An ultimately periodic chain in the integral Lie ring of partitions

Abstract

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in $\{1,2,\dots , n-1\}$. Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic.

1 Introduction and preliminaries

Given an integer $n\ge 3$, we defined in a recent work a Lie ring structure on the set of partitions with parts in $\{1,2,\dots ,n-1\}$ and where each part is allowed to have at most $m-1$ repetitions, for some given $m > 2$. Within the acquired structure, here denoted as ${{\,\mathrm{{\mathfrak {L}}}\,}}_m(n)$, we recursively defined a chain of idealizers $({\mathfrak {N}}_i)_{i \ge -1}$ originating from an abelian subring ${\mathcal {T}}$. We proved that the rank of ${\mathfrak {N}}_{i}/{\mathfrak {N}}_{i-1}$ as free ${{\,\mathrm{{\mathbb {Z}}}\,}}_m$-module is $q_{i+1}$, where $q_i$ is the partial sum of sequence counting the number of partitions of i into at least two parts, each allowed to be repeated at most $m-1$ times [1]. The objective was to tackle the unresolved problem posed by Aragona et al. [3] of computing the growth of a chain of normalizers in a Sylow m-subgroup starting from an elementary abelian regular subgroup $T <{{\,\textrm{Sym}\,}}(m^n)$, when m is an odd prime. Indeed, ${{\,\mathrm{{\mathfrak {L}}}\,}}_m(n)$ was constructed as the iterated wreath product of Lie rings of rank one [1] and, when m is prime, this corresponds exactly to the construction of the graded Lie algebra associated with the lower central series of a Sylow m-subgroup of ${{\,\textrm{Sym}\,}}(m^n)$ [7]. In this construction, the abelian subring ${\mathfrak {N}}_{-1} = {\mathfrak {T}}$ at the base of the idealizer chain corresponds to the elementary abelian regular subgroup T at the base of normalizer chain. It is important to stress that the combinatorial equality mentioned above is valid only for the first $n-2$ terms of the chain, and the problem of understanding the general behavior of the chain is out of reach at the time of writing.

In this work, we address a similar problem in the case of characteristic zero, i.e., in a Lie ring ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ with integer coefficients of partitions with parts in $\{1,2, \dots , n-1\}$. The main combinatorial difference between the two settings is that now each part is allowed to have an unbounded number of repetitions. We show that this significantly affects how the idealizer chain grows. In particular, we prove here that the sequence of consecutive quotient ranks in the idealizer chain depends on the second partial sum of the sequence of the integer partitions and is (ultimately) periodic.

We conclude this section by introducing the construction of the partition Lie ring with integer coefficients as a Lie subring of the Witt algebra and by showing some preliminary properties.

1.1 Preliminaries on the Witt algebra

Let ${\mathfrak {L}}$ and ${\mathfrak {H}}$ be finite dimensional Lie algebras over a field $K$, and let ${{\,\textrm{U}\,}}({\mathfrak {L}})$ be the universal enveloping algebra of ${\mathfrak {L}}$. It is well known that ${{\,\textrm{U}\,}}({\mathfrak {L}})$ is a Hopf algebra (see Kassel [4, Proposition V.2.4]) and that its dual space ${{\,\textrm{U}\,}}({\mathfrak {L}})^*={{\,\textrm{Hom}\,}}_K({\mathfrak {L}},K)$ is an associative algebra on which ${\mathfrak {L}}$ acts by way of the coadjoint action as a derivation Lie algebra. In particular, the tensor product ${{\,\textrm{U}\,}}({\mathfrak {L}})^* \otimes _K{\mathfrak {H}}$ inherits from ${\mathfrak {H}}$ a Lie algebra structure on which ${\mathfrak {L}}$ acts as a Lie algebra of derivations via its action on ${{\,\textrm{U}\,}}({\mathfrak {L}})^*$. The standard wreath product ${\mathfrak {H}}\wr {\mathfrak {L}}$ is then defined to be the semidirect product $({{\,\textrm{U}\,}}({\mathfrak {L}})^* \otimes _K{\mathfrak {H}}) \rtimes {\mathfrak {L}}$ [6]. In the special case, when ${\mathfrak {L}}=Ku$ is a one-dimensional Lie algebra and $K$ is a field of characteristic $0$, the algebra ${{\,\textrm{U}\,}}({\mathfrak {L}})=K[u^{(n)}]_{n\ge 0}$ is the associative $K$-algebra of the divided powers $u^{(n)}=\frac{u^n}{n!}$, where $u^{(i)}u^{(j)}=\left( {\begin{array}{c}i+j\\ i\end{array}}\right) u^{(i+j)}$. Its dual ${{\,\textrm{U}\,}}({\mathfrak {L}})^*$ can be then identified with the formal power series ring $K[[x]]$ on which ${\mathfrak {L}}$ acts as an algebra of standard derivations, i.e., $u\cdot f(x)=\frac{\partial }{\partial x}f(x)$.

We recall (see Strade [8, Chapter 2]) that the $n$-th Witt Lie algebra is defined as

$$\begin{aligned}{{\,\textrm{W}\,}}(n):=\left\{ \sum _{k=1}^n f_k\partial _k \mid f_k\in K[[x_1,\dots ,x_n]] \right\} \end{aligned}$$

with bracket operation defined on its basis via

$$\begin{aligned}:=\left( f \frac{\partial }{\partial _k}g\right) \partial _h - \left( g \frac{\partial }{\partial _h}f\right) \partial _k. \end{aligned}$$

The iterated (restricted) wreath product

$$\begin{aligned}(Ku)^{\wr n}:= \underbrace{Ku \wr \cdots \wr Ku}_{n \text { times}}\end{aligned}$$

can be identified with the subalgebra of ${{\,\textrm{W}\,}}(n)$ having the monomial elements $x_{i_1}^{\lambda _{i_1}}\cdots x_{i_k}^{\lambda _{i_h}}\partial _k$, where $1\le i_1< i_2< \dots< i_h < k\le n$, as a basis. A similar construction, with $K={\mathbb {F}}_p$ and divided powers of exponent at most $p-1$, has been used to describe the graded Lie algebra associated with the lower central series to the iterated wreath product of $n$ copies of the cyclic group of order $p$, i.e., the Sylow $p$-subgroup of the symmetric group ${{\,\textrm{Sym}\,}}(p^n)$ [7].

1.2 The integral ring of partitions

In the present paper, we consider the integral ring of partitions, i.e., the Lie subring of the iterated wreath product $({{\,\mathrm{{\mathbb {Q}}}\,}}u)^{\wr n}$, generated over ${{\,\mathrm{{\mathbb {Z}}}\,}}$ by the monomial elements described above. We now give a description of this ring starting from the construction of the Lie ring over ${{\,\mathrm{{\mathbb {Z}}}\,}}_m$ of partitions with bounded maximal part [1], customizing it for $m=0$. We advise the reader to also refer to the cited paper for further details and to the original problem of characterizing the chain of normalizers originating from a regular elementary abelian group of order $2^n$ in the Sylow $2$-subgroup of the symmetric group [2, 3].

Let $\Lambda = \{\lambda _i\}_{i=1}^\infty$ be a sequence of non-negative integers with finite support, i.e.,

$$\begin{aligned} {{\,\textrm{wt}\,}}(\Lambda )= \sum _{i=1}^\infty i\lambda _i < \infty . \end{aligned}$$

The sequence $\Lambda$ defines a partition of $N = {{\,\textrm{wt}\,}}(\Lambda )$. Each nonzero i is a part of the partition, and the integer $\lambda _i$ is the multiplicity of the part $i$ in $\Lambda$. The maximal part of $\Lambda$ is the number $\max \bigl (\{i \mid \lambda _i \ne 0\}\bigr )$. The set of the partitions whose maximal part less than or equal to $j$ is denoted by ${{\,\mathrm{{\mathcal {P}}}\,}}(j)$.

The power monomial $x^\Lambda$, where $\Lambda$ is a partition, is defined as $x^\Lambda = \prod _{i} x_i^{\lambda _i}$. Given a positive integer $n$ and denoting by $\partial _k$ the derivation given by the standard partial derivative with respect to $x_k$, where $1 \le k \le n$, we define by ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ the free ${{\,\mathrm{{\mathbb {Z}}}\,}}$-module spanned by the basis

$$\begin{aligned}{\mathcal {B}}:=\left\{ x^\Lambda \partial _k \mid 1\le k \le n \text { and } \Lambda \in {{\,\mathrm{{\mathcal {P}}}\,}}(k-1) \right\} , \end{aligned}$$

and we set ${\mathcal {B}}_u:= \left\{ x^\Lambda \partial _k \in {\mathcal {B}} \mid k=u \right\}$. The module ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ is endowed with a structure of Lie ring, where the Lie bracket is defined on the basis ${\mathcal {B}}$ by

$$\begin{aligned} \left[ x^\Lambda \partial _k, x^\Theta \partial _j\right] :=&\partial _{j}(x^\Lambda ) x^\Theta \partial _k - x^\Lambda \partial _{k}(x^\Theta ) \partial _j \nonumber \\ =&{\left\{ \begin{array}{ll} \partial _{j}(x^\Lambda ) x^\Theta \partial _k &{} \text {if j<k}, \\ - x^\Lambda \partial _{k}(x^\Theta ) \partial _j &{} \text {if j>k},\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

This operation is then extended to a Lie product on ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ by bilinearity, and the resulting structure ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ is called the integral Lie ring of partitions with parts in $\{1,2,\dots , n-1\}$.

In the remainder of the paper, we deal with homogeneous subrings of ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$, which are defined as follows.

Definition 1.1

A Lie subring ${\mathfrak {H}}$ of ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ is said to be homogeneous if it is the free ${{\,\mathrm{{\mathbb {Z}}}\,}}$-module spanned by some subset ${\mathcal {H}}$ of ${\mathcal {B}}$.

The following result on homogeneous subrings can be proved as in the case of the modular ring ${{\,\mathrm{{\mathfrak {L}}}\,}}_m(n)$ [1, Theorem 2.5]. Here and the remainder of the paper, if ${\mathcal {H}}$ is a subset of ${\mathcal {B}}$, then its idealizer is defined as

$$\begin{aligned} N_{\mathcal {B}}({\mathcal {H}}):= \left\{ b\in {\mathcal {B}} \mid [b,h] \in {{\,\mathrm{{\mathbb {Z}}}\,}}{\mathcal {H}} \text { for all } h\in {\mathcal {H}} \right\} . \end{aligned}$$

Theorem 1.2

Let ${\mathfrak {H}}$ be a homogeneous subring of ${{\,\mathrm{{\mathfrak {L}}}\,}}$ having basis ${\mathcal {H}} \subseteq {\mathcal {B}}$. The idealizer of ${\mathfrak {H}}$ in ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ is the homogeneous subring of ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ spanned by $N_{\mathcal {B}}({\mathcal {H}})$ as a free ${{\,\mathrm{{\mathbb {Z}}}\,}}$-module.

The chain of idealizers in ${{\,\mathrm{{\mathfrak {L}}}\,}}_m(n)$ originated from the abelian homogeneous Lie subring ${\mathfrak {T}}=\left\langle \partial _1,\dots ,\partial _n\right\rangle$ [1]. Analogously, here we deal with the idealizer chain defined in ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ starting from ${\mathfrak {T}}$ and defined as follows:

$$\begin{aligned} {\mathfrak {N}}_i = {\left\{ \begin{array}{ll} {\mathfrak {T}} &{} i=-1,\\ N_{{{\,\mathrm{{\mathfrak {L}}}\,}}(n)}({\mathfrak {T}}) &{} i=0,\\ N_{{{\,\mathrm{{\mathfrak {L}}}\,}}(n)}({\mathfrak {N}}_{i-1}) &{} i \ge 1. \end{array}\right. } \end{aligned}$$

(1)

1.3 Organization of the paper

The remainder of the paper is organized as follows: Sect. 2 is devoted to combinatoric aspects of ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$: we introduce a chain of subsets $({\mathcal {N}}_i)_{i \ge -1}$, and we show that the cardinalities of ${\mathcal {N}}_i\setminus {\mathcal {N}}_{i-1}$ depend, up to periodicity, on the second partial sum of the sequence of integer partitions (see Corollary 2.12 and 2.13). In Sect. 3, we show (see Theorem 3.4) that the free ${{\,\mathrm{{\mathbb {Z}}}\,}}$-modules spanned by the sets ${\mathcal {N}}_i$s coincide with the idealizers of Eq. (1), yielding our main contribution of Corollary 3.6, which connects, up to periodicity, the rank of the free ${{\,\mathrm{{\mathbb {Z}}}\,}}$-modules ${\mathfrak {N}}_i/{\mathfrak {N}}_{i-1}$ with the second partial sum of the sequence of integer partitions.

2 Combinatorics of the integral Lie ring of partitions

The methods crafted for positive characteristic scenarios are not well-suited for the integral Lie ring of partitions. This situation calls for the introduction of novel combinatorial tools that we are in the process of defining.

2.1 Levels of the basis elements

As we will show in the remainder of this paper, the combinatorial properties of ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ depend on the behavior of the functions defined below.

Definition 2.1

Let $i\ge -1$ be an integer and let $1\le r_i \le n-1$ be such that $i\equiv r_i \bmod (n-1)$. Write

$$\begin{aligned} i=(h_i-1)(n-1)+r_i, \end{aligned}$$

where

$$\begin{aligned} h_i:=\left\lfloor \dfrac{i-1}{n-1}\right\rfloor +1. \end{aligned}$$

The weight-degree function is defined as ${{\,\textrm{wd}\,}}:{{\,\mathrm{{\mathbb {Z}}}\,}}{\mathcal {B}} \rightarrow {{\,\mathrm{{\mathbb {Z}}}\,}}$ by

$$\begin{aligned} {{\,\textrm{wd}\,}}(c \cdot x^\Lambda \partial _{k}):= {{\,\textrm{wt}\,}}(\Lambda )-\deg (x^\Lambda )+n-k, \end{aligned}$$

and the $i$-th level function ${{\,\textrm{lev}\,}}_i:{{\,\mathrm{{\mathbb {Z}}}\,}}{\mathcal {B}} \rightarrow {{\,\mathrm{{\mathbb {Z}}}\,}}$ by

$$\begin{aligned} {{{\,\textrm{lev}\,}}_i(c \cdot x^\Lambda \partial _{k}):= h_i{{\,\textrm{wd}\,}}(x^\Lambda \partial _k)+\deg (x^\Lambda )-1.} \end{aligned}$$

We now show that the weight-degree function is bounded.

Lemma 2.2

Let $x^\Lambda \partial _{k} \in {\mathcal {B}}$. Then ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) \ge n-k$ and ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) = n-k$ if and only if $x^\Lambda \partial _{k}=x_1^{\lambda _1}\partial _k$. Moreover, assume that ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k})\le i$ for some $i$. Then ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) \le n-1$ and ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) = n-1$ if and only if $x^\Lambda \partial _k=\partial _1$.

Proof

Note that ${{\,\textrm{wd}\,}}(x^\Lambda \partial _k)= \left( {{\,\textrm{wt}\,}}(\Lambda )-\deg (x^\Lambda )\right) + n-k > n-k$ unless ${{\,\textrm{wt}\,}}(\Lambda )=\deg (x^\Lambda )$, which is equivalent to $x^\Lambda \partial _{k}=x_1^{\lambda _1}\partial _k$.

Let now i be such that ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k})\le i$ and assume ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) \ge n-1$. Then

$$\begin{aligned} i&\ge {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k}) \\&= h_i{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})+\deg (x^\Lambda )-1 \\&\ge h_i(n-1)+\deg (x^\Lambda )-1 \\&= (n-1)+(h_i-1)(n-1)+r_i+\deg (x^\Lambda )-r_i-1\\&=i+(n-1)-r_i+\deg (x^\Lambda )-1. \end{aligned}$$

This implies that $(n-1)-r_i+\deg (x^\Lambda )-1 \le 0$. Since $r_i\le n-1$, this is possible only if either $r_i=n-1$ and $\deg (x^\Lambda ) \le 1$ or $r_i=n-2$ and $\deg (x^\Lambda )=0$.

In the first case, we have either $k=1$ and hence $x^\Lambda \partial _k=\partial _1$, as required, or $x^\Lambda =x_j^{\lambda _j}$, where $1\le j<k$ and $\lambda _j\le 1$. This implies $n-1\le {{\,\textrm{wd}\,}}(x^\Lambda \partial _k)= (j-1)\lambda _j+n-k \le j+n-k-1$ and consequently $k \le j$, which contradicts $j<k$.

In the second case, we have $x^\Lambda \partial _k=\partial _k$ and $n-1 \le {{\,\textrm{wd}\,}}(x^\Lambda \partial _k)=n-k$. This leads again to $k=1$ and $x^\Lambda \partial _k=\partial _1$. $\square$

Proposition 2.3

If $x^\Lambda \partial _k, x^\Theta \partial _u \in {\mathcal {B}}$ do not commute, then

$$\begin{aligned} \begin{aligned} {{\,\textrm{lev}\,}}_i([x^\Lambda \partial _{k}, x^\Theta \partial _{u}])&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k}) + {{\,\textrm{lev}\,}}_i(x^\Theta \partial _{u})-h_i(n-1)\\ \end{aligned} \end{aligned}$$

(2)

and

$$\begin{aligned} \begin{aligned} {{\,\textrm{wd}\,}}([x^\Lambda \partial _{k}, x^\Theta \partial _{u}])&= {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) + {{\,\textrm{wd}\,}}(x^\Theta \partial _{u}) -(n-1). \end{aligned} \end{aligned}$$

(3)

Moreover, if ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)\le i$ and ${{\,\textrm{lev}\,}}_j(x^\Theta \partial _u)\le j$ for some $i,j\ge -1$, then

$$\begin{aligned} \begin{aligned} {{\,\textrm{wd}\,}}([x^\Lambda \partial _{k}, x^\Theta \partial _{u}]) \le \min ({{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}), {{\,\textrm{wd}\,}}(x^\Theta \partial _{u})), \end{aligned} \end{aligned}$$

(4)

with equality if and only if one of $x^\Lambda \partial _{k}$ or $x^\Theta \partial _{u}$ is $\partial _1$.

Proof

Let $x^\Gamma \partial _u=[x^\Lambda \partial _{k}, x^\Theta \partial _{u}]$. We have

$$\begin{aligned} {{\,\textrm{wd}\,}}(x^\Gamma \partial _u)&={{\,\textrm{wt}\,}}(\Lambda )+{{\,\textrm{wt}\,}}(\Theta )-k -\deg (x^\Lambda )- \deg (x^\Theta ) + 1 +n-u\\&={{\,\textrm{wd}\,}}(x^\Lambda \partial _k)+{{\,\textrm{wd}\,}}(x^\Theta \partial _u) -(n -1). \end{aligned}$$

As a consequence,

$$\begin{aligned} {{\,\textrm{lev}\,}}_i(x^\Gamma \partial _u)=&h_i{{\,\textrm{wd}\,}}(x^\Gamma \partial _u) + \deg (x^\Gamma )-1 \\ =&h_i\bigl ({{\,\textrm{wd}\,}}(x^\Lambda \partial _k)+{{\,\textrm{wd}\,}}(x^\Theta \partial _u) -(n -1)\bigr ) + \deg (x^\Lambda ) +\deg (x^\Theta )-2 \\ =&{{\,\textrm{lev}\,}}_i(x^\Lambda \partial _u)+{{\,\textrm{lev}\,}}_i(x^\Theta \partial _k) -(n-1)h_i. \end{aligned}$$

The last part of the claim is a straightforward consequence of Lemma 2.2. $\square$

Remark 1

A direct check shows that

$$\begin{aligned} {{\,\textrm{lev}\,}}_j(x^\Lambda \partial _{k}) = {{\,\textrm{lev}\,}}_{i}(x^\Lambda \partial _{k}) +(h_j-h_i){{\,\textrm{wd}\,}}(x^\Lambda \partial _k). \end{aligned}$$

Lemma 2.4

If ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) \le i$, then there exists $j\le i$ such that ${{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k)=j$.

Proof

We argue by induction on $i$. If $i=-1$, then $h_{-1}=0$ and hence

$$\begin{aligned}{{\,\textrm{lev}\,}}_{-1}(x^\Lambda \partial _{k}) = \deg (x^\Lambda )-1 \le -1\end{aligned}$$

so $\deg (x^\Lambda )=0$ and ${{\,\textrm{lev}\,}}_{-1}(x^\Lambda \partial _{k}) = -1$.

Let now $i\ge 0$. We can assume that $\ell :={{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) < i$. Note that

$$\begin{aligned}{{\,\textrm{lev}\,}}_{\ell }(x^\Lambda \partial _{k})= {{\,\textrm{lev}\,}}_{i}(x^\Lambda \partial _{k})+ (h_\ell -h_i){{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) \le {{\,\textrm{lev}\,}}_{i}(x^\Lambda \partial _{k})=\ell .\end{aligned}$$

If ${{\,\textrm{lev}\,}}_{\ell }(x^\Lambda \partial _{k})=\ell$, then we have proved the claim, otherwise ${{\,\textrm{lev}\,}}_{\ell }(x^\Lambda \partial _{k})< {{\,\textrm{lev}\,}}_{i}(x^\Lambda \partial _{k})=\ell < i$ and the induction hypothesis is satisfied. Therefore, ${{\,\textrm{lev}\,}}_{j}(x^\Lambda \partial _{k}) = j$ for some $j\le \ell <i$. $\square$

Lemma 2.5

If ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) \le i$, then ${{\,\textrm{lev}\,}}_{i+(n-1)}(x^\Lambda \partial _k) \le i+(n-1)$. In particular, ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) > i$ for all $i\le h(n-1)$ if and only if ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) > i$ for all $i$ such that $(h-1)(n-1)+1 \le i\le h(n-1)$.

Proof

By Lemma 2.2, we have that ${{\,\textrm{wd}\,}}(x^\Lambda \partial _k)\le n-1$. Thus, $h_{i+(n-1)}=h_{i}+1$ and

$$\begin{aligned}{{\,\textrm{lev}\,}}_{i+(n-1)}(x^\Lambda \partial _k) = {{\,\textrm{wd}\,}}(x^\Lambda \partial _k) + {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) \le (n-1)+i.\end{aligned}$$

$\square$

2.2 Introducing the chain

We are now ready to introduce a chain of subsets of ${\mathcal {B}}$. We will prove later that they span, as free modules, the idealizers of Eq. (1).

Definition 2.6

For $i \ge -1$ let

$$\begin{aligned} {\mathcal {N}}_i:=\left\{ x^\Lambda \partial _{k} \mid {{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k) \le j \text { for some } j\le i \right\} . \end{aligned}$$

Moreover, for $i \ge 0$ let

$$\begin{aligned} {\mathcal {L}}_i:={\mathcal {N}}_i\setminus {\mathcal {N}}_{i-1}.\end{aligned}$$

Remark 2

Note that ${\mathcal {N}}_{-1}=\left\{ \partial _1,\dots ,\partial _n \right\}$, that ${\mathfrak {T}}$ is the free ${{\,\mathrm{{\mathbb {Z}}}\,}}$-module spanned by ${\mathcal {N}}_{-1}$, and that the subsets $\left\{ {\mathcal {N}}_i \right\} _{i\ge -1}$ constitute an ascending chain of ${\mathcal {B}}$. Note also that an easy application of Lemma 2.4 gives

$$\begin{aligned} {\mathcal {N}}_i=\left\{ x^\Lambda \partial _{k} \mid {{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k) = j \text { for some } j\le i \right\} . \end{aligned}$$

Furthermore, $x^{\Lambda }\partial _k\in {\mathcal {L}}_i$ if and only if $i$ is minimum such that ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) = i$, i.e.,

$$\begin{aligned} {\mathcal {L}}_i = \left\{ x^{\Lambda }\partial _{k} \in {\mathcal {B}} \Big {|} i= \min _j \left\{ j={{\,\textrm{lev}\,}}_j(x^{\Lambda }\partial _{k}) \right\} \right\} . \end{aligned}$$

Proposition 2.7

Let $i\ge 0$. If $x^\Lambda \partial _{k}\in {\mathcal {L}}_i$, then $r_i > {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})$. In particular if $k < n-r_i+1$, then ${\mathcal {L}}_i\cap {\mathcal {B}}_k=\emptyset$.

Proof

Assume by contradiction that $x^\Lambda \partial _{k}\in {\mathcal {L}}_i$ and $r_i \le {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})$. We have that $i$ is minimum such that $i={{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)$. Thus, $j:=i-{{\,\textrm{wd}\,}}(x^\Lambda \partial _k) > i -(n-1)$ is such that

$$\begin{aligned}(h_i-2)(n-1) < j=(h_i-1)(n-1)+r_i -{{\,\textrm{wd}\,}}(x^\Lambda \partial _k) \le (h_i-1)(n-1).\end{aligned}$$

Hence $h_j=(h_i-1)$ and, by Remark 1,

$$\begin{aligned}j=i-{{\,\textrm{wd}\,}}(x^\Lambda \partial _k)={{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) - {{\,\textrm{wd}\,}}(x^\Lambda \partial _k)= {{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k),\end{aligned}$$

where $j<i$, a contradiction.

If $k < n-r_i+1$, then $r_i\le n-k \le {{\,\textrm{wd}\,}}(x^\Lambda \partial _k)$. From above, we have ${\mathcal {L}}_i\cap {\mathcal {B}}_k=\emptyset$. $\square$

The following corollary is straightforward.

Corollary 2.8

If $r_t=1$ then ${\mathcal {L}}_t = \left\{ x_1^{t+1}\partial _{n} \right\}$.

2.3 Periodicity

We can now define a concept of periodicity for the sequence $\left\{ {\mathcal {L}}_i \right\} _{i\ge 0}$. The following definition is crucial.

Definition 2.9

The period function ${{\,\mathrm{\nu }\,}}:{\mathcal {B}} \rightarrow {\mathcal {B}}$ is defined by

$$\begin{aligned} {{\,\mathrm{\nu }\,}}(x^\Lambda \partial _k):= x_1^{n-1-{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})}x^\Lambda \partial _k. \end{aligned}$$

Remark 3

Note that if $x^\Lambda \partial _k\in {\mathcal {L}}_i$, then

$$\begin{aligned} {{\,\textrm{lev}\,}}_{i+n-1}({{\,\mathrm{\nu }\,}}(x^\Lambda \partial _k))&={{\,\textrm{lev}\,}}_{i}({{\,\mathrm{\nu }\,}}(x^\Lambda \partial _k))+ {{\,\textrm{wd}\,}}({{\,\mathrm{\nu }\,}}(x^\Lambda \partial _k)) \\&={{\,\textrm{lev}\,}}_{i}({{\,\mathrm{\nu }\,}}(x^\Lambda \partial _k))+ {{\,\textrm{wd}\,}}(x^\Lambda \partial _k) \\&={{\,\textrm{lev}\,}}_{i}(x^\Lambda \partial _k)+ \bigl (n-1 -{{\,\textrm{wd}\,}}(x^\Lambda \partial _k)\bigr )+ {{\,\textrm{wd}\,}}(x^\Lambda \partial _k) \\&={{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k}) +n-1=i+n-1, \end{aligned}$$

so ${{\,\mathrm{\nu }\,}}(x^\Lambda \partial _k) \in {\mathcal {L}}_{i+n-1}$, as $t=i+n-1$ is trivially minimum such that ${{\,\textrm{lev}\,}}_t({{\,\mathrm{\nu }\,}}(x^\Lambda \partial _k))=t$. Moreover, since the period map is injective, we have that $\left|{\mathcal {L}}_{i} \right| \le \left|{\mathcal {L}}_{i+n-1} \right|$.

Remark 4

Let $i\ge 1$ and $k> u\ge 1$. Suppose that $x^\Lambda \partial _{k}\in {\mathcal {L}}_i$ and $x^\Theta \partial _{u}\in {\mathcal {L}}_{h_i(n-1)}$. Equation (2) shows that $\lambda _u x^\Gamma \partial _k:=[x^\Lambda \partial _{k}, x^\Theta \partial _{u}]\in {{\,\mathrm{{\mathbb {Z}}}\,}}{\mathcal {L}}_i$, as ${{\,\textrm{lev}\,}}_i(x^\Gamma \partial _k)={{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k})$, whereas Eq. (3) shows that ${{\,\textrm{wd}\,}}(x^\Gamma \partial _k) < {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})$. Note also that the element $b_u=x_1^{h_i}x_{u-1}\partial _{u} \in {\mathcal {L}}_{h_i(n-1)}$. Hence, given $x^\Lambda \partial _{k}\in {\mathcal {L}}_i$, there exists a sequence $u_1,u_2,\dots ,u_s$ such that

$$\begin{aligned} = \lambda x_1^{i-h_i(n-k)+1}\partial _k\in {{\,\mathrm{{\mathbb {Z}}}\,}}{\mathcal {L}}_i\end{aligned}$$

for a suitable non-zero $\lambda \in {{\,\mathrm{{\mathbb {Z}}}\,}}$. Equation (3) shows that ${{\,\textrm{wd}\,}}([x^\Lambda \partial _{k}, b_{u}])={{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})-1$. Hence, from Lemma 2.2, we have $s\le k-2$. Moreover, if $\lambda _{u} \ne 0$, then

$$\begin{aligned}{}[x^\Lambda \partial _{k}, \underbrace{b_{u},\dots ,b_{u}}_{\lambda _{u}\text { times}},\underbrace{b_{u-1}, \dots ,b_{u-1}}_{\lambda _{u}\text { times}},\dots ,\underbrace{b_{2},\dots ,b_{2}}_{\lambda _{u}\text { times}}] = {\lambda x_u^{-\lambda _{u}}x_1^{\lambda _{u}(u+(h_i-1)(u-1))}x^\Lambda \partial _k} \end{aligned}$$

for some nonzero $\lambda \in {{\,\mathrm{{\mathbb {Z}}}\,}}$, so that $\sum \lambda _{u}(u-1) =s \le k-2$, a condition which is trivially equivalent to ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) \le n-2$, as already seen in Lemma 2.2.

The previous remark together with Eq. (3) and Lemma 2.2 yields the following result.

Proposition 2.10

If ${\mathcal {L}}_i \cap {\mathcal {B}}_k\ne \emptyset$, then there exists at most one element in ${\mathcal {L}}_i \cap {\mathcal {B}}_k$ of the form $x_1^t\partial _k$, in which case it is the unique element having minimum weight degree in ${\mathcal {L}}_i \cap {\mathcal {B}}_k$. The exponent $t=i-h_i(n-k)+1$ is determined by the condition $x_1^t\partial _k\in {\mathcal {L}}_{i}$.

Now we can prove one of the main contributions of this work where we give a precise characterization of the elements of ${\mathcal {L}}_i$.

Theorem 2.11

Let $i\ge 0$. A basis element $x^\Lambda \partial _{k}$ belongs to ${\mathcal {L}}_{i}\cap {\mathcal {B}}_k$ if and only if the following conditions are satisfied:

• (a) $n-k \le {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) < r_i$,

• (b) $i={{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)$.

Proof

Suppose first that (a) and (b) are satisfied. We have ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k}) = i$, and so $x^\Lambda \partial _{k}\in {\mathcal {L}}_j\cap {\mathcal {B}}_k$, for some $j\le i$. By contradiction, assume that $j={{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k)$ and that $i\ne j$. We have $i> j=(h_j-1)(n-1)+r_j = {{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k) = h_j{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) + \deg (x^\Lambda )-1$. Note that at least one of the two conditions $h_i >h_j$ or $r_i>r_j$ must be satisfied, so we have

$$\begin{aligned} r_i&= i-(h_i-1)(n-1)\\&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)-(h_i-1)(n-1)\\&= h_i{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) + \deg (x^\Lambda )-1 -(h_i-1)(n-1)\\&= (h_i-h_j)({{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})-n+1) +{{\,\textrm{lev}\,}}_j(x^\Lambda \partial _{k}) - (h_j-1)(n-1)\\&= (h_i-h_j)({{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})-n+1) +j - (h_j-1)(n-1)\\&= (h_i-h_j)({{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})-n+1) +r_j\\&\le {\left\{ \begin{array}{ll} {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) +(r_j-n+1)\le {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})< r_i &{} \text {if }h_i > h_j\\ r_j < r_i &{} \text {if }{h_i = h_j}, \end{array}\right. } \end{aligned}$$

which is, in both the cases, a contradiction. Hence $j=i$.

Conversely, suppose that $x^\Lambda \partial _{k}\in {\mathcal {L}}_i\cap {\mathcal {B}}_k$, so $i= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k})$. By Proposition 2.7 and Lemma 2.2, we obtain that $n-k \le {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) < r_i$.

$\square$

2.4 Connection to the sequence of partitions

Let us give a description of the behavior of the chain $\left\{ {\mathcal {N}}_i \right\} _{i}$ in terms of integer partitions. Let $\{a_n\}_{n=0}^\infty$ be the sequence whose term $a_n$ is equal to the number of partitions of $n$. Let also $b_n=\sum _{i=0}^na_i$ be the partial sum of $\{a_i\}$ and $c_n=\sum _{i=0}^nb_n$ be the partial sum of $\{b_i\}$, or the second partial sum of $\{a_i\}$. The first values of the sequences are displayed in Table 1, which also contains the corresponding OEIS references [5].

Table 1 First values of the sequences $\{a_i\}, \{b_i\}$ and $\{c_i\}$

We are now ready to show that, after a threshold value with depends quadratically on n, the sequence $\{ \left|{\mathcal {L}}_{i} \right|\}$ is periodic and how it depends on $\{c_i\}$. Here the value $r_i$ is as in Definition 2.1.

Corollary 2.12

If $i > (n-4)(n-1)$ and $1 \le k \le n$, then

$$\begin{aligned} \left|{\mathcal {L}}_{i}\cap {\mathcal {B}}_k \right|&=b_{r_i+k-n-1}\\ \text {and} \left|{\mathcal {L}}_{i} \right|&= c_{r_i-1}. \end{aligned}$$

Proof

Let $x^\Lambda \partial _{k}\in {\mathcal {L}}_i\cap {\mathcal {B}}_k$. By Theorem 2.11, $x^\Lambda \partial _{k}$ satisfies

• (a) $n-k\le {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) < r_i$,

• (b) $i={{\,\textrm{lev}\,}}_i(x^\Lambda \partial _{k})$.

Letting $\theta _u=\lambda _{u+1}$, the condition (a) can be rewritten as $0\le \sum _{u\ge 1} u\theta _u < r_i+k-n$, where $\theta _u$ is a non-negative integer. There are exactly $b_{r_i+k-n-1}$ sequences $\Theta =\{\theta _u\}_{u=1}^\infty$, including the trivial one, satisfying this condition and determining the values of $\lambda _u$ for $u\ge 2$. The value of $\lambda _1$ is uniquely determined by the condition (b). Indeed, ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})$ does not depend on $\lambda _1$ and by (b) we have $i=h_i{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})+\lambda _1+\sum _{u\ge 2}\lambda _u-1$. Moreover, by the hypotheses we have $h_i\ge n-3$. Thus,

$$\begin{aligned} \lambda _1&=i - h_i{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) - \sum _{u\ge 2}\lambda _u+1 \\&=(h_i-1)(n-1)+r_i-h_i{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k}) - \sum _{u\ge 2}\lambda _u+1\\&=h_i\left( n-1-{{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})\right) -n+1 +r_i - \sum _{u\ge 2}\lambda _u+1 \\&\ge h_i\left( n-1-r_i+1\right) -n+1 +r_i - r_i-k+n+2\\&= h_i\left( n-r_i\right) +3 -k \ge h_i +3 -k \ge n-k \ge 0 \end{aligned}$$

is uniquely determined and non-negative.

Finally, the equality $\left|{\mathcal {L}}_{i} \right|= c_{r_i-1}$ is obtained computing $\sum _{k=2}^{n} \left|{\mathcal {L}}_{i}\cap {\mathcal {B}}_k \right|$. $\square$

A straightforward consequence is the following result, which proves that the sequence $\{\left|{\mathcal {L}}_i \right|\}_{i\ge 0}$ is ultimately periodic, i.e., there exist integers $k$ and $j$ such that $\left|{\mathcal {L}}_i \right|=\left|{\mathcal {L}}_{i+k} \right|$ for all $i\ge j$.

Corollary 2.13

If $i > (n-4)(n-1)$, then the period function ${{\,\mathrm{\nu }\,}}$ is a bijection from ${\mathcal {L}}_i$ to ${\mathcal {L}}_{i+n-1}$.

We conclude this section with an explicit example where we highlight the periodic structure of the sequence $\left\{ {\mathcal {L}}_i \right\}$.

Example 2.14

Let $n=5$ and $i\ge 5$. We list the non-empty sets ${\mathcal {L}}_i\cap {\mathcal {B}}_k$.

$r_i=1$:

$$\begin{aligned}{\mathcal {L}}_i\cap {\mathcal {B}}_5 = \left\{ x_1^{i+1}\partial _5 \right\} ;\end{aligned}$$

$r_i=2$:

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_i\cap {\mathcal {B}}_5&= \left\{ x_1^{i+1}\partial _5,\, x_1^{i-h_i}x_2\partial _5 \right\} \\ {\mathcal {L}}_i\cap {\mathcal {B}}_{4}&= \left\{ x_1^{i+1-h_i}\partial _{4} \right\} ; \end{aligned} \end{aligned}$$

$r_i=3$:

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_i\cap {\mathcal {B}}_5&= \left\{ x_1^{i+1}\partial _5,\, x_1^{i-h_i}x_2\partial _5,\, x_1^{i+1-2h_i}x_2^2\partial _5,\, x_1^{i-2h_i}x_3\partial _5 \right\} \\ {\mathcal {L}}_i\cap {\mathcal {B}}_4&= \left\{ x_1^{i+1-h_i}\partial _4,\, x_1^{i-2h_i}x_2\partial _4 \right\} \\ {\mathcal {L}}_i\cap {\mathcal {B}}_{3}&= \left\{ x_1^{i+1-2h_i}\partial _{3} \right\} ; \end{aligned} \end{aligned}$$

$r_i=4$:

$$\begin{aligned} \begin{aligned} {\mathcal {L}}_i\cap {\mathcal {B}}_5&= \left\{ x_1^{i+1}\partial _5,\, x_1^{i-h_i}x_2\partial _5,\, x_1^{i+1-2h_i}x_2^2\partial _5,\, x_1^{i-2h_i}x_3\partial _5,\,\right. \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \,\left. x_1^{i-2-3h_i}x_2^3\partial _5,\, x_1^{i-1-3h_i}x_2x_3\partial _5,\, x_1^{i-3h_i}x_4\partial _5 \right\} \\ {\mathcal {L}}_i\cap {\mathcal {B}}_4&= \left\{ x_1^{i+1-h_1}\partial _4,\, x_1^{i-2h_i}x_2\partial _4,\, x_1^{i+1-3h_i}x_2^2\partial _4,\, x_1^{i-3h_i}x_3\partial _4 \right\} \\ {\mathcal {L}}_i\cap {\mathcal {B}}_3&= \left\{ x_1^{i+1-2h_i}\partial _3,\, x_1^{i+1-3h_i}x_2\partial _3 \right\} \\ {\mathcal {L}}_i\cap {\mathcal {B}}_{2}&= \left\{ x_1^{i+1-3h_i}\partial _{2} \right\} . \end{aligned} \end{aligned}$$

3 The idealizer chain over the ring of integers

We conclude the paper by proving that the idealizer chain is generated by the subsets ${\mathcal {N}}_i$ of Definition 2.6. We start by describing the commutator structure of the chain $\left\{ {\mathcal {N}}_i \right\} _{i\ge -1}$.

Lemma 3.1

If $i < j$, then $[{\mathcal {N}}_i,{\mathcal {N}}_j] \subseteq {{\,\mathrm{{\mathbb {Z}}}\,}}{\mathcal {N}}_{j-1}$.

Proof

Let $x^\Lambda \partial _k\in {\mathcal {N}}_i$ and $x^\Theta \partial _u\in {\mathcal {N}}_j$ be such that $[x^\Lambda \partial _{k}, x^\Theta \partial _{u}] \ne 0$. By definition of ${\mathcal {N}}_j$ there exist $\ell \le j$ and $m\le i$ such that ${{\,\textrm{lev}\,}}_\ell (x^\Theta \partial _u) \le \ell$ and ${{\,\textrm{lev}\,}}_m(x^\Lambda \partial _{k})\le m$. We may assume $m\le \ell$, otherwise we interchange the roles of $i$ and $j$ in the statement and we argue by induction.

If $x^\Lambda \partial _k= \partial _1$, then

$$\begin{aligned} {{\,\textrm{lev}\,}}_{\ell -1}([x^\Lambda \partial _{k}, x^\Theta \partial _{u}]) \le {{\,\textrm{lev}\,}}_\ell ([x^\Lambda \partial _{k}, x^\Theta \partial _{u}])= {{\,\textrm{lev}\,}}_\ell (x^\Theta \partial _{u})-1 \le \ell -1.\end{aligned}$$

Thus, $[x^\Lambda \partial _{k}, x^\Theta \partial _{u}]\in {\mathcal {N}}_{j-1}$.

Assume now that $x^\Lambda \partial _k\ne \partial _1$.

By Proposition 2.3, ${{\,\textrm{wd}\,}}(x^\Lambda \partial _k) < n-1$. If $h_m=h_\ell$, then

$$\begin{aligned} {{\,\textrm{lev}\,}}_\ell ([x^\Lambda \partial _{k}, x^\Theta \partial _{u}])&= {{\,\textrm{lev}\,}}_\ell (x^\Lambda \partial _{k}) + {{\,\textrm{lev}\,}}_\ell (x^\Theta \partial _{u})-h_\ell (n-1) \\&= {{\,\textrm{lev}\,}}_m(x^\Lambda \partial _{k}) + {{\,\textrm{lev}\,}}_\ell (x^\Theta \partial _{u})-h_\ell (n-1)\\&\le {{\,\textrm{lev}\,}}_m(x^\Lambda \partial _{k}) +\ell -h_\ell (n-1) = {{\,\textrm{lev}\,}}_m(x^\Lambda \partial _{k}) +r_\ell -(n-1) \le m. \end{aligned}$$

If $m=\ell$, then $\ell \le j-1$ and so $[x^\Lambda \partial _{k}, x^\Theta \partial _{u}]\in {\mathcal {N}}_{j-1}$. Otherwise $m<\ell \le j$ and

$$\begin{aligned}{{\,\textrm{lev}\,}}_m([x^\Lambda \partial _{k}, x^\Theta \partial _{u}]) \le {{\,\textrm{lev}\,}}_\ell ([x^\Lambda \partial _{k}, x^\Theta \partial _{u}])\ \le m < j,\end{aligned}$$

which implies $[x^\Lambda \partial _{k}, x^\Theta \partial _{u}]\in {\mathcal {N}}_{j-1}$.

The remaining possibility is that $h_m<h_\ell$. In this case $m<\ell$ and ${{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})(h_\ell -h_m)\le (n-1)(h_\ell -h_m) -1$, thus

$$\begin{aligned} {{\,\textrm{lev}\,}}_\ell ([x^\Lambda \partial _{k}, x^\Theta \partial _{u}])&= {{\,\textrm{lev}\,}}_\ell (x^\Lambda \partial _{k}) + {{\,\textrm{lev}\,}}_\ell (x^\Theta \partial _{u})-h_\ell (n-1) \\&={{\,\textrm{lev}\,}}_m(x^\Lambda \partial _{k}) + {{\,\textrm{lev}\,}}_\ell (x^\Theta \partial _{u})-h_\ell (n-1)+ {{\,\textrm{wd}\,}}(x^\Lambda \partial _{k})(h_\ell -h_m) \nonumber \\&\le {{\,\textrm{lev}\,}}_m(x^\Lambda \partial _{k}) + {{\,\textrm{lev}\,}}_\ell (x^\Theta \partial _{u})-h_\ell (n-1)+ (n-1)(h_\ell -h_m) -1\\&\le m-h_m(n-1) +\ell -1= r_m-(n-1) +\ell -1 \le \ell -1. \end{aligned}$$

Hence, ${{\,\textrm{lev}\,}}_{\ell -1}([x^\Lambda \partial _{k}, x^\Theta \partial _{u}]) \le {{\,\textrm{lev}\,}}_\ell ([x^\Lambda \partial _{k}, x^\Theta \partial _{u}])\le \ell -1$, therefore $[x^\Lambda \partial _{k}, x^\Theta \partial _{u}] \in {\mathcal {N}}_{j-1}$.

$\square$

As straightforward consequence is the following corollary.

Corollary 3.2

${\mathcal {N}}_j\subseteq N_{{\mathcal {B}}}({{\mathcal {N}}_{j-1}})$.

We prove now the opposite inclusion.

Proposition 3.3

${\mathcal {N}}_j= N_{{\mathcal {B}}}({{\mathcal {N}}_{j-1}})$.

Proof

By the previous corollary it suffices to show that ${\mathcal {N}}_j\supseteq N_{{\mathcal {B}}}({\mathcal {N}}_{j-1})$. Looking for a contradiction, we assume that $x^\Lambda \partial _k\in N_{{\mathcal {B}}}({\mathcal {N}}_{j-1})$ is such that ${{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)> i$ for all $i\le j$. Set

$$\begin{aligned}x^\Theta \partial _u=:x_1^{(h_{j}-1)(u-\ell )}x_{\ell }\partial _{u},\end{aligned}$$

where $1 \le \ell < u$, and $s:=(h_{j}-1)(n-1)\le j-1$. We have $h_s=h_{j}-1$ and

$$\begin{aligned} {{\,\textrm{lev}\,}}_s(x^\Theta \partial _u)=(h_{j}-1)(\ell -1+n-u)+(h_{j}-1)(u-\ell )=(h_{j}-1)(n-1)=s. \end{aligned}$$

Hence, both $x^\Theta \partial _u$ and $[x^\Lambda \partial _k,x^\Theta \partial _u]$ belong to ${\mathcal {N}}_{j-1}$. Assume that either $\lambda _u\ne 0$ for some $1<u<n$ or $\ell =k$, so $[x^\Lambda \partial _k,x^\Theta \partial _u]\ne 0$. For some $i \le j-1$, we have ${{\,\textrm{lev}\,}}_i([x^\Lambda \partial _k,x^\Theta \partial _u])= i$. We start with assuming that $i>s$ and that $h_j=h_{j-1}$, which in turn implies $h_i=h_{j-1}=h_j$. We have

$$\begin{aligned} \begin{aligned} i&={{\,\textrm{lev}\,}}_i([x^\Lambda \partial _k,x^\Theta \partial _u])\\&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)+{{\,\textrm{lev}\,}}_i(x^\Theta \partial _u)- h_i(n-1)\\&={{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k) + h_{i}(\ell -1+n-u)+ (h_{j}-1)(u-\ell ) - h_i(n-1)\\&={{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k) + h_{j}(\ell -1+n-u)+ (h_{j}-1)(u-\ell ) - h_{j}(n-1)\\&= {{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k) + (\ell -u) > j + (\ell -u). \end{aligned} \end{aligned}$$

As a consequence, we have $j-1\ge i > j + (\ell -u)$. Since we may alternatively choose $\ell$ or $u$, with respect to our assumption on $x^\Theta \partial _u$ such that $u-\ell =1$, we have a contradiction.

Suppose now that $i\le s$ and that $h_j=h_{j-1}$, and so $h_i+1\le h_{j-1}=h_j$. We have

$$\begin{aligned} \begin{aligned} i&={{\,\textrm{lev}\,}}_i([x^\Lambda \partial _k,x^\Theta \partial _u])\\&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)+{{\,\textrm{lev}\,}}_i(x^\Theta \partial _u)- h_i(n-1)\\&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)+{{\,\textrm{lev}\,}}_s(x^\Theta \partial _u) +(h_i-h_s){{\,\textrm{wd}\,}}(x^\Theta \partial _u)- h_i(n-1)\\&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)+{{\,\textrm{lev}\,}}_s(x^\Theta \partial _u) +(h_i-h_s)(\ell -1+n-u)- h_i(n-1)\\&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)+h_s(n-1) +(h_i-h_s)(\ell -1+n-u)- h_i(n-1)\\&= {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k) +(h_i-h_s)(\ell -u) \ge {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k), \\ \end{aligned} \end{aligned}$$

a contradiction.

Suppose now that $j-1=s$ so that $h_{j-1}=h_s=h_j-1$. By a repeated use of Lemma 2.5, we may assume ${{\,\textrm{lev}\,}}_t([x^\Lambda \partial _k,x^\Theta \partial _u])\le t$ for some $t$ such that $s-(n-1)< t \le s=j-1$. Hence,

$$\begin{aligned} t&\ge {{\,\textrm{lev}\,}}_t([x^\Lambda \partial _k,x^\Theta \partial _u])\\&= {{\,\textrm{lev}\,}}_{s}([x^\Lambda \partial _k,x^\Theta \partial _u])\\&= {{\,\textrm{lev}\,}}_{s}(x^\Lambda \partial _k)+{{\,\textrm{lev}\,}}_{s}(x^\Theta \partial _u)- h_{s}(n-1)\\&= {{\,\textrm{lev}\,}}_{j-1}(x^\Lambda \partial _k) + s -s = {{\,\textrm{lev}\,}}_{j-1}(x^\Lambda \partial _k) \end{aligned}$$

giving the contradiction ${{\,\textrm{lev}\,}}_{j-1}(x^\Lambda \partial _k) \le t \le s= j-1$.

We are now left with the case $x^\Lambda \partial _k= x_1^{\lambda _1}\partial _n$. In order to have $[x^\Lambda \partial _k,x^\Theta \partial _u]\ne 0$, $x^\Theta \partial _u$ must be set to $\partial _1$. In particular, $[x^\Lambda \partial _k,x^\Theta \partial _u]= \lambda _1 x_1^{\lambda _1-1}\partial _n\in {{\,\mathrm{{\mathbb {Z}}}\,}}{\mathcal {N}}_{j-1}$, and also ${{\,\textrm{lev}\,}}_i(x_1^{\lambda _1-1}\partial _n)=\lambda _1-2$ is independent of $i$. Thus, for some $i\le j-1$ we have

$$\begin{aligned}{{\,\textrm{lev}\,}}_{j-1}(x_1^{\lambda _1-1}\partial _n)={{\,\textrm{lev}\,}}_i(x_1^{\lambda _1-1}\partial _n)=\lambda _1-2 \le i \le j-1.\end{aligned}$$

Hence, $j \ge \lambda _1-1={{\,\textrm{lev}\,}}_j(x^\Lambda \partial _k)>j$, a contradiction. $\square$

We are now able to prove the claimed result on the idealizer chain.

Theorem 3.4

Let ${\mathfrak {M}}_i$ be the free ${{\,\mathrm{{\mathbb {Z}}}\,}}$-module spanned by ${\mathcal {N}}_i$, for $i \ge -1$. Then ${\mathfrak {M}}_i$ is a homogeneous subring and

$$\begin{aligned}{\mathfrak {M}}_i=N_{{\mathfrak {L}}(n)}({\mathfrak {M}}_{i-1}) = {\mathfrak {N}}_i.\end{aligned}$$

Proof

The statement follows directly from Theorem 1.2 and Proposition 3.3 noting that ${\mathfrak {T}}={{\,\mathrm{{\mathbb {Z}}}\,}}\partial _1+\dots + {{\,\mathrm{{\mathbb {Z}}}\,}}\partial _n ={\mathfrak {N}}_{-1}$ is an abelian homogeneous subring. $\square$

Remark 5

A straightforward consequence of Lemma 2.2 is that ${{\,\mathrm{{\mathfrak {L}}}\,}}{(n)} \ne \bigcup _{i\ge -1}{\mathfrak {N}}_i$ for $n\ge 3$. For example, the element $x_2^{3}\partial _3$, which has weight-degree $n$, cannot belong to any of the ${\mathfrak {N}}_{i}$s. We point out that, unlike the case of the Lie ring ${{\,\mathrm{{\mathfrak {L}}}\,}}_{m}(n)$ ($m>0$), this shows that ${{\,\mathrm{{\mathfrak {L}}}\,}}(n)$ is not nilpotent, beside not being finitely generated.

We can now conclude the paper with the characterization of the idealizers of Eq. (1).

Theorem 3.5

If $i\ge 0$, then $x^\Lambda \partial _k \in {\mathfrak {N}}_i{\setminus } {\mathfrak {N}}_{i-1}$ if and only if $i$ is the least non-negative integer such that $i = {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)$.

Proof

The proof follows by noticing that, by Remark 2, we have that $x^\Lambda \partial _k \in {\mathcal {N}}_i{\setminus } {\mathcal {N}}_{i-1}$ if and only if $i$ is the least non-negative integer such that $i = {{\,\textrm{lev}\,}}_i(x^\Lambda \partial _k)$. $\square$

A trivial consequence of Corollary 2.12 is the following conclusive result.

Corollary 3.6

For $i > (n-4)(n-1)$, the ${{\,\mathrm{{\mathbb {Z}}}\,}}$-module ${\mathfrak {N}}_i/{\mathfrak {N}}_{i-1}$ is free of rank $c_{r_i-1}$.