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

In the particular case $q=1$, we recover a result of [3]:

$$\Psi_n = \sum_{\substack{\sigma \in S_n \\ \sigma(1)=1}} M_{\sigma}, \qquad (71)$$

where $\Psi_n$ is the noncommutative power sum associated with Dynkin's idempotent [11, Prop. 5.2].
Acknowledgements

This work started during a stay of the first author at the University Paris-Est Marne-la-Vallée. The authors would also like to thank the contributors of the MuPAD project, and especially those of the combinat package, for providing the development environment for their research (see [10] for an introduction to MuPAD-Combinat).

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