# Harmonics for deformed Steenrod operators
(Extended Abstract)

François Bergeron<fnref n="1" />†, Adriano Garsia<fnref n="2" />‡ and Nolan Wallach<fnref n="2" />§

received 4 Nov 2009,

**Abstract.** We explore in this paper the spaces of common zeros of several deformations of Steenrod operators. Proofs are omitted in view of pages limitation for the extended abstract.

**Résumé.** Nous explorons dans cet article l'espace des zéros communs de plusieurs déformations d'opérateurs de Steenrod. Faute de place, les preuves sont omises.

**Keywords:** Harmonic polynomials, Steenrod Operators

## 1 Introduction

In recent years many authors have studied variations on a striking classical result of invariant theory holding for any finite group W of real n × n matrices generated by reflections. Roughly stated, this result asserts that there is a natural decomposition

$$ \mathbb{R}[\mathbf{x}] \simeq \mathbb{R}[\mathbf{x}]^W \otimes \mathbb{R}[\mathbf{x}]_W \quad (1) $$

of the ring of polynomials $\mathbb{R}[\mathbf{x}]$, in n variables $\mathbf{x} = x_1, x_2, \dots, x_n$, as a tensor product of the ring $\mathbb{R}[\mathbf{x}]^W$ of W-invariant polynomials, and the “W-coinvariant-space” $\mathbb{R}[\mathbf{x}]_W$. This last is simply the space obtained as the quotient of the ring $\mathbb{R}[\mathbf{x}]$ by the ideal generated by constant-term-free W-invariant polynomials. It is well known that $\mathbb{R}[\mathbf{x}]_W$ is isomorphic as a W-module to the space $\mathcal{H}_W$ of W-harmonic polynomials, i.e.: the set of polynomials $f(\mathbf{x})$ that satisfy all partial differential equations of the form $p(\partial_\mathbf{x})f(\mathbf{x}) = 0$, where $p(\partial_\mathbf{x})$ any constant-term-free W-invariant polynomial in the partial derivatives $\partial_i$.

The purpose of this work is to study twisted versions of this setup. Typically, we replace symmetric operators $\partial_1^k + \dots + \partial_n^k$, by operators of the form

$$ D_k := \sum_{i=1}^{n} a_{i,k} x_i \partial_i^{k+1} + b_{i,k} \partial_i^k, \quad (2) $$

<footnote n="1">¹Dépt. de Math., UQAM, Montréal, Québec, H3C 3P8, CANADA.</footnote>

<footnote n="2">²UCSD, 9500 Gilman Drive # 0112 La Jolla, CA 92093-0112, USA.</footnote>

<footnote>†supported by NSERC-Canada and FQRNT-Québec</footnote>

<footnote>‡supported by NSF</footnote>

<footnote>§supported by NSF</footnote>
with some parameters $a_{i,k}$ and $b_{i,k}$. We then consider the solution set $\mathcal{H}_x$ of the system of partial differential equations $D_k f(\mathbf{x}) = 0$, for $k \ge 1$. Observe that the operators $D_k$ are homogeneous. We say that they are of degree $-k$ since they lower degree of polynomials by $k$. It follows that $\mathcal{H}_x$ is graded by degree. In particular, it makes sense to consider the *Hilbert series*

$$H_n(t) := \sum_{d \ge 0} t^d \dim(\pi_d(\mathcal{H}_x)), \qquad (3)$$

with $\pi_d$ denoting the projection onto the homogeneous component of degree $d$. Clearly the right-hand side of (3) depends on the choice of the parameters $a_{i,k}$ and $b_{i,k}$. Recall that the Hilbert series of the space $\mathcal{H}_{\mathfrak{S}_n}$, of $\mathfrak{S}_n$-harmonic polynomials (which corresponds to setting $a_{i,k} = 0$ and $b_{i,k} = 1$) is the classical $t$-analog of $n!$. As we will see later, this is a “generic” value for $H_n(t)$.

Before going on with our discussion, let us consider an interesting dual point of view. Following a terminology of Wood [5], we shall say that a polynomial is a *hit-polynomial* if it can be expressed in the form

$$f(\mathbf{x}) = \sum_k D_k^* g_k(\mathbf{x}), \qquad (4)$$

for some polynomials $g_k(\mathbf{x})$, with $D_k^*$ standing for the dual operator of $D_k$ with respect to the following scalar product on the ring of polynomials.

For two polynomials $f$ and $g$ in $\mathbb{R}[\mathbf{x}]$, one sets

$$\langle f, g \rangle := f(\partial_{\mathbf{x}})g(\mathbf{x})|_{\mathbf{x}=0}. \qquad (5)$$

In other words, this corresponds to the constant term of the polynomial resulting from the application of the differential operator $f(\partial_{\mathbf{x}})$ to $g(\mathbf{x})$. A straightforward computation reveals that, for two monomials $\mathbf{x}^a$ and $\mathbf{x}^b$, we have $\langle \mathbf{x}^a, \mathbf{x}^b \rangle = a!$, if $\mathbf{a} = \mathbf{b}$, and 0 otherwise. Here, as is now almost usual, $a!$ stands for $a_1! a_2! \cdots a_n!$. This observation makes it clear that (5) indeed defines a scalar product on $\mathbb{R}[\mathbf{x}]$. Moreover, the dual of the operator $\partial_i^k$ is easily checked to be multiplication by $x_i^k$. It follows that

$$D_k^* = \sum_{i=1}^{n} a_{i,k} x_i^{k+1} \partial_i + b_{i,k} x_i^k.$$

From general basic linear algebra principles, it follows that the space of $\mathcal{H}_x$, of general harmonic polynomials, is orthogonal to the space of hit-polynomials. Moreover, since the subspace of hit-polynomials is homogeneous, the corresponding quotient $\mathbb{C}$ of $\mathbb{R}[\mathbf{x}]$, by this subspace, is isomorphic to $\mathcal{H}_x$ as a graded space.

For, the special case corresponding to setting $a_{i,k} = q$, and $b_{i,k} = 1$, for all $i$ and $k$, we denote $\mathcal{H}_{\mathbf{x};q}$ resulting space which has been considered by Hivert and Thiéry (in [3]). Using the notation

$$D_{k;q} := \sum_{i=1}^{n} q x_i \partial_i^{k+1} + \partial_i^k,$$

Using a simply Lie-bracket calculation, Hivert and Thiéry have observed that $\mathcal{H}_{\mathbf{x};q}$ is simply characterized as the common solutions of the two equations $D_{1;q}f(\mathbf{x}) = 0$, and $D_{2;q}f(\mathbf{x}) = 0$. Recall that the ring of polynomials $\mathbb{R}[\mathbf{x}]$ can be considered as a $\mathfrak{S}_n$-module for the action that corresponds to permutation of
the variables. This action restricts to a natural $\mathfrak{S}_n$-action on the space $\mathcal{H}_{x;q}$, since the operators $D_{k;q}$ are symmetric. It is classical that $\mathcal{H}_{\mathfrak{S}_n} = \mathcal{H}_{x;0}$ is isomorphic, as a $\mathfrak{S}_n$-module, to the regular representation of $\mathfrak{S}_n$. Hivert-Thiéry go on to state that

**Conjecture 1 (Hivert-Thiéry)** As $\mathfrak{S}_n$-modules, the spaces $\mathcal{H}_{x;q}$ is isomorphic to $\mathcal{H}_{\mathfrak{S}_n}$, when $q > 0$. In particular, this implies that the Hilbert series of $\mathcal{H}_{x;q}$ is $[n]!_t$.

It follows from (1) that the graded Frobenius characteristic $F_n(t)$ of $\mathcal{H}_{x;q}$ (and $\mathcal{H}_{\mathfrak{S}_n}$) is

$$F_n(t) = [n]!_t (1-t)^n \sum_{\lambda \vdash n} \prod_{k=1}^{n} \frac{1}{d_k!} \left( \frac{p_k}{k(1-t^k)} \right)^{d_k}, \quad (6)$$

where $d_k = d_k(\lambda)$ is the number of size $k$ parts of $\lambda$.

In this work we generalize and extend the scope of the above conjecture to include the more general operators of (2). Along the way we prove several related results.

## 2 Tilde-Harmonics and Hat-Harmonics

We first consider another interesting special case of (2). Namely, we suppose that all $b_{i,k}$'s vanish, and all $a_{i,k}$'s are equal to 1. Thus, we consider the space of common zeros of the operators $\tilde{D}_k := \sum_{i=1}^n x_i \partial_i^{k+1}$, which is called the space of tilde-harmonics, and denoted $\tilde{\mathcal{H}}_x$. We easily check that

$$[\tilde{D}_k, \tilde{D}_j] = (k-j)\tilde{D}_{k+j}, \quad (7)$$

hence $\tilde{\mathcal{H}}_x$ is simply the set of common zeros of the two equations $\tilde{D}_1 f(x) = 0$, and $\tilde{D}_2 f(x) = 0$. The space $\tilde{\mathcal{H}}_x$ affords a natural action of the symmetric group, and the associated graded Frobenius characteristic is denoted $\tilde{F}_n(t)$. Computer experimentations suggest that the Hilbert series of $\tilde{\mathcal{H}}_x$ seems to be

$$\tilde{\mathcal{H}}_n(t) = \sum_{k=0}^{n} \binom{n}{k} t^k [k]_t!. \quad (8)$$

Modulo a natural conjecture, this follows from a very explicit description of $\tilde{\mathcal{H}}_x$ outlined below. To state it we need one more family of operators and yet another version of harmonic polynomials. For each $k \ge 1$, consider the operator $\hat{D}_k = \sum_{i=1}^n x_i \partial_i^{k+1} + (k+1) \partial_i^k$, and introduce the space

$$\hat{\mathcal{H}}_x := \{ f(x) \in \mathbb{R}[x] \mid \hat{D}_k f(x) = 0, \quad \forall k \ge 1 \},$$

whose elements are said to be “hat-harmonics”. We will soon relate the two notions of tilde and hat harmonics. Experimentation suggest that $\hat{\mathcal{H}}_x$ has dimension $n!$, and that even more precisely we have the following.

**Conjecture 2** As a graded $\mathfrak{S}_n$-module, $\hat{\mathcal{H}}_x$ is isomorphic to the space of $\mathfrak{S}_n$-harmonics.

Now, for any given $k$-subset $y$ of the $n$ variables $x$, let us consider the space $\hat{\mathcal{H}}_y$, and write

$$e_y := \prod_{x \in y} x,$$
for the elementary symmetric polynomial of degree $k$ in the variables $y$. As usual, we define the support of a monomial to be the set of variable that appear in it, with non-zero exponent. Clearly, $y^a$ has support $y$ if and only if $y^a = e_y y^b$, for some $b$. Observe that we have the operator identity we can easily check the operator identity

$$ \tilde{D}_k e_x = e_x \tilde{D}_k, \qquad (9) $$

where $e_x$ stands for the operator of multiplication by $e_x$. We can now state the following remarkable fact.

**Theorem 1** The space of tilde-harmonics has the direct sum decomposition

$$ \tilde{\mathcal{H}}_x = \bigoplus_{y \subseteq x} e_y \hat{\mathcal{H}}_y, \qquad (10) $$

if we consider that hat-harmonics for $y = \emptyset$ are simply the scalars.

The same holds for the more general case of operators $a_k \tilde{D}_k$ and $a_k \tilde{D}_k$, with the $a_k$'s equal to 0 or 1. The intent here is to restrict the set of equations considered to those $k$ for which $a_k$ takes the value 1. The corresponding spaces are denoted $\tilde{\mathcal{H}}_x^a$ and $\tilde{\mathcal{H}}_x^b$, with similar convention for the corresponding Hilbert series and graded Frobenius characteristics. It follows that, even in this more general context, we have

**Corollary 2** For all choices of $a_k$,

$$ \tilde{\mathcal{H}}_n^a(t) = \sum_{k=0}^{n} \binom{n}{k} t^k \tilde{\mathcal{H}}_k^a(t). \qquad (11) $$

In particular, if conjecture 2 holds then (8) holds. There is an even finer corollary of Theorem 1.

**Corollary 3** The graded Frobenius characteristic of $\tilde{\mathcal{H}}_x^a$ is given by the symmetric function

$$ \tilde{F}_n^a(t) = \sum_{k=0}^{n} t^k \tilde{F}_k^a(t) h_{n-k}(z) \qquad (12) $$

A conjecture of Wood [5, conjecture 7.3] is thus partially addressed in a very explicit manner. Indeed, in view of Theorem 1, Wood's conjecture is a consequence of Conjecture 2 and the fact that $\tilde{C}$ is isomorphic to $\tilde{\mathcal{H}}_x$ as a graded $\mathfrak{S}_n$-module.

## 3 More on $q$-harmonics

We now link the study of harmonics of the $\tilde{D}_k$ to further our understanding of the common zeros of the operators $D_{k:q}$, in the case when $q$ is considered as a formal parameter. Our point of departure is the following important fact. Denote by $\nabla_k := \sum_{i=1}^n \partial_i^k$ the generalized Laplacian, and observe that $D_{k:q} = q\tilde{D}_k + \nabla_k$, then we get the following.

**Theorem 4** Up to a power of $q$, every $q$-harmonic polynomial $f$ may be written in the form

$$ f = f_0 + qf_1 + q^2 f_2 + \cdots + q^m f_m \qquad (13) $$
with $f_i \in \mathbb{R}[\mathbf{x}]$, and such that for all $k \ge 1$ we have

$$
\begin{align}
(a) \quad & \nabla_k f_0 = 0 \nonumber \\
(b) \quad & \nabla_k f_i = -\tilde{D}_k f_{i-1}, \quad \text{for all } i = 2, \dots, m-1, \tag{14} \\
(c) \quad & \tilde{D}_k f_m = 0. \nonumber
\end{align}
$$

$$
(c) \quad \tilde{D}_k f_m = 0.
$$

In particular, it follows that for any $r \ge 0$, and any choice of $k_1, k_2, \dots, k_r \ge 1$, the element

$$
\nabla_{k_1} \nabla_{k_2} \cdots \nabla_{k_r} f_r \qquad (15)
$$

is a $\mathfrak{S}_n$-harmonic polynomial in the usual sense.

Let us now reformulate the expansion of (13) in the form

$$
f = q^r (f_0 + q f_1 + \dots + q^m f_m) \quad (\text{with each } f_i \in \mathbb{R}[\mathbf{x}], f_i \neq 0)
$$

We call $f_0$ the first term of $f$ and denote it “FT($f$)”. Analogously we say that $f_m$ is the last term of $f$
and denote it “LT($f$)”. The integer $m$ will be called the length of $f$. We also set

$$
\mathcal{H}_{\mathbf{x}}^{F} := \mathcal{L}[\text{FT}(f) \mid f \in \mathcal{H}_{\mathbf{x};q}] \quad \text{and} \quad \mathcal{H}_{\mathbf{x}}^{L} := \mathcal{L}[\text{LT}(f) \mid f \in \mathcal{H}_{\mathbf{x};q}] \qquad (16)
$$

to respectively stand for the span of first terms of *q*-harmonics and last terms. Using a theorem of [3] we
then get the following remarkable corollary.

**Corollary 5** The three spaces $\mathcal{H}_{\mathbf{x}}^F$, $\mathcal{H}_{\mathbf{x}}^L$ and $\mathcal{H}_{\mathbf{x};q}$ are isomorphic as graded $\mathfrak{S}_n$-modules and therefore they are all isomorphic to a submodule of the Harmonics of $\mathfrak{S}_n$.

Since the dimension of $\mathcal{H}_{\mathbf{x};q}$ is thus bounded above, the single equality dim $\mathcal{H}_{\mathbf{x};q} = n!$ would imply that $\mathcal{H}_{\mathbf{x};q}$ affords the regular representation of $\mathfrak{S}_n$. In particular this would yield that $\mathcal{H}_{\mathbf{x}}^F$ is none other than the space of harmonics of $\mathfrak{S}_n$. Since $\mathcal{H}_{\mathbf{x};q}$ is isomorphic to $\mathcal{H}_{\mathbf{x}}^F$, as a graded $\mathfrak{S}_n$-module, it would follow that $\mathcal{H}_{\mathbf{x};q}$ itself is isomorphic to the space of harmonics of $\mathfrak{S}_n$ (as a graded $\mathfrak{S}_n$-module). Thus the Hivert-Thiéry conjecture would result.

4 The Kernel of $D_k$

To compute the general space $\mathcal{H}_x$ of harmonic polynomials, we need to find common solutions of the
differential equations $D_k f(x) = 0$, for $k > 0$. For each $k$, the kernel of the operator $D_k$ may be given a
precise explicit description whenever $a_{i,k} d + b_{i,k} \neq 0$, for all $d \in \mathbb{N}$. We lighten the notation by writing
simply $a_i$ instead of $a_{i,k}$.

The case $k=1$ illustrates all aspects of the method. We construct a set

$$
\{\mathbf{y}^{\mathbf{r}} + \Psi_1(\mathbf{y}^{\mathbf{r}})\}_{\mathbf{r} \in \mathbb{N}^{n-1}} \quad (17)
$$

which is a basis of the solution set of $D_1 f(\mathbf{x}) = 0$. Here, $\Psi_1$ is a linear operator described below. Simply
writing $x$ for $x_n$, and $y$ for the set of variables $x_1, \dots, x_{n-1}$, we expand $f \in \mathbb{R}[\mathbf{x}]$ as polynomials in $x$:

$$
f = \sum_{d} f_d \frac{x^d}{d!}, \quad \text{with } f_d \in \mathbb{R}[\mathbf{y}]. \tag{18}
$$
The effect of $D_1$ can then be described in the format

$$D_1 \left( \sum_d f_d \frac{x^d}{d!} \right) = \sum_d [D_1(f_d) + (d a_n + b_n) f_{d+1}] \frac{x^d}{d!} \quad (19)$$

Setting $a := a_n$ and $b := b_n$, we now assume that $a d + b \neq 0$, for all $d \in \mathbb{N}$. Then, the right-hand side of (19) vanishes if and only we choose $f$ to be such that

$$f_{d+1} = \frac{-1}{a d + b} D_1(f_d), \quad (20)$$

for all $d \ge 0$. Unfolding this recurrence for the $f_d$'s, we find that every element of the kernel of $D_1$ can be written as $f_0 + \Psi_1(f_0)$, if we define the linear operator $\Psi_1$ as

$$\Psi_1(g) := \sum_{m \ge 1} (-1)^m \frac{D_1^m(g)}{[a; b]_m} \frac{x^m}{m!}, \quad \text{for } g \in \mathbb{R}[\mathbf{y}]. \quad (21)$$

Here we use the notation $[a; b]_m := b(a+b)(2a+b)\cdots((m-1)a+b)$. This leads to the following theorem.

**Theorem 6** The collection of polynomials $\mathbf{y}^r + \Psi_1(\mathbf{y}^r)$ is a basis for the kernel of $D_1$. In fact, given any polynomial $f$ in the kernel of $D_1$, its expansion in terms of this basis is simply obtained as

$$f = \sum_{\mathbf{r}} a_{\mathbf{r}}(\mathbf{y}^{\mathbf{r}} + \Psi_1(\mathbf{y}^{\mathbf{r}})) \quad (22)$$

with $(f \bmod x) = \sum_{\mathbf{r}} a_{\mathbf{r}} \mathbf{y}^{\mathbf{r}}$.

It follows readily that, whenever $a d + b \neq 0$ for all $d \in \mathbb{N}$, the Hilbert series of the dimension of the kernel of $D_1$ is $(1-t)^{1-n}$. In view of Theorem 1, this implies that the Hilbert series of the kernel of $\tilde{D}_1$ is

$$1 + \sum_{k=1}^{n} \binom{n}{k} t^k \frac{1}{(1-t)^{k-1}}. \quad (23)$$

In fact, we can get an explicit description of this kernel using (9).

We can generalize formula (21) to get a description of the kernel of $D_k$ as follows. Observe as before that

$$D_k \left( \sum_d f_d \frac{x^d}{d!} \right) = \sum_d [D_k(f_d) + (a+d+b)f_{d+k}] \frac{x^d}{d!} \quad (24)$$

For this expression to be zero, we must have

$$f_{d+k} = \frac{-1}{a d + b} D_k(f_d),$$

with the same conditions as before on $a$ and $b$. This recurrence has a unique solution given initial values for $f_d$, $0 \le d \le k-1$. Clearly these can be fixed at leisure. Substituting the solution of the recurrence in $f$, we get an element of the kernel of $D_k$ if and only if $f$ is of the form

$$f = (f \bmod x^k) + \Psi_k(f \bmod x^k),$$
with $\Psi_k$ the linear operator defined as

$$ \Psi_k \left( \sum_{r=0}^{k-1} f_r \frac{x^r}{r!} \right) := \sum_{m \ge 1} \sum_{r=0}^{k-1} (-1)^m \frac{D_k^m(f_r)}{[a k; a r + b]_m} \frac{x^{k m + r}}{(k m + r)!} \quad (25) $$

In particular, it follows that the Hilbert series of the kernel of $D_k$ is $(1 + t + \dots t^{k-1})(1 - t)^{1-n}$.

# 5 Some explicit harmonic polynomials

Common zeros of all $D_k$'s are exactly what we are looking for. Some of these are easy to find when the $D_k$'s are symmetric. Let $\lambda$ be any partition of $n$, and consider a tableau $\tau$ of shape $\lambda$, this is to say a bijection

$$ \tau : \lambda \to \{1, 2, \dots, n\}, $$

with $\lambda$ identified with the set of cells of its Ferrers diagram. Recall that, for $\lambda = \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_k > 0$, the cells of $\lambda$ are the $n$ pairs $(i, j)$ in $\mathbb{N}^2$, such that

$$ 1 \le i \le \lambda_j, \quad 1 \le j \le k. $$

The value $\tau(i, j)$ is called an entry of $\tau$, and it is said to lie in column $i$ of $\tau$. The Garnir polynomial of a $\lambda$-shape tableau $\tau$, is defined to be

$$ \Delta_{\tau}(\mathbf{x}) := \prod_{i, j < k} (x_{\tau(i,j)} - x_{\tau(i,k)}). $$

In other terms, the factors that appear in $\Delta_{\tau}(\mathbf{x})$ are differences of entries of $\tau$ that lie in the same column.

Now, define $\mathcal{V}_{\lambda}$ to be the linear span of the polynomials $\Delta_{\tau}$, for $\tau$ varying in the set of tableaux of shape $\lambda$. In formula,

$$ \mathcal{V}_{\lambda} := \mathbb{R}\{\Delta_{\tau} | \tau \text{ tableau of shape } \lambda\}. $$

In other words, $\mathcal{V}_{\lambda}$ is the linear span of the $\Delta_{\tau}$. It is well known that this homogeneous (invariant) subspace is an irreducible representation of $\mathfrak{S}_n$ of dimension equal to the number of standard Young tableaux. Moreover, in the ring $\mathbb{R}[\mathbf{x}]$, there exists no isomorphic copy of this irreducible representation lying in some homogeneous component of degree lower then that in which lies $\mathcal{V}_{\lambda}$. It is easy to check that the degrees of all of the $\Delta_{\tau}$'s, for a tableau of shape $\lambda$, are all equal to

$$ \sum_{i=1}^{\ell(\lambda)} (i-1) \lambda_i, $$

which is usually denoted $n(\lambda)$ in the literature (see [4]). This is the smallest possible value for the cocharge of a standard tableau of shape $\lambda$. This fact has the following easy implication.

**Proposition 7** For any tableau $\tau$ of shape $\lambda$, the Garnir polynomial $\Delta_{\tau}(\mathbf{x})$ is a zero of $D_k$, for $k \ge 1$, whenever $D_k$ is symmetric.
A direct consequence of this is that there is at least one copy of each irreducible representation of $\mathfrak{S}_n$ in $\mathcal{H}_x$, when the $D_k$'s are all symmetric. Moreover, under the same conditions, we have

$$ \sum_{\lambda \vdash n} f_{\lambda} t^{n(\lambda)} \ll H_n(t), $$

with “$\ll$” denoting coefficientwise inequality, and $H_n(t)$ as in (3).

# 6 A new regular sequence and a universal dimension bound

The goal of this section is to establish a bound for the dimension of $\mathcal{H}_{x;q}$ which is valid for all values of $q$. To carry this out we need some auxiliary results from commutative algebra. Let $\mathbb{F}$ be an algebraically closed field and let $\theta_1(\mathbf{x})$, $\theta_2(\mathbf{x})$, ..., $\theta_n(\mathbf{x})$ be homogeneous polynomials of $\mathbb{F}[\mathbf{x}]$ of respective degrees $d_1, d_2, \dots, d_n$. The following result is basic.

**Proposition 8** The polynomials $\theta_1(\mathbf{x})$, $\theta_2(\mathbf{x})$, ..., $\theta_n(\mathbf{x})$ form a regular sequence in $\mathbb{F}[\mathbf{x}]$ if and only if the system of equations

$$ \theta_1(\mathbf{x}) = 0, \theta_2(\mathbf{x}) = 0, \dots, \theta_n(\mathbf{x}) = 0 $$

has, for $\mathbf{x} \in \mathbb{F}^n$, the unique solution

$$ x_1 = 0, x_2 = 0, \dots, x_n = 0. $$

We next make use of this proposition to study the sequence of polynomials

$$ \varphi_m(\mathbf{x}) := \sum_{i=1}^{n} a_i x_i^m, $$

for $m \ge 0$. More precisely we seek to obtain conditions on the coefficient sequence

$$ \mathbf{a} = (a_1, a_2, \dots, a_n) \in \mathbb{F}^n \qquad (26) $$

which assure that, for a given $k \ge 1$, that the polynomials

$$ \varphi_k(\mathbf{x}), \varphi_{k+1}(\mathbf{x}), \dots, \varphi_{k+n-1}(\mathbf{x}) $$

form a regular sequence in $\mathbb{F}[\mathbf{x}]$.

We first observe that the polynomials $\varphi_m(\mathbf{x})$, for $m > n$, may be expressed in term of the $\varphi_k(\mathbf{x})$'s, for $1 \le k \le n$. Indeed, recall that the ordinary elementary symmetric functions $e_r(\mathbf{x})$ may be presented in the form of the identity

$$ (t - x_1)(t - x_2) \cdots (t - x_n) = \sum_{r=0}^{n} (-1)^r e_r(\mathbf{x}) t^{n-r}. $$

Setting $t = x_i$, we obtain

$$ \sum_{r=0}^{n} (-1)^r e_r(\mathbf{x}) x_i^{n-r} = 0. $$
Multiplying both sides by $a_i x_i^{m-n}$ and isolating $a_i x_i^m$, we get

$$a_i x_i^m = - \sum_{r=1}^{n} (-1)^r e_r(\mathbf{x}) a_i x_i^{m-r}.$$

Thus, summing up on $i$, the following recurrence results

$$\varphi_m(\mathbf{x}) = \sum_{r=1}^{n} (-1)^{r+1} e_r(\mathbf{x}) \varphi_{m-r}(\mathbf{x}). \qquad (27)$$

Unfolding this recurrence, we conclude that $\varphi_m$ lies in the ideal $(\varphi_1, \varphi_2, \dots, \varphi_n)_{\mathbb{F}[\mathbf{x}]}$, for all $m \ge 1$.

**Remark 1** It is interesting to observe that identity (27) yields that

$$\varphi_1(\mathbf{x}), \varphi_2(\mathbf{x}), \dots, \varphi_n(\mathbf{x}) \qquad (28)$$

is never a regular sequence when $a_1 + a_2 + \dots + a_n = 0$. Indeed, setting $m = n$ in (27), we get

$$\varphi_m(\mathbf{x}) = \sum_{r=1}^{n-1} \varphi_{m-r}(\mathbf{x}) (-1)^{r+1} e_r(\mathbf{x}) + (-1)^{n+1} e_n(\mathbf{x}) (a_1 + a_2 + \dots + a_n)$$

and thus the vanishing of $a_1 + a_2 + \dots + a_n$ forces $\varphi_n(\mathbf{x})$ to vanish modulo the ideal

$$(\varphi_1, \varphi_2, \dots, \varphi_{n-1})_{\mathbb{F}[\mathbf{x}]}.$$

Let us now denote

$$\Phi_n^k := (\varphi_k, \varphi_{k+1}, \dots, \varphi_{k+n-1})_{\mathbb{F}[\mathbf{x}]},$$

the ideal in $\mathbb{F}[\mathbf{x}]$ generated by the $n$ polynomials $\varphi_\ell(\mathbf{x})$, with $k \le \ell \le k+n-1$. We also write $\Phi_n$ for $\Phi_n^1$. Proposition 8 and (27) combine to yield the following remarkable result.

**Theorem 9** For any $k \ge 1$ the sequence

$$\varphi_k(\mathbf{x}), \varphi_{k+1}(\mathbf{x}), \dots, \varphi_{k+n-1}(\mathbf{x}), \qquad (29)$$

is regular if and only if the sequence

$$\varphi_1(\mathbf{x}), \varphi_2(\mathbf{x}), \dots, \varphi_n(\mathbf{x}), \qquad (30)$$

is regular.

This given, here and after we need only be concerned with finding conditions on $a_1, a_2, \dots, a_n$ that assure the regularity of sequence $\varphi_1, \varphi_2, \dots, \varphi_n$. The following result offers a useful criterion.

**Theorem 10** In the ring $\mathbb{F}[\mathbf{x}]$, the polynomials

$$\varphi_1, \varphi_2, \dots, \varphi_n$$
form a regular sequence if and only if we have

$$x_i^{\binom{n}{2}+1} \in \Phi_n. \qquad (31)$$

When this happens we have the Hilbert series equalities

$$F_{\mathbb{F}[x]/\Phi_n^k}(t) = [k]_t[k+1]_t \cdots [k+n-1]_t \qquad (32)$$

and, in particular,

$$\dim \mathbb{F}[x]/\Phi_n^k = (k)(k+1)\cdots(k+n-1).$$

Going along the lines of Remark 1, we are now ready to assert the following characterization of the $a_i$'s for which we have regularity.

**Theorem 11** For $k > 1$, the sequences

$$\varphi_k, \varphi_{k+1}, \dots, \varphi_{k+n-1} \qquad (33)$$

is regular if and only if we have

$$a_{i_1} + a_{i_2} + \cdots + a_{i_k} \neq 0, \qquad (34)$$

for all $1 \le i_1 < i_2 < \cdots < i_k \le n$.

We intend to derive the consequences of this assumption in the theory of $q$-harmonics. First, we simply reformulated every statement modulo the substitution of variables

$$\begin{aligned} (a_1, a_2, \dots, a_n) &\mapsto \mathbf{x} = (x_1, x_2, \dots, x_n), \\ \mathbf{x} &\mapsto \xi = (\xi_1, \xi_2, \dots, \xi_n), \end{aligned}$$

and we now have

$$\Phi_n^k = (\varphi_k(\xi), \varphi_{k+1}(\xi), \dots, \varphi_{k+n-1}(\xi))_{\mathbb{F}_x[\xi]}.$$

This given, from Theorem (10) we can derive the following facts about the ring

$$\mathbb{F}_x[\xi_1, \xi_2, \dots, \xi_n],$$

where now, $\mathbb{F}_x$ denotes the field of rational functions in $x$ with coefficients in $\mathbb{F}$.

**Theorem 12** Let

$$u_1(\xi), u_2(\xi), \dots, u_{(n+1)!}(\xi)$$

be a monomial basis for the quotient

$$\mathbb{F}_x[\xi]/\Phi_n^k,$$

and let $\deg(u_i) = d_i$. Then every polynomial $f(\xi) \in \mathbb{F}_x[\xi]$, which is homogeneous of degree $d$, has a unique expansion of the form

$$f(\xi) = \sum_{i=1}^{(n+1)!} u_i(\xi) \sum_{\sum_k r_k (k+1) = d - d_i} a_{i;r}(\mathbf{x}) \varphi_1^{r_1}(\xi), \varphi_2^{r_2}(\xi) \cdots \varphi_n^{r_n}(\xi), \qquad (35)$$

where the coefficients $a_{i;r}(\mathbf{x})$ are rational functions of $\mathbf{x}$, for $\mathbf{r} \in \mathbb{N}^n$. In particular if $d > \binom{n+1}{2}$ then

$$f(\xi) \equiv 0 \mod \Phi_n^k. \qquad (36)$$
Let us now denote by $\mathcal{D}(\mathbf{x})$ the algebra of differential operators with coefficients in $\mathbb{F}_x$. Moreover, let $\mathcal{D}_d(\mathbf{x})$ denote the subspace of $\mathcal{D}(\mathbf{x})$ consisting of operators of order $d$. More precisely we have $D \in \mathcal{D}_d(\mathbf{x})$ if and only if $D$ may be expanded in the form

$$D = \sum_{|\mathbf{r}| \le d} a_{\mathbf{r}}(\mathbf{x}) \partial_{\mathbf{x}}^{\mathbf{r}} \qquad (37)$$

with coefficients $a_{\mathbf{r}}(\mathbf{x}) \in \mathbb{F}_x$ such that $a_{\mathbf{r}}(\mathbf{x}) \neq 0$ at least once when $|\mathbf{r}| = d$. We are here extending our vectorial notation to operators, so that

$$\partial_{\mathbf{x}}^{\mathbf{r}} = \partial_{1}^{r_{1}} \partial_{2}^{r_{2}} \cdots \partial_{n}^{r_{n}}$$

is an operator of order $|\mathbf{r}| = r_1 + r_2 + \cdots + r_n$. The degree condition in (37) imply that the polynomial

$$\sigma(D) := \sum_{|\mathbf{r}|=d} a_{\mathbf{r}}(\mathbf{x}) \xi^{\mathbf{r}}.$$

does not identically vanish. We will refer to $\sigma(D)$ as the “symbol” of $D$.

This given, as a corollary of Theorem (10), we obtain the following basic result for Steenrod operators

**Theorem 13** Every operator $D \in \mathcal{D}_d(\mathbf{x})$ has an expansion of the form

$$D = \sum_{i=1}^{(n+1)!} \sum_{\sum_l r_k (k+1) \le d-d_i} a_{i;\mathbf{r}}(\mathbf{x}) u_i(\partial_{\mathbf{x}}) D_{1;q}^{r_1} D_{2;q}^{r_2} \cdots D_{n;q}^{r_n}$$

where $d_i = \deg(u_i)$ and $a_{i;\mathbf{r}}(\mathbf{x}) \in \mathbb{F}_x$. Note that this holds true for any rational value of $q$.

We may now establish the main goal of this section.

**Theorem 14** For any value of $q$ the dimension of the space of $q$-Harmonic polynomials in $\mathbf{x}$ does not exceed $(n+1)!$

# 7 Last Considerations

Further computer experiments suggest that we have

**Conjecture 3** The set $\mathcal{D}_n^a$ of common polynomial zeros of the operators

$$\sum_{i=1}^{n} a_i \partial_{x_i}^k \partial_{y_i}^j,$$

for all $k, j \in \mathbb{N}$ such that $k + j > 0$, is of a bigraded space of dimension $(n+1)^{n-1}$, whenever we have $\mathbf{a} = (a_1, \dots, a_n)$ such that

$$\sum_{k \in K} a_k \neq 0, \qquad (38)$$

for all nonempty subsets $K$ of $\{1, \dots, n\}$.
Another interesting experimental observation concerning the space of common zeros of $D_1$ and $D_2$ with general operators

$$D_1 := \sum_{i=1}^{n} a_i x_i \partial_i^2 + b_i \partial_i,$$

$$D_2 := \sum_{i=1}^{n} c_i x_i \partial_i^3 + d_i \partial_i^2,$$

is that there seem to be conditions, similar to (38), for which this space is always $n$-dimensional.

## References

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[3] F. Hivert and N. Thiéry, *Non-commutative deformation of symmetric functions and the integral Steenrod algebra*, in: Invariant theory in all characteristics, volume 35 of CRM Proc. Lecture Notes, pages 91-125, Amer. Math. Soc., Providence, RI, 2004.

[4] I. G. Macdonald, *Symmetric functions and Hall polynomials*, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications.

[5] R.M.W. Wood, *Problems in the Steenrod algebra*, Bull. London Math. Soc. 30 (1998) 449–517.