On the S_n-module structure of the noncommutative harmonics

TL;DR

This paper investigates the structure of noncommutative harmonics under the symmetric group action, deriving Frobenius series for these modules and related algebraic structures using a noncommutative Chevalley decomposition.

Contribution

It introduces a noncommutative analog of Chevalley's decomposition to compute Frobenius series for noncommutative harmonics and related algebraic objects.

Findings

Computed graded Frobenius series for noncommutative harmonics

Derived Frobenius series for the enveloping algebra of the derived free Lie algebra

Extended classical symmetric polynomial decompositions to noncommutative setting

Abstract

Using a noncommutative analog of Chevalley's decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius series for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables.

Equations

$$⟨{\partial }_{x}f,g⟩$$

$$⟨d_{x}f,g⟩$$

$$MHa{r}_n$$

$$NCHa{r}_n$$

$$\mathbb{Q}⟨X_n⟩$$

$$(q;q{)}_k$$

$$\{q;q{\}}_k$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(NCSy{m}_n)$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(Sy{m}_n)=h_n\left[X\right]\frac{1}{(q;q{)}_n}$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(\mathbb{Q}⟨X_n⟩)=\sum _{d=0}^n\frac{q^d}{\{q,q{\}}_d}{h}_{(n-d,1^d)}[X].$$

$$\mathbb{Q}⟨X_n⟩\simeq \underset{d=0}{\overset{n}{⨁}}\underset{A:\ell (A)=d}{⨁}{M}^A$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(R{\uparrow }_{{\mathfrak{S}}_d}^{{\mathfrak{S}}_n})=h_{n-d}[X]{\mathcal{F}rob}_{{\mathfrak{S}}_d}(R),$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(\mathbb{Q}⟨X_n⟩)=\sum _{d=0}^n\sum _{A:\ell (A)=d}{q}^{|A|}{h}_{(n-d,1^d)}[X]=\sum _{d=0}^n\frac{q^d}{\{q,q{\}}_d}{h}_{(n-d,1^d)}[X].$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(MHa{r}_n)=(q;q{)}_n\sum _{d=0}^n\frac{q^d}{\{q,q{\}}_d}{h}_{(n-d,1^d)}[X]$$

$${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(NCHar_{n})=\Big{(}\sum_{d=0}^{n}\frac{q^{d}}{\{q,q\}_{d}}\Big{)}^{-1}\sum_{d=0}^{n}\frac{q^{d}}{\{q,q\}_{d}}h_{(n-d,1^{d})}[X].$$

$$\sum _{d=0}^n\frac{q^d}{\{q,q{\}}_d}{h}_{(n-d,1^d)}[X]$$

$$di{m}_q(MHa{r}_n)$$

$$di{m}_q(NCHa{r}_n)$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(MHa{r}_n)$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(NCHa{r}_n)$$

$$di{m}_q(MHa{r}_n)$$

$$di{m}_q(NCHa{r}_n)$$

$${\mathcal{A}}_n^{\prime }=\bigcap _{x\in X_n}\ker {\partial }_x.$$

$$MHa{r}_n\simeq {\mathcal{A}}_n^{\prime }\otimes {\mathcal{H}}_n.$$

$$\mathbb{Q}⟨X_n⟩\simeq \mathbb{Q}[X_n]\otimes {\mathcal{A}}_n^{\prime }.$$

$$\mathbb{Q}⟨X_n⟩=u.e.a.({\mathcal{L}}_n)=u.e.a(\mathbb{Q}{X}_n\oplus {\mathcal{L}}_n^{\prime })\simeq \mathbb{Q}[X_n]\otimes {\mathcal{A}}_n^{\prime }.$$

$$P_w=P_{{\ell }_1}{P}_{{\ell }_2}\cdots P_{{\ell }_k}.$$

$$cha{r}_{G{L}_n(\mathbb{Q})}({\mathcal{A}}_n^{\prime })(a_1,a_2,\cdots ,a_n)$$

$$\sum _T{s}_{shape(T)}(a_1,a_2,\dots ,a_n)$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}({\mathcal{A}}_n^{\prime })=\sum _{d=0}^n\frac{q^d}{\{q;q{\}}_d}{h}_{(n-d,1^d)}[X(1-q)].$$

$$f[X]\odot h_n\left[\frac{X}{1-q}\right]=f\left[\frac{X}{1-q}\right].$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}(\mathbb{Q}⟨X_n⟩)$$

$${\mathcal{F}rob}_{{\mathfrak{S}}_n}({\mathcal{A}}_n^{\prime }\otimes {\mathcal{H}}_n)$$

Full text

Expanded version of a paper to appear in Journal of Combinatorial theory, series A, http://www.elsevier.com/locate/jcta.

Emmanuel Briand is supported by a contract Juan de la Cierva, MEC. Mercedes Rosas is supported by a contract Ramón y Cajal, MEC. Mike Zabrocki is supported by NSERC

In honor of Manfred Schocker (1970-2006). The authors would also like to acknowledge the contributions that he made to this paper.

1. Introduction

A central result of Claude Chevalley [3] decomposes the ring of polynomials in $n$ variables (as graded representation of the symmetric group ${\mathfrak{S}}_{n}$) as the tensor product of the symmetric polynomials times the coinvariants of ${\mathfrak{S}}_{n}$ (i.e., polynomials modulo symmetric polynomials with no constant term).

The coinvariants of the symmetric group can also be defined as its harmonics (the polynomials annhilated by all symmetric polynomial differential operators with no constant term). They admit as a basis the famous Schubert polynomials of Schubert calculus, that play an important role in algebraic combinatorics, see for instance [6].

The space of invariant polynomials in noncommutative variables was introduced in 1936 by Wolf [16] where she found a noncommutative version of the fundamental theorem of symmetric functions. This space has been studied from a modern perspective in [13, 1, 2]. On the other hand, two sets of noncommutative harmonics for the symmetric group were introduced in [1] that translated into two noncommutative analogues of Chevalley decomposition for the ring of polynomials in noncommuting variables. The question of decomposing as ${\mathfrak{S}}_{n}$–modules both kinds of noncommutative harmonics was left open. This is the starting point in our investigations.

We begin the present work with the computation of the graded Frobenius characteristic of noncommutative harmonics. We then use these calculations to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in $n$ variables, ${\mathcal{A}}_{n}^{\prime}$. This last computation is achieved by using the existence of an isomorphism of $GL_{n}({\mathbb{Q}})$–modules between the space of polynomials in noncommutative variables, and the tensor product of the space of commuting polynomials with ${\mathcal{A}}^{\prime}_{n}$. Such an isomorphism is presented explicitly in the last section.

We conclude this introduction with some basic definitions and results that we will be using in the following sections. Let ${\mathfrak{S}}_{n}$ denote the symmetric group in $n$ letters. Denote by ${\mathbb{Q}}[X_{n}]={\mathbb{Q}}[x_{1},x_{2},\ldots,x_{n}]$ the space of polynomials in $n$ commuting variables and by ${\mathbb{Q}}{\langle}X_{n}{\rangle}={\mathbb{Q}}{\langle}x_{1},x_{2},\ldots,x_{n}{\rangle}$ the space of polynomials in $n$ noncommutative variables.

The space of symmetric polynomials in $n$ variables will be denoted by $Sym_{n}$ and the space of noncommutative polynomials which are invariant under the canonical action of the symmetric group ${\mathfrak{S}}_{n}$ will be denoted by $NCSym_{n}$.

Given any polynomial $f(X_{n})\in{\mathbb{Q}}[X_{n}]$, the notation $f(\partial_{X_{n}})$ represents the polynomial turned into an operator with each of the variables replaced by its corresponding derivative operator. Analogous notation will also hold for $f(X_{n})\in{\mathbb{Q}}{\langle}X_{n}{\rangle}$ except that there are two types of differential operators acting on words in noncommutative variables. The first is the Hausdorff derivative, $\partial_{x}$, whose action on a word $w$ is defined to be the sum of the subwords of $w$ with an occurrence of the letter $x$ deleted. The second derivative is the twisted derivative, $d_{x}$, which is defined on $w$ to be $w^{\prime}$ if $w=xw^{\prime}$, and [math] otherwise. Both derivations are extended to polynomials by linearity.

It is interesting to remark (as does Lenormand in [8], section Séries comme opérateurs) that these two operations are dual to the shuffle and concatenation products respectively, with respect to a scalar product where the noncommutative monomials are self dual. That is,

[TABLE]

Following [1], we introduce the following two sets of noncommutative analogues of the harmonic polynomials. The canonical action of the symmetric group endow them with the structure of ${\mathfrak{S}}_{n}$–modules.

[TABLE]

where $\mathfrak{M}_{n}=\{p\in NCSym_{n}\hbox{ with }p(0)=0\}$.

We are now ready to state the two decompositions of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ as the tensor product (over ${\mathbb{Q}}$) of its invariants times its coinvariants that we have described.

Proposition 1** ([1], Theorems 6.8 and 8.8).**

As graded ${\mathfrak{S}}_{n}$–modules,

[TABLE]

2. The Frobenius characteristic of noncommutative harmonics

In this section we compute the Frobenius characteristic of both kinds of noncommutative harmonics. This section is based of the observation that the graded Frobenius series for each of the ${\mathfrak{S}}_{n}$–modules appearing in Proposition 1 is either known or can be deduced from the existence of the isomorphisms described there.

The expressions for Frobenius images and characters will require a little use of symmetric function notation and identities. We will follow Macdonald [9] for the notation of the $s_{\lambda}$ Schur, $h_{\lambda}$ homogeneous, $e_{\lambda}$ elementary and $p_{\lambda}$ power sums bases for the ring of symmetric functions Sym, that we identify with ${\mathbb{Q}}[p_{1},p_{2},p_{3},\ldots]$. For convenience we will make use of some plethystic notation.

For a symmetric function $f$, $f[X]$ represents the symmetric function evaluated at an unspecified (possibly infinite) alphabet $X$. Then, $f[X(1-q)]$ is the image of $f$ under the algebra automorphism sending the power sum symmetric function $p_{k}$ to $(1-q^{k})p_{k}[X]$. Similarly, $f\left[\frac{X}{1-q}\right]$ is the image of the symmetric function $f$ under the inverse automorphism (sending the power sum $p_{k}$ to $p_{k}/(1-q^{k})$).

In our calculations, we use the Kronecker product $\odot$ of symmetric functions. This operation on symmetric functions corresponds, under the Frobenius map, to the inner tensor product of representations of the symmetric group (tensor product of representations with the diagonal action on the tensors). It can also be defined directly on symmetric functions by the equation $p_{\lambda}\odot p_{\mu}=\delta_{\lambda,\mu}\left(\prod_{i}n_{i}({\lambda})!i^{n_{i}({\lambda})}\right)p_{\lambda}$ where $n_{i}({\lambda})$ is the number of parts of size $i$ in ${\lambda}$, and then extended by bilinearity.

We introduce the notations

[TABLE]

Then $q^{d}/\{q;q\}_{d}$ is the generating function for the set partitions with length $d$ and $q^{d}/(q;q)_{d}$ is the generating function for partitions with length $d$, [15]. Finally, since $Sym_{n}$ and $NCSym_{n}$ are made of graded copies of the trivial ${\mathfrak{S}}_{n}$-module we conclude that

[TABLE]

In the following lemma we compute the graded Frobenius characteristic for the module ${\mathbb{Q}}{\langle}X_{n}{\rangle}$.

Lemma 2** (The Frobenius characteristic of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$).**

[TABLE]

Proof.

For each monomial $x_{i_{1}}\cdots x_{i_{r}}$, we define its type $\nabla(x_{i_{1}}\cdots x_{i_{r}})$ to be the set partition of $[r]=\{1,2,\ldots,r\}$ such that $a$ and $b$ are in the same part of the set partition if and only if $i_{a}=i_{b}$ in the monomial. For a set partition $A$ with at most $n$ parts, we will let $M^{A}$ equal the ${\mathfrak{S}}_{n}$ submodule of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ spanned by all monomials of type $A$. As ${\mathfrak{S}}_{n}$–module,

[TABLE]

where the second direct sum is taken over all set partitions $A$ with $d$ parts.

Fix a set partition $A$, and let $d$ be the number of parts of $A$, and ${\bf x}_{\vec{i}}=x_{i_{1}}x_{i_{2}}...x_{i_{r}}$ be the smallest monomial in lex order in $M^{A}$. It involves only the variables $x_{1}$, $x_{2}$, …, $x_{d}$. The representation $M^{A}$ is the representation of ${\mathfrak{S}}_{n}$ induced by the action of the subgroup ${\mathfrak{S}}_{d}\times{\mathfrak{S}}_{1}^{n-d}\simeq{\mathfrak{S}}_{d}$ on the subspace ${\mathbb{Q}}[{\mathfrak{S}}_{d}]\cdot{\bf x}_{\vec{i}}$. The representation ${\mathbb{Q}}[{\mathfrak{S}}_{d}]\cdot{\bf x}_{\vec{i}}$ of ${\mathfrak{S}}_{d}$ is isomorphic to the regular representation. We use the rule for a representation $R$ of ${\mathfrak{S}}_{d}$ induced to ${\mathfrak{S}}_{n}$,

[TABLE]

and conclude that the Frobenius characteristic of $M^{A}$ is $h_{(n-d,1^{d})}[X].$ Hence the graded Frobenius characteristic of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ is

[TABLE]

∎

We are now able to compute the Frobenius characteristic for $MHar_{n}$ and $NCHar_{n}$.

Theorem 3** (The Frobenius characteristic of the noncommutative harmonics).**

[TABLE]

and

[TABLE]

Proof.

This follows since ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(MHar_{n}\otimes Sym_{n})={{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(MHar_{n})\odot{{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(Sym_{n})$. Since $h_{n}[X]$ is the unity for the Kronecker product on symmetric functions of degree $n$, and since ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(Sym_{n})=h_{n}[X]/(q;q)_{n}$, we conclude that ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(MHar_{n})/(q;q)_{n}={{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}({\mathbb{Q}}{\langle}X_{n}{\rangle})$. We can now solve for ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(MHar_{n})$.

A similar argument demonstrates the formula for ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(NCHar_{n})$. We have from Proposition 1 and Lemma 2,

[TABLE]

From this equation we can solve for ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}(NCHar_{n})$. ∎

As a corollary, we obtain the generating functions for the graded dimensions of these spaces.

Corollary 4** (The Hilbert series of the noncommutative harmonics).**

[TABLE]

Proof.

After Theorem 3,

[TABLE]

This implies

[TABLE]

since the Hilbert series of a graded ${\mathfrak{S}}_{n}$–module is obtained by coefficient extraction from the graded Frobenius characteristic (of the coefficient of $p_{(1^{n})}[X]/n!\,$ in the expansion in power sum symmetric functions). Last, the Hilbert series of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ is $\frac{1}{1-nq}$. ∎

The graded dimensions of $MHar_{n}$ for $2\leq n\leq 5$ are listed in [14] as sequences $A122391$ through $A122394$. The sequences of graded dimensions of $NCHar_{n}$ for $3\leq n\leq 8$ are listed in [14] as sequences $A122367$ through $A122372$.

3. Non–commutative harmonics and the enveloping algebra of the derived free Lie algebra

Let ${\mathcal{L}}_{n}$ be the canonical realization of the free Lie algebra inside the ring of polynomials in noncommuting variables ${\mathbb{Q}}{\langle}X_{n}{\rangle}$. More precisely, ${\mathcal{L}}_{n}$ is the linear span of the minimal set of polynomials in ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ that includes ${\mathbb{Q}}$ and the variables $X_{n}$, and is closed under the bracket operation $[x,y]=xy-yx$. Let ${\mathcal{L}}^{\prime}_{n}=[{\mathcal{L}}_{n},{\mathcal{L}}_{n}]$ be the derived free Lie algebra. Remark that ${\mathcal{L}}_{n}={\mathcal{L}}^{\prime}_{n}\oplus{\mathbb{Q}}X_{n}$, where ${\mathbb{Q}}X_{n}$ denotes the space of linear polynomials. The enveloping algebra ${\mathcal{A}}_{n}^{\prime}$ of ${\mathcal{L}}_{n}$ can be realized as a subalgebra of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ as follows (see [12] 1.6.5):

[TABLE]

More explicitly, ${\mathcal{A}}_{n}^{\prime}$ is the subalgebra of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ generated by all the brackets under concatenation.

In [1] it was established that there is an isomorphism of vector spaces between $MHar_{n}$ and ${\mathcal{A}}_{n}^{\prime}\otimes{\mathcal{H}}_{n}$. In this section we will show the following result.

Theorem 5**.**

As ${\mathfrak{S}}_{n}$–modules,

[TABLE]

The proposition will be established by comparing the Frobenius image of $MHar_{n}$ (known from Theorem 3) to ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}({\mathcal{A}}_{n}^{\prime}\otimes{\mathcal{H}}_{n})$, which is equal to ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}({\mathcal{A}}_{n}^{\prime})\odot{{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}({\mathcal{H}}_{n})$. We will determine ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}({\mathcal{A}}_{n}^{\prime})$ in Theorem 8 below. An intermediate step will make use the following Theorem due to V. Drensky.

Proposition 6** (Drensky, [5] Theorem 2.6).**

As $GL_{n}({\mathbb{Q}})$–modules (and consequently as ${\mathfrak{S}}_{n}$–modules),

[TABLE]

Drensky proved Proposition 6 by exhibiting an explicit isomorphism between these two representations.

We will look at it in the next section. For now, we will provide a non–constructive proof of the theorem.

Before, we need to introduce some notation.

It is known that ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ is the universal enveloping algebra (u.e.a) of the free Lie algebra, ${\mathcal{L}}_{n}$. Using the Poincaré-Birkhoff-Witt theorem, a linear basis for ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ is given by decreasing products of elements of ${\mathcal{L}}_{n}$. Since we can choose an ordering of the elements of ${\mathcal{L}}_{n}$ so that the space of linear polynomials is smallest and decreasing products of linear polynomials are isomorphic to ${\mathbb{Q}}[X_{n}]$ (as a vector space), we note that as vector spaces

[TABLE]

To distinguish between the commutative elements of ${\mathbb{Q}}[X_{n}]$ and the noncommutative words of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$, we will place a dot over the variables (as in $\dot{x}_{i}$) to indicate the commutative variables.

Let $[n]=\{1,2,\ldots,n\}$ and let $[n]^{r}$ denote the words of length $r$ in the alphabet of the numbers $1,2,\ldots,n$. A word $w\in[n]^{r}$ is called a Lyndon word if $w<w_{k}w_{k+1}\cdots w_{r}$ for all $2\leq k\leq r$ where $<$ represents lexicographic order on words.

Every word $w\in[n]^{r}$ is equal to a unique product $w=\ell_{1}\ell_{2}\cdots\ell_{k}$ such that $\ell_{1}\geq\ell_{2}\geq\cdots\geq\ell_{k}$ and each $\ell_{i}$ is Lyndon (e.g. Corollary 4.4 of [12]).

Let $\ell$ be a Lyndon word of length greater than $1$. We say that $\ell=uv$ is the standard factorization of $\ell$ if $v$ is the smallest nontrivial suffix in lexicographic order. It follows that $u$ and $v$ are Lyndon words and $u<v$.

For a Lyndon word $\ell$, if $\ell$ is a single letter $a$ then define $P_{a}=x_{a}\in{\mathbb{Q}}{\langle}X_{n}{\rangle}$. If $\ell=uv$ is the standard factorization of $\ell$, then $P_{\ell}=[P_{u},P_{v}]$. For any $w\in[n]^{r}$ with Lyndon decomposition $w=\ell_{1}\ell_{2}\cdots\ell_{k}$, define

[TABLE]

The set $\{P_{w}\}_{w\in[n]^{r}}$ forms a basis for the noncommutative polynomials of degree $r$ ([12], Theorem 5.1). The elements $P_{w}$ with Lyndon decomposition $w=\ell_{1}\ell_{2}\cdots\ell_{k}$ such that each Lyndon factor has degree at least $2$ are a basis of ${\mathcal{A}}^{\prime}_{n}$.

Proof.

To prove that ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ and ${\mathbb{Q}}[X_{n}]\otimes{\mathcal{A}}_{n}^{\prime}$ are isomorphic as $GL_{n}({\mathbb{Q}})$–modules, we use the fact that two polynomial $GL_{n}({\mathbb{Q}})$–modules with the same character are isomorphic (see for instance the notes by Kraft and Procesi, [7]). The character of a $GL_{n}({\mathbb{Q}})$–module is the trace of the action of the diagonal matrix $diag(a_{1},a_{2},\ldots,a_{n})$.

A basis for ${\mathbb{Q}}[X_{n}]\otimes{\mathcal{A}}_{n}^{\prime}$ are the elements ${\bf\dot{x}}^{\alpha}\otimes P_{\ell_{1}}\cdots P_{\ell_{k}}$ with $\ell_{1}\geq\ell_{2}\geq\cdots\geq\ell_{k}$ and $|\ell_{i}|\geq 2$. The action of the diagonal matrix $diag(a_{1},a_{2},\ldots,a_{n})$ on this basis element is the same as the action on the noncommutative polynomial $x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{n}^{\alpha_{n}}P_{\ell_{1}}P_{\ell_{2}}\cdots P_{\ell_{k}}$ (in both cases: multiplication by $a_{1}^{\alpha_{1}+m_{1}}a_{2}^{\alpha_{2}+m_{2}}\cdots a_{n}^{\alpha_{n}+m_{n}}$ where $m_{i}$ is the number of occurrences of $i$ in the word $\ell_{1}\ell_{2}\cdots\ell_{k}$). By the Poincaré-Birkhoff-Witt theorem, these polynomials form a basis for ${\mathbb{Q}}\left<X_{n}\right>$, hence the trace of the action of $diag(a_{1},a_{2},\ldots,a_{n})$ acting on ${\mathbb{Q}}\left<X_{n}\right>$ and ${\mathbb{Q}}[X_{n}]\otimes{\mathcal{A}}_{n}^{\prime}$ are equal. Since their characters are equal, we conclude that they are isomorphic as $GL_{n}({\mathbb{Q}})$ modules. ∎

The $GL_{n}({\mathbb{Q}})$–character of ${\mathbb{Q}}[X_{n}]$ is $\prod_{i=1}^{n}\frac{1}{1-a_{i}}$, and the $GL_{n}({\mathbb{Q}})$–character of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ is $\frac{1}{1-(a_{1}+a_{2}+\cdots+a_{n})}$. Therefore, the existence of a $GL_{n}({\mathbb{Q}})$-module isomorphism between ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ and ${\mathbb{Q}}[X_{n}]\otimes{\mathcal{A}}_{n}^{\prime}$ implies the following result.

Corollary 7** (The $GL_{n}({\mathbb{Q}})$–character of ${\mathcal{A}}_{n}^{\prime}$).**

[TABLE]

Moreover this last sum is equal to

[TABLE]

where the sum is over all standard tableaux $T$ such that the smallest integer which does not appear in the first column of $T$ is odd.

By Schur-Weyl duality, the above formula also describes the decomposition of the subspace of multilinear polynomials (i.e. with distinct occurrences of the variables) of ${\mathcal{A}}_{n}^{\prime}$. That is, if $n$ is the number of variables, the the multilinear polynomials of degree $n$ will be an ${\mathfrak{S}}_{n}$-module with Frobenius image equal to $\sum_{i=2}^{n}(-1)^{i}e_{(i,1^{n-i})}[X]$. This decomposition was considered in the papers [4], [10], [11] where an expression was given degree by degree up to $n=7$. The expansion of this formula in the Schur basis provided in the Theorem agrees with the computations in those papers.

We can derive a formula for the Frobenius characteristic of ${\mathcal{A}}^{\prime}$ by using a similar technique.

Theorem 8** (The Frobenius characteristic of ${\mathcal{A}}_{n}^{\prime}$).**

[TABLE]

Proof.

For any symmetric function $f[X]$ of degree $n$, we have that

[TABLE]

In particular, since ${{\mathcal{F}rob}_{{\mathfrak{S}}_{n}}}({\mathbb{Q}}[X_{n}])=h_{n}\left[\frac{X}{1-q}\right],$ we conclude that

[TABLE]

This implies that if we make the plethystic substitution $X{\rightarrow}X(1-q)$ into both sides of this equation and using Lemma 2 we arrive at the stated formula. ∎

We can now prove Theorem 5.

Proof.

From Theorem 3 we know the Frobenius image of $MHar_{n}$, we compare this to

[TABLE]

Since the two ${\mathfrak{S}}_{n}$–modules have the same Frobenius image, we conclude that they must be isomorphic. ∎

4. An explicit isomorphism between ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ and ${\mathbb{Q}}[X_{n}]\otimes{\mathcal{A}}_{n}^{\prime}.$

Let $V$ be a finite–dimensional vector space over ${\mathbb{Q}}$. Let $S(V)$ and $T(V)$ be its symmetric algebra and tensor algebra respectively. There exists a unique embedding $\varphi$ of $GL(V)$–modules of $S(V)$ into $T(V)$ such that

[TABLE]

Its image is the subspace of the symmetric tensors. In the case $V=\bigoplus_{i=1}^{n}{\mathbb{Q}}x_{i}$, we have $S(V)={\mathbb{Q}}[X_{n}]$ and $T(V)={\mathbb{Q}}{\langle}X_{n}{\rangle}$. Then the embedding $\varphi$ and the inclusion ${\mathcal{A}}^{\prime}_{n}\subset{\mathbb{Q}}{\langle}X_{n}{\rangle}$ induce a map of $GL_{n}({\mathbb{Q}})$–modules $\Phi:{\mathbb{Q}}[X_{n}]\otimes{\mathcal{A}}^{\prime}_{n}\longrightarrow{\mathbb{Q}}{\langle}X_{n}{\rangle}$ characterized by $\Phi(f\otimes a)=\varphi(f)a$ for all $f\in{\mathbb{Q}}[X_{n}]$ and all $a\in{\mathcal{A}}^{\prime}_{n}$. Then,

Proposition 9** (Drensky, [5] Theorem 2.6).**

The map $\Phi$ is a $GL_{n}({\mathbb{Q}})$ equivariant isomorphism from ${\mathbb{Q}}[X_{n}]\otimes{\mathcal{A}}^{\prime}_{n}$ to ${\mathbb{Q}}{\langle}X_{n}{\rangle}$.

Indeed, Drensky showed that given an arbitrary homogeneous basis of $\mathcal{G}$ of ${\mathcal{A}}^{\prime}_{n}$, the elements $\Phi(m\otimes g)$ for $m$ monomial and $g\in\mathcal{G}$, are a basis of ${\mathcal{A}}^{\prime}_{n}$ ([5] Lemma 2.4). We refine Drensky’s proof by considering for $\mathcal{G}$ the bracket basis $\{P_{w}\}_{w\in[n]^{r}}$ of ${\mathcal{A}}^{\prime}_{n}$ (introduced before the proof of Proposition 6) and the shuffle basis (see below) to realize ${\mathbb{Q}}[X_{n}]$ in ${\mathbb{Q}}{\langle}X_{n}{\rangle}$. We show that the elements $\Phi(m\otimes g)$ form a basis ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ (the hybrid basis) that is triangularly related and expands positively in the bracket basis of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ (Theorem 10 below).

We follow the book of Reutenauer [12] for the classical definitions and results used in this section. The bracket basis $P_{w}$ has been introduced in the previous section (before the proof of Proposition 6). Before presenting the hybrid basis we introduce another classical basis of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$: the shuffle basis.

The shuffle basis of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$.

Consider two monomials, $x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}$ and $x_{j_{1}}x_{j_{2}}\cdots x_{j_{r^{\prime}}}$ in ${\mathbb{Q}}{\langle}X_{n}{\rangle}$. For a subset

[TABLE]

and the complement subset $T=\{t_{1},t_{2},\ldots,t_{r^{\prime}}\}=[r+r^{\prime}]\backslash S$, we let

[TABLE]

be the unique monomial in ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ of length $r+r^{\prime}$ such that $w_{s_{1}}w_{s_{2}}\cdots w_{s_{r}}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}$ and $w_{t_{1}}w_{t_{2}}\cdots w_{t_{r^{\prime}}}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{r^{\prime}}}$.

The shuffle of any two monomials is defined as

[TABLE]

This shuffle of monomials is then extended to a bilinear operation on any two elements of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ The shuffle product is a commutative and associative operation on ${\mathbb{Q}}{\langle}X_{n}{\rangle}$.

Let $w$ be a word in $[n]^{r}$ and let $w=\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}\cdots\ell_{k}^{i_{k}}$ be the factorization of $w$ into decreasing products of Lyndon words $\ell_{1}>\ell_{2}>\cdots>\ell_{k}$. For a Lyndon word $\ell=i_{1}i_{2}\cdots i_{r}$, let $S_{\ell}$ be the corresponding monomial in ${\mathbb{Q}}{\langle}X_{n}{\rangle}$, that is $S_{\ell}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}$. If $w$ is not a single Lyndon word then define

[TABLE]

The set $\{S_{w}\}_{w\in[n]^{r}}$ forms a basis for the noncommutative polynomials of degree $r$ ([12], Corollary 5.5).

It is interesting to note that the bracket basis $P_{w}$ and the shuffle basis $S_{w}$ are dual with respect to the scalar product where the noncommutative monomials are self-dual.

The hybrid basis of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$.

We are now ready to introduce the hybrid basis.

Given a word $w\in[n]^{r}$ with a factorization into decreasing products of Lyndon words $w=\ell_{1}^{i_{1}}\ell_{2}^{i_{2}}\cdots\ell_{k}^{i_{k}}$, then let $\ell_{j_{1}},\ell_{j_{2}},\cdots,\ell_{j_{r}}$ be the Lyndon words of length $1$ in this decomposition and set

[TABLE]

Observe that $M(w)$ is the image under the embedding $\varphi$ of the monomial $X(w)=\dot{x}_{\ell_{j_{1}}}^{i_{j_{1}}}\dot{x}_{\ell_{j_{2}}}^{i_{j_{2}}}\cdots\dot{x}_{\ell_{j_{r}}}^{i_{j_{r}}}$. For all of the remaining Lyndon words $\ell_{{a}_{1}}$, $\ell_{{a}_{2}}$, …, $\ell_{{a}_{k-r}}$ with length greater than $1$ we define the Lie portion of the word to be $L(w)=P_{\ell_{{a}_{1}}^{i_{{a}_{1}}}\ell_{{a}_{2}}^{i_{{a}_{2}}}\cdots\ell_{{a}_{k-r}}^{i_{{a}_{k-r}}}}$. We will define the hybrid elements to be $H_{w}:=M(w)L(w)=\Phi(X(w)\otimes L(w))$.

The result of this section is:

Theorem 10**.**

The noncommutative polynomials $H_{w}$ are triangularly related to and expand positively in the $P_{u}$ basis. Precisely, for $w$ of length $r$,

[TABLE]

As a consequence, the set $\{H_{w}\}_{w\in[n]^{r}}$ is a basis for the noncommutative polynomials of ${\mathbb{Q}}{\langle}X_{n}{\rangle}$ of degree $r$.

We require a few facts about Lyndon words and the lexicographic ordering which can be found in [12].

1. (1)

If $u$ and $v$ are Lyndon words and $u<v$ then $uv$ is a Lyndon word. ([12], (5.1.2)) 2. (2)

If $u<v$ and $u$ is not a prefix of $v$, then $ux<vy$ for all words $x,y$. ([12], Lemma 5.2.(i)) 3. (3)

If $w=\ell_{1}\ell_{2}\cdots\ell_{k}$ with $\ell_{1}\geq\ell_{2}\geq\cdots\geq\ell_{k}$ then $\ell_{k}$ is the smallest (with respect to the $>$ order) nontrivial suffix of $w$. ([12], Lemma 7.14) 4. (4)

If $\ell^{\prime}<\ell$, are both Lyndon words, then $\ell^{\prime}\ell<\ell\ell^{\prime}$ (follows from (1)). As a consequence, for $\ell_{1}\geq\ell_{2}\geq\cdots\geq\ell_{k}$, $\ell_{1}\ell_{2}\cdots\ell_{k}\geq\ell_{\sigma(1)}\ell_{\sigma(2)}\cdots\ell_{\sigma(k)}$ for any permutation $\sigma\in{\mathfrak{S}}_{k}$ with equality if and only $\ell_{i}=\ell_{\sigma(i)}$ for all $1\leq i\leq k$.

Proof.

To see that (4) holds consider a weakly decreasing product of Lyndon words $\ell_{1}\ell_{2}\cdots\ell_{k}$. If $id\rightarrow\sigma^{(1)}\rightarrow\sigma^{(2)}\rightarrow\cdots\rightarrow\sigma$ is a chain in the weak right order then we have just shown that

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with equality if and only if the two Lyndon factors which were transposed are equal. Therefore there exists a chain of words one greater than or equal to the next with $\ell_{1}\ell_{2}\cdots\ell_{k}$ on one end and $\ell_{\sigma(1)}\ell_{\sigma(2)}\cdots\ell_{\sigma(k)}$ on the other. ∎

Theorem 10 will be established after the following lemma.

Lemma 11**.**

Let $w$ be a word and $\ell_{1}\ell_{2}\cdots\ell_{r}$ the decomposition of $w$ into a decreasing product of Lyndon words. Let $\ell$ be a Lyndon word such that $\ell=af_{1}f_{2}\cdots f_{k}$ with $a$ one of the variables, each $f_{i}$ a Lyndon word and $f_{i}\geq f_{i+1}$ and $f_{k}\geq\ell_{1}$. Let $u=\ell_{1}\cdots\ell_{d}\ell\ell_{d+1}\cdots\ell_{r}$ where $\ell_{d}>\ell\geq\ell_{d+1}$ or $d=0$ and $\ell\geq\ell_{1}$. Then

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Proof.

Assume that $r=1$, and we have that either $\ell\geq\ell_{1}$ and $P_{\ell}P_{\ell_{1}}=P_{\ell\ell_{1}}$ and we are done, or $\ell<\ell_{1}$ and

[TABLE]

In this case $P_{\ell_{1}}P_{\ell}=P_{\ell_{1}\ell}$. By (1) we know that $\ell\ell_{1}$ is Lyndon. Moreover, $\ell\ell_{1}$ is its standard factorization (this follows from (3), since the nontrivial suffixes of $\ell\ell_{1}$ are all suffixes of $f_{1}f_{2}\cdots f_{k}\ell_{1}$, which is a nonincreasing product of Lyndon words). Therefore $P_{\ell\ell_{1}}=[P_{\ell},P_{\ell_{1}}]$ and $P_{\ell}P_{\ell_{1}}=P_{\ell_{1}\ell}+P_{\ell\ell_{1}}$. By (4), $\ell\ell_{1}<\ell_{1}\ell$ so the triangularity relation holds.

Now for an arbitrary $r>1$ we have the same two cases. Either $\ell\geq\ell_{1}$ and $P_{\ell}P_{\ell_{1}}P_{\ell_{2}}\cdots P_{\ell_{r}}=P_{\ell w}$, or $\ell<\ell_{1}$ and

[TABLE]

Our induction hypothesis holds for $P_{\ell}P_{\ell_{2}}\cdots P_{\ell_{r}}$ since $f_{k}\geq\ell_{1}\geq\ell_{2}$, hence $P_{\ell}P_{\ell_{2}}\cdots P_{\ell_{r}}=P_{u^{\prime}}+\sum_{v^{\prime}<u^{\prime}}c^{\prime}_{v^{\prime}}P_{v^{\prime}}$ where $u^{\prime}=\ell_{2}\cdots\ell_{d}\ell\ell_{d+1}\cdots\ell_{r}$. Moreover, $P_{\ell_{1}}P_{u^{\prime}}=P_{\ell_{1}u^{\prime}}=P_{u}$ since $\ell_{1}\geq\ell_{2}$ and $P_{\ell_{1}}P_{v^{\prime}}=P_{\ell_{1}v^{\prime}}$ since $\ell_{1}\geq$ any Lyndon prefix of $v^{\prime}$.

Since $[P_{\ell},P_{\ell_{1}}]=P_{\ell\ell_{1}}$ by (3), and $\ell_{1}\geq\ell_{2}$, we have by the induction hypothesis that $P_{\ell\ell_{1}}P_{\ell_{2}}\cdots P_{\ell_{r}}=P_{u^{\prime\prime}}+\sum_{v^{\prime\prime}<u^{\prime\prime}}c_{v^{\prime\prime}}^{\prime\prime}P_{v^{\prime\prime}}$ where

[TABLE]

with $\ell_{d^{\prime}}>\ell\ell_{1}\geq\ell_{d^{\prime}+1}$. In order to justify the induction step we also need to have that $u^{\prime\prime}<u$. This follows from (4) since $u^{\prime\prime}$ is a permutation of the factors of $u$ and $\ell_{1}>\ell$ and $\ell$ lies to the left of $\ell_{1}$ in $u^{\prime}$. ∎

We are now in a position to prove Theorem 10.

Proof.

$H_{w}$ is defined as the product $M(w)L(w)$ where $M(w)$ is a a shuffle of monomials. It expands as $M(w)=\sum{\tilde{c}}_{b}{\bf x}_{b}$ with $\sum{\tilde{c}}_{b}=r!$ and where each monomial in $M(w)$ has the same number of $x_{1}$s, $x_{2}$s, etc. We fix one such monomial ${\bf x}_{b}$ that as follows by indexing its letters backwards: ${\bf x}_{b}=x_{i_{k}}x_{i_{k-1}}\cdots x_{i_{1}}$. We define inductively words $w[k]$, …, $w[1]$, $w[0]$ as follows: $w[k]:=w$ and $w[j-1]$ is the word obtained from $w[j]$ by removing one of its Lyndon factors of length $1$ equal to $x_{i_{j}}$. Remark that $L(w[j])=L(w)$ for all $j$. Then we establish by induction on $j$ that

[TABLE]

by applying Lemma 11 with $x_{i_{j}}$ for $\ell$ and $w[j]$ for $w$. ∎