The center and invariants of standard filiform Lie algebras

Vanderlei Lopes de Jesus Departamento de Matemática
Instituto de Ciências Exatas
Universidade Federal de Minas Gerais
Av. Antônio Carlos 6627
Belo Horizonte, MG, Brazil. Email: [email protected] and Csaba Schneider Departamento de Matemática
Instituto de Ciências Exatas
Universidade Federal de Minas Gerais
Av. Antônio Carlos 6627
Belo Horizonte, MG, Brazil. Email: [email protected], URL: schcs.github.io/WP/

Abstract.

This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit generators for the quotient fields and also a compact form for the generators for the invariants rings. We prove several combinatorial results concerning the Hilbert series of these algebras.

Key words and phrases:

Lie algebras, nilpotent Lie algebras, filiform Lie algebras, universal enveloping algebras, invariants, Hilbert series

2010 Mathematics Subject Classification:

17B35, 17B30, 16U70, 16W22, 17-08

1. Introduction

The problem of describing the center $Z(\mathfrak{g})$ of the universal enveloping algebra for nilpotent Lie algebras $\mathfrak{g}$ has received quite a bit of attention. Dixmier [Dix58] determined $Z(\mathfrak{g})$ for nilpotent Lie algebras of dimension up to 5, and Ooms in [Oom09, Oom12] extended this work to algebras of dimension at most 7. Considering the descriptions of the centers $Z(\mathfrak{g})$ for these small-dimensional nilpotent Lie algebras, it becomes apparent that the most complicated structure is exhibited in the cases when $\mathfrak{g}$ is standard filiform. So we decided that the study of the center $Z(\mathfrak{g})$ for standard filiform Lie algebras $\mathfrak{g}$ is worth special attention. In this paper we consider both the cases of characteristic zero and prime characteristic.

The standard filiform Lie algebra $\mathfrak{g}(n+2)$ is the Lie algebra over a field $\mathbb{F}$ with basis $\{x,y_{0},y_{1},\ldots,y_{n}\}$ and nonzero brackets $[x,y_{i}]=y_{i-1}$ for $i\in\{1,\ldots,n\}$ (in particular, $[x,y_{0}]=0$ and $y_{0}$ is central). This is a nilpotent Lie algebra of dimension $n+2$ and nilpotency class $n+1$ ; such a Lie algebra is also referred to as a Lie algebra of maximal class. The center of $\mathfrak{g}(n+2)$ is one-dimensional and is generated by $y_{0}$ .

As it turns out (and explained in Section 2), the center $Z(\mathfrak{g})$ of the standard filiform Lie algebra $\mathfrak{g}=\mathfrak{g}(n+2)$ (see Section 2 for the notation) is equal to the algebra of polynomial solutions $f\in\mathbb{F}[y_{0},\ldots,y_{n}]$ of the partial differential equation

\sum_{i\geqslant 0}y_{i}\frac{\partial f}{\partial y_{i+1}}=0.

The operator $f\mapsto\sum y_{i}\partial f/\partial y_{i+1}$ is a locally nilpotent derivation, also known as a Weitzenböck derivation, on the polynomial algebra $\mathbb{F}[y_{0},\ldots,y_{n}]$ and was considered in [Bed08, Bed07, Bed09, Bed11b, Bed11a, BI15]. The same operator on the polynomial ring $\mathbb{F}[y_{0},y_{1},\ldots]$ on infinitely many generators is referred to as the down operator, see [Fre13]. The crucial observation made by [Bed05, Bed10] is that the action of this operator on linear polynomials can be extended to an irreducible representation of the simple Lie algebra $\mathfrak{sl}_{2}$ and one can use the representation theory of $\mathfrak{sl}_{2}$ to obtain more information on the center $Z(\mathfrak{g})$ . In fact, considering the standard basis $\{e,f,h\}$ of $\mathfrak{sl}_{2}$ , the homogeneous elements of $Z(\mathfrak{g})$ are homogeneous polynomials in $y_{0},y_{1},\ldots,y_{n}$ , which are $h$ -eigenvectors and are annihilated by $e$ . Thus, $Z_{n}$ is linked to the algebra of $\mathfrak{sl}_{2}$ -covariants of the binary form of degree $n$ .

For $n\geqslant 1$ , let $Z_{n}$ denote $Z(\mathfrak{g}(n+2))$ . If the characteristic of the field is zero, then $Z_{n}$ can be viewed as a graded subalgebra of the polynomial algebra $\mathbb{F}[y_{0},y_{1},\ldots,y_{n}]$ . For $k\geqslant 0$ , let $Z_{n,k}$ denote the degree- $k$ homogeneous component of $Z_{n}$ .

Theorem 1.1.

If $\mathbb{F}$ has characteristic zero, then the algebra $Z_{n}$ has Krull dimension $n$ and the following are valid.

(1)

The fraction field $\textrm{Frac}(Z_{n})$ can be written as

$\textrm{Frac}(Z_{n})=\mathbb{F}(z_{1},z_{2},\ldots,z_{n})=\mathbb{F}(w_{1},w_{2},\ldots,w_{n})$

where the elements $z_{1},z_{2},\ldots,z_{n}$ and $w_{1},w_{2},\ldots,w_{n}$ are defined in Section 5.
(2)

$Z_{n}\subseteq\mathbb{F}[y_{0}^{-1},z_{1},z_{2},\ldots,z_{n}]$ and $Z_{n}\subseteq\mathbb{F}[y_{0}^{-1},w_{1},\ldots,w_{n}]$ .

In Section 5 we define an operation $\circ_{d}$ between two homogeneous elements of $Z_{n}$ that are also $h$ -eigenvectors for the generator $h$ of the Cartan subalgebra of $\mathfrak{sl}_{2}$ . We show that $Z_{n}$ is linearly spanned by elements of the form $y_{0}\circ_{d_{1}}y_{0}\circ_{d_{2}}\cdots\circ_{d_{k}}y_{0}$ and one can choose minimal algebra generating sets out of such elements. Using computer calculation, we explicitly determine such generating sets for $Z_{n}$ for $n\leqslant 6$ and for $n=8$ in the Appendix A.

In Section 6, we consider the Hilbert series

H_{n}(t)=\sum_{k\geqslant 0}\dim Z_{n,k}t^{k}.

It is interesting that the dimensions $\dim Z_{n,k}$ occur also in the context of representation theory and combinatorics.

Theorem 1.2.

Assuming that $\mathbb{F}$ has characteristic zero, that $n\geqslant 1$ and $k\geqslant 1$ , the following are true for $\dim Z_{n,k}$ :

(1)

$\dim Z_{n,k}$ coincides with the number of irreducible components of the $\mathfrak{sl}_{2}$ -module $S^{k}(U_{n})$ where $U_{n}$ is the $(n+1)$ -dimensional irreducible representation of $\mathfrak{sl}_{2}$ and $S^{k}$ is the $k$ -th symmetric power.
(2)

$\dim Z_{n,k}$ is equal to the number of partitions of $\lfloor kn/2\rfloor$ into $k$ blocks each of size at most $n$ (permitting block size zero).
(3)

$\dim Z_{n,k}=\dim Z_{k,n}$ .

Theorem 1.2 follows from Theorem 6.1 (part (1)) and from Theorem 6.4 (parts (2) and (3)).

By the Mauer–Weitzenböck Theorem [Fre17, Theorem 6.1], $Z_{n}$ is finitely generated and so $H_{n}(t)$ is a rational function. The combinatorial description of $\dim Z_{n,k}$ in Theorem 1.2 and the computations in [EZ22] make it possible to determine the Hilbert series $H_{n}(t)$ as rational expressions for $n\leqslant 18$ . As in the situation considered by Almkvist [Alm80a, Alm80b], one also has the following closed formula for $H_{n}(t)$ .

Theorem 1.3.

If $\mathbb{F}$ has characteristic zero, then, for $n\geqslant 1$ ,

(1)

H_{n}(t)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1+\exp(i\varphi)}{\prod_{k=0}^{n}(1-t\exp(i(n-2k)\varphi))}\,d\varphi.

The integral formula in Theorem 1.3 appears in [Alm80a], since in that situation, the counting argument also boils down to counting the partitions in Theorem 1.2(2). We include a proof in Section 6 for easier reference.

In the final Section 7, we consider the case when the characteristic of the field is a prime. In this case, a well-known theorem of Zassenhaus states that $Z(\mathfrak{g})$ is a normal domain and our main result states that the generators $z_{2},\ldots,z_{n}$ in Theorem 1.1 generate $Z_{n}$ over the $p$ -center $Z_{p}(\mathfrak{g})$ up to normal closure.

Theorem 1.4.

Suppose that $\mathbb{F}$ has characteristic $p$ and also that $p\geqslant n+1$ . Set $\mathfrak{g}=\mathfrak{g}(n+2)$ . Then $Z(\mathfrak{g})$ is the integral closure of $Z_{p}(\mathfrak{g})[z_{2},\ldots,z_{n}]$ in its field of fractions, and it also coincides with the integral closure of $Z_{p}(\mathfrak{g})[z_{2},\ldots,z_{n}]$ in the localization $Z_{p}(\mathfrak{g})[y_{0}^{-1},z_{2},\ldots,z_{n}]$ . In particular, $Z(\mathfrak{g})\subseteq Z_{p}(\mathfrak{g})[y_{0}^{-1},z_{2},\ldots,z_{n}]$ .

The first author was financially supported by a PhD scholarship awarded by CNPq (Brazil). The second author acknowledges the financial support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.: 421624/2018-3 and 402934/2021-0) and the FAPEMIG project Universal (project no.: APQ-00818-23). We thank Lucas Calixto for several useful comments on the earlier drafts, to Dmitry Shcheglov and to Csaba Noszály for help with Theorem 1.3. We appreciate the useful observations of the referee.

2. The standard filiform Lie algebra

The adjoint action of the Lie algebra $\mathfrak{g}=\mathfrak{g}(n+2)$ on itself can be extended to the polynomial ring $\mathbb{F}[\mathfrak{g}]=\mathbb{F}[x,y_{0},y_{1},\ldots,y_{n}]$ , and we define the invariant ring $\mathbb{F}[\mathfrak{g}]^{\mathfrak{g}}$ as

\mathbb{F}[\mathfrak{g}]^{\mathfrak{g}}=\{f\in\mathbb{F}[\mathfrak{g}]\mid u(f)=0\mbox{ for all }u\in\mathfrak{g}\}.

The commutative algebra $\mathbb{F}[\mathfrak{g}]^{\mathfrak{g}}$ is referred to as the algebra of polynomial invariants of $\mathfrak{g}$ .

Let $\mathbb{F}[\mathbf{y}]$ denote the polynomial algebra $\mathbb{F}[y_{0},y_{1}\ldots]$ in infinitely many variables and let $\mathbb{F}[\mathbf{y}_{n}]=\mathbb{F}[y_{0},\ldots,y_{n}]$ . Then the one-dimensional Lie algebra $\mathfrak{n}=\left<x\right>$ acts on the vector space $V=\left<y_{0},y_{1},\ldots\right>$ by mapping $y_{i}$ to $y_{i-1}$ for $i\geqslant 1$ and $y_{0}$ to $0$ . Set $V_{n}=\left<y_{0},\ldots,y_{n}\right>$ and note that $V_{n}$ is an $\mathfrak{n}$ -submodule of $V$ . The Lie algebra $\mathfrak{g}(n+2)$ can also be viewed as the semidirect product $V_{n}\rtimes\mathfrak{n}$ .

The $\mathfrak{n}$ -action on $V$ can be extended to $\mathbb{F}[\mathbf{y}]$ by the Leibniz rule and $\mathbb{F}[\mathbf{y}_{n}]$ is $\mathfrak{n}$ -invariant for all $n\geqslant 1$ . We denote by $x(f)$ the image of $f\in\mathbb{F}[\mathbf{y}]$ under this action. Simple computation shows that

(2)

x(f)=\sum_{i\geqslant 0}y_{i}\frac{\partial f}{\partial y_{i+1}}.

The algebra of $x$ -invariants in $\mathbb{F}[\mathbf{y}]$ is denoted by $\mathbb{F}[\mathbf{y}]^{x}$ ; more precisely,

\mathbb{F}[\mathbf{y}]^{x}=\{f\in\mathbb{F}[\mathbf{y}]\mid x(f)=0\}.

The operator $f\mapsto x(f)$ on $\mathbb{F}[\mathbf{y}]$ is often referred to as the down operator; see [Fre13]. The algebra of $x$ -invariants can also be described as the set of polynomial solutions of the partial differential equation

(3)

\sum_{i\geqslant 0}y_{i}\frac{\partial f}{\partial y_{i+1}}=0.

For a Lie algebra $\mathfrak{g}$ , let $Z(\mathfrak{g})$ denote the center of the universal enveloping algebra $U(\mathfrak{g})$ . Since $y_{0},\ldots,y_{n}$ commute in $\mathfrak{g}(n+2)$ , the Poincaré–Birkhoff–Witt Theorem gives an embedding of the polynomial algebra $\mathbb{F}[\mathbf{y}_{n}]$ into $U(\mathfrak{g}(n+2))$ . Further, if $\mathbb{F}$ has prime characteristic $p$ and $p\geqslant n+1$ , then $x^{p}\in Z(\mathfrak{g}(n+2))$ and $\mathbb{F}[x^{p},y_{0},\ldots,y_{n}]$ is a subalgebra of $U(\mathfrak{g}(n+2))$ . Note that the action of $x$ on $\mathbb{F}[\mathbf{y}_{n}]$ is the same whether it is considered inside $U(\mathfrak{g}(n+2))$ or it is considered as a stand-alone polynomial algebra under the action defined in (2).

Lemma 2.1.

Let $\mathfrak{g}=\mathfrak{g}(n+2)$ be a standard filiform Lie algebra. If $\mathbb{F}$ has characteristic zero, then

Z(\mathfrak{g})\subseteq\mathbb{F}[\mathbf{y}_{n}]

and

Z(\mathfrak{g})=\mathbb{F}[\mathfrak{g}]^{\mathfrak{g}}=\mathbb{F}[\mathbf{y}_{n}]^{x}.

If the characteristic of $\mathbb{F}$ is a prime $p$ , then

Z(\mathfrak{g})\subseteq\mathbb{F}[x^{p},y_{0},\ldots,y_{n}]

and, if in addition $p\geqslant n+1$ , then

Z(\mathfrak{g})=\mathbb{F}[\mathfrak{g}]^{\mathfrak{g}}=\mathbb{F}[x^{p},y_{0},\ldots,y_{n}]^{x}.

Proof.

Easy induction shows, in $U(\mathfrak{g})$ , that

[x^{k},y_{n}]=x^{k}y_{n}-y_{n}x^{k}=kx^{k-1}y_{n-1}+h_{k}

where $h_{k}$ is a linear combination of monomials whose degree in $x$ is smaller than $k-1$ . Suppose that $z\in Z(\mathfrak{g})$ and let $m=x^{\alpha}y_{0}^{\alpha_{0}}\cdots y_{n}^{\alpha_{n}}$ be the leading term of $z$ in the lexicographic monomial order. Then the last displayed equation implies that the leading term of $[z,y_{n}]$ is

\alpha x^{\alpha-1}y_{0}^{\alpha_{0}}\cdots y_{n-1}^{\alpha_{n-1}+1}y_{n}^{\alpha_{n}}.

But, as $z$ is central, $[z,y_{n}]=0$ , and so either $\alpha=0$ or the characteristic of $\mathbb{F}$ is $p$ and $p\mid\alpha$ . Thus, the inclusions for $Z(\mathfrak{g})$ in $\mathbb{F}[\mathbf{y}_{n}]$ and, in the case of characteristic $p$ , in $\mathbb{F}[x^{p},y_{0},\ldots,y_{n}]$ is verified. One can similarly verify the same containment $\mathbb{F}[\mathfrak{g}]^{\mathfrak{g}}\subseteq\mathbb{F}[\mathbf{y}_{n}]$ and $\mathbb{F}[\mathfrak{g}]^{\mathfrak{g}}\subseteq\mathbb{F}[x^{p},y_{0},\ldots,y_{n}]$ . For the equations regarding $Z(\mathfrak{g})$ , note that if $\mathbb{F}$ has prime characteristic $p$ and $p\geqslant n+1$ , then $x^{p}\in Z(\mathfrak{g})$ . The rest of the lemma follows from the observation above that $\mathbb{F}[\mathbf{y}_{n}]$ and $\mathbb{F}[x^{p},y_{0},\ldots,y_{n}]$ are embedded into $U(\mathfrak{g})$ under the Poincaré–Birkhoff–Witt Theorem and the $x$ -action on these subalgebras is the same as in (2). ∎

From now on, we set $Z_{n}=Z(\mathfrak{g}_{n+2})$ considered as a subalgebra of $\mathbb{F}[\mathbf{y}_{n}]$ . Note that the $x$ -action on $\mathbb{F}[\mathbf{y}_{n}]$ preserves the grading, and hence $Z_{n}$ is a graded subalgebra.

Example 2.2.

Suppose that the characteristic of $\mathbb{F}$ is zero. The ring $Z_{1}$ is the center of $U(\mathfrak{g}(3))$ which coincides with the invariant algebra of $\mathfrak{g}(3)$ . Note that $\mathfrak{g}(3)$ is the three-dimensional Heisenberg Lie algebra. By Lemma 2.1,

Z_{1}=\left\{f\in\mathbb{F}[y_{0},y_{1}]\mid y_{0}\frac{\partial f}{\partial y_{1}}=0\right\}=\left\{f\in\mathbb{F}[y_{0},y_{1}]\mid\frac{\partial f}{\partial y_{1}}=0\right\}=\mathbb{F}[y_{0}].

3. The elements of $\mathbb{F}[\mathbf{y}_{n}]^{x}$

In this section, we prove some general facts concerning the elements of $\mathbb{F}[\mathbf{y}_{n}]^{x}$ and of the fraction field $\textrm{Frac}(\mathbb{F}[\mathbf{y}_{n}]^{x})$ . We will use these facts to determine explicit generators in Section 3. In this section $\mathbb{F}$ is a field of characteristic zero.

Lemma 3.1.

Suppose that $z\in\mathbb{F}[\mathbf{y}_{n}]$ and write

z=\sum_{i=0}^{k}y_{n}^{i}g_{i}

where $g_{i}\in\mathbb{F}[\mathbf{y}_{n-1}]$ . Then $z\in\mathbb{F}[\mathbf{y}_{n}]^{x}$ if and only if

x(g_{k})=0\quad\mbox{and}\quad x(g_{i})=-(i+1)y_{n-1}g_{i+1}\quad\mbox{for all}\quad i\leqslant k-1.

Proof.

First, we compute

	$\displaystyle x(z)$	$\displaystyle=x\left(\sum_{i=0}^{k}y_{n}^{i}g_{i}\right)=\sum_{i=1}^{k}iy_{n}^{i-1}y_{n-1}g_{i}+\sum_{i=0}^{k}y_{n}^{i}\cdot x(g_{i})$
		$\displaystyle=y_{n}^{k}\cdot x(g_{k})+\sum_{i=0}^{k-1}y_{n}^{i}\left((i+1)y_{n-1}g_{i+1}+x(g_{i})\right).$

Since $g_{i},x(g_{i})\in\mathbb{F}[\mathbf{y}_{n-1}]$ for all $i\in\{0,\ldots,k\}$ , we obtain that the coefficient of $y_{n}^{k}$ in $x(z)$ is $x(g_{k})$ and, for $i\in\{0,\ldots,k-1\}$ , the coefficient of $y_{n}^{i}$ is $(i+1)y_{n-1}g_{i+1}+x(g_{i})$ . Thus, $x(z)=0$ if and only if $x(g_{k})=0$ and $(i+1)y_{n-1}g_{i+1}+x(g_{i})=0$ for $i\in\{0,\ldots,k-1\}$ . ∎

Corollary 3.2.

If $z\in\mathbb{F}[\mathbf{y}_{n}]^{x}$ is written as in the previous lemma, then $z$ is determined by the term $g_{0}\in\mathbb{F}[\mathbf{y}_{n-1}]$ .

In the following lemma, we consider monomials ordered lexicographically considering the exponents of $y_{n},\ldots,y_{1},y_{0}$ in this order.

Lemma 3.3.

Suppose that $z\in\mathbb{F}[\mathbf{y}_{n}]^{x}$ and write

z=\sum_{i=0}^{k}y_{n}^{i}g_{i}

where $g_{i}\in\mathbb{F}[\mathbf{y}_{n-1}]$ for all $i\in\{0,\ldots,k\}$ . Then

(1)

$g_{k}\in\mathbb{F}[\mathbf{y}_{n-1}]^{x}$ ;
(2)

the leading monomial of $z$ does not contain $y_{1}$ .

Proof.

Statement (1) follows from Lemma 3.1. Let us prove (2) by induction on $n$ . If $n=1$ , then the statement is clear, since $Z_{1}=\mathbb{F}[\mathbf{y}_{1}]^{x}=\mathbb{F}[y_{0},y_{1}]^{x}=\mathbb{F}[y_{0}]$ (see Example 2.2). Suppose that the statement holds for $n-1\geqslant 1$ and consider $z\in\mathbb{F}[\mathbf{y}_{n}]$ . If $z$ does not contain the variable $y_{n}$ , then $z\in\mathbb{F}[\mathbf{y}_{n-1}]^{x}$ and we are done by the induction hypothesis. Otherwise, write $z=\sum_{i\leqslant k}y_{n}^{i}g_{i}$ . Then the leading monomial of $z$ is $y_{n}^{k}m$ where $m$ is the leading monomial of $g_{k}$ . By Statement (1), $g_{k}\in\mathbb{F}[\mathbf{y}_{n-1}]^{x}$ , and so the induction hypothesis implies that $m$ does not contain $y_{1}$ . Thus, $y_{n}^{k}m$ does not contain $y_{1}$ also. ∎

Theorem 3.4.

Suppose that $Z\subseteq\mathbb{F}[\mathbf{y}_{n}]$ is a graded subalgebra (with respect to the grading by degree) such that if $z\in Z$ then the leading monomial of $z$ is not divisible by $y_{1}$ . Suppose that $z_{1},\ldots,z_{n}\in Z$ are homogeneous elements such that $z_{1}=y_{0}$ and, for each $i\in\{2,\ldots,n\}$ ,

z_{i}=y_{i}y_{0}^{k_{i}}+z^{\prime}_{i}\mbox{ with }k_{i}\geqslant 0\mbox{ and }z_{i}^{\prime}\in\mathbb{F}[\mathbf{y}_{i-1}].

Then the following are valid.

(1)

$Z\subseteq\mathbb{F}[y_{0}^{-1},z_{1},\ldots,z_{n}]$ .
(2)

The fraction field of $Z$ is generated by $z_{1},\ldots,z_{n}$ .
(3)

The elements $z_{1},\ldots,z_{n}$ are algebraically independent over $\mathbb{F}$ .
(4)

$\dim Z=n$ (Krull dimension).

Proof.

(1) Suppose that $z\in Z$ ; we need to show that $z$ can be written as a polynomial expression in $y_{0}^{-1}$ and in the given $z_{i}$ . We show this by induction on the leading monomial of $z$ . The base case of the induction is when $z=1$ and the statement in this case is obvious. Suppose now that $z$ is a homogeneous element of $Z$ and its leading monomial is

m=y_{n}^{\alpha_{n}}\cdots y_{2}^{\alpha_{2}}y_{0}^{\alpha_{0}}

with $m\neq 1$ . Note, for $i\geqslant 2$ , that the leading monomial of $z_{i}$ is $y_{i}y_{0}^{k_{i}}$ . Consider the element

z^{\prime}=z-z_{2}^{\alpha_{2}}\cdots z_{n}^{\alpha_{n}}y_{0}^{\alpha_{0}-\alpha_{2}k_{2}-\cdots-\alpha_{k}k_{n}}

Then the leading monomial of $z^{\prime}$ is smaller than $m$ in the monomial ordering and $z^{\prime}\in Z$ . By the induction hypothesis, $z^{\prime}\in\mathbb{F}[y_{0}^{-1},z_{1},\ldots,z_{n}]$ which implies that $z\in\mathbb{F}[y_{0}^{-1},z_{1},\ldots,z_{n}]$ .

(2) follows from (1). To prove (3), take a polynomial $g\in\mathbb{F}[t_{1},\ldots,t_{n}]$ such that $g(z_{1},\ldots,z_{n})=0$ . If $g\neq 0$ , then, supposing that the leading term of $g$ is $t_{1}^{\alpha_{1}}\cdots t_{n}^{\alpha_{n}}$ , the leading term of $g(z_{1},\ldots,z_{n})$ is $y_{n}^{\alpha_{n}}\ldots y_{2}^{\alpha_{2}}y_{0}^{k}$ for some $k\geqslant 0$ , which is nonzero. Thus, $g=0$ must hold and $z_{1},\ldots,z_{n}$ are algebraically independent.

(4) follows from (3). ∎

4. Some facts concerning the representations of $\mathfrak{sl}_{2}$

Suppose in this section that $\mathbb{F}$ is a field of characteristic zero. It is well known that the Lie algebra $\mathfrak{sl}_{2}=\left<e,f,h\right>$ , where $h=[e,f]$ , acts on the polynomial ring $\mathbb{F}[x_{1},x_{2}]$ by extending the action

	$\displaystyle e$	$\displaystyle:x_{1}\mapsto 0,\quad x_{2}\mapsto x_{1},$
	$\displaystyle f$	$\displaystyle:x_{1}\mapsto x_{2},\quad x_{2}\mapsto 0,$
	$\displaystyle h$	$\displaystyle:x_{1}\mapsto x_{1},\quad x_{2}\mapsto-x_{2}$

using the Leibniz rule. This action on $\mathbb{F}[x_{1},x_{2}]$ can be described, for $g\in\mathbb{F}[x_{1},x_{2}]$ , as

e(g)=x_{1}\frac{\partial g}{\partial x_{2}}\qquad\mbox{and}\qquad f(g)=x_{2}\frac{\partial g}{\partial x_{1}}.

Let $U_{n}$ denote the space of homogeneous polynomials in $\mathbb{F}[x_{1},x_{2}]$ of degree $n$ with basis $x_{1}^{n},x_{1}^{n-1}x_{2},\ldots,x_{1}x_{2}^{n-1},x_{2}^{n}$ . Then the action of $e$ and $f$ can be described as

e(x_{1}^{i}x_{2}^{n-i})=(n-i)x_{1}^{i+1}x_{2}^{n-i-1}\quad\mbox{and}\quad f(x_{1}^{i}x_{2}^{n-i})=ix_{1}^{i-1}x_{2}^{n-i+1}.

The basis $x_{1}^{n},x_{1}^{n-1}x_{2},\ldots,x_{2}^{n}$ consists of eigenvectors for the operator $h$ with eigenvalues $n,n-2,\ldots,-n+2,-n$ , respectively. The eigenvectors of $h$ in a representation of $\mathfrak{sl}_{2}$ are often referred to as weight vectors and the corresponding eigenvalues as weights. The following result is well-known.

Theorem 4.1.

For $n\geqslant 0$ , the space $U_{n}$ is irreducible considered as an $\mathfrak{sl}_{2}$ -module. Furthermore, each finite-dimensional irreducible $\mathfrak{sl}_{2}$ -module is isomorphic to $U_{n}$ for some $n\geqslant 0$ .

Note that the $\mathfrak{sl}_{2}$ -module $U_{n}$ can also be recognized by the weights (that is, the $h$ -eigenvalues) which are $n,n-2,\ldots,-n+2,-n$ . The reason why we are interested in the $\mathfrak{sl}_{2}$ -modules $U_{n}$ is because the action of $e$ on $U_{n}$ is isomorphic to the action of $x$ on $V_{n}=\left<y_{0},\ldots,y_{n}\right>$ defined in Section 2 in the context of standard filiform Lie algebras.

Lemma 4.2.

Suppose that $n\geqslant 0$ and, for $i\in\{0,\ldots,n\}$ , let $w_{i}=(1/i!)x_{1}^{n-i}x_{2}^{i}$ . Then $e(w_{0})=0$ and $e(w_{i})=w_{i-1}$ for $i\in\{1,\ldots,n\}$ .

Proof.

This is easy: just calculate for $i\geqslant 0$ that

e(w_{i})=\frac{1}{i!}e(x_{1}^{n-i}x_{2}^{i})=i\cdot\frac{1}{i!}x_{1}^{n-i+1}x_{2}^{i-1}=\frac{1}{(i-1)!}x_{1}^{n-i+1}x_{2}^{i-1}=w_{i-1}.

∎

For a Lie algebra $\mathfrak{g}$ and for a $\mathfrak{g}$ -module $U$ , let $U^{\mathfrak{g}}$ denote the space

U^{\mathfrak{g}}=\{u\in U\mid a(u)=0\mbox{ for all }a\in\mathfrak{g}\}

of $\mathfrak{g}$ -invariants in $U$ . If $\mathfrak{g}=\left<x\right>$ , then we write $U^{x}$ for $U^{\mathfrak{g}}$ and $U^{x}$ is the zero eigenspace of $x$ acting on $U$ .

Corollary 4.3.

The following are valid for a finite-dimensional $\mathfrak{sl}_{2}$ -module $U$ written as a direct sum $U=W_{1}\oplus\cdots\oplus W_{d}$ of simple $\mathfrak{sl}_{2}$ -submodules.

(1)

For $i\in\{1,\ldots,d\}$ , let $z_{i}\in W_{i}^{e}\setminus\{0\}$ (which is unique up scalar multiple). Then $\{z_{1},\ldots,z_{d}\}$ is a basis of $U^{e}$ .
(2)

$\dim U^{e}=\dim U^{f}=d$ .
(3)

$\dim U^{e}=\dim Y_{0}+\dim Y_{1}$ where $Y_{0}$ and $Y_{1}$ are the eigenspaces in $U$ of the operator $h$ corresponding to the eigenvalues $0$ and $1$ , respectively.

Proof.

By Weyl’s Theorem (see [Jac79, Theorem 8, Section III.7]),

(4)

U=W_{1}\oplus\cdots\oplus W_{d}

where the $W_{i}$ are irreducible $\mathfrak{sl}_{2}$ -modules. Then each $W_{i}$ is isomorphic to $U_{k_{i}}$ where $k_{i}\geqslant 0$ (Theorem 4.1). By Lemma 4.2, the matrix of $e$ on $W_{i}$ is conjugate to a single Jordan block that corresponds to the eigenvalue zero. Thus, the matrix of $e$ on $U$ is conjugate to a block-diagonal matrix with $d$ Jordan blocks each of which corresponds to the eigenvalue zero. Noting that $U^{e}$ is the zero eigenspace of $e$ considered as an operator on $U$ , we obtain assertions (1) and (2). For the assertion that $d=\dim Y_{0}+\dim Y_{1}$ , choose for each $W_{i}$ a basis consisting of $h$ -eigenvectors and note that each irreducible component $W_{i}$ contributes with dimension one to either $Y_{0}$ (if $k_{i}$ is even) or to $Y_{1}$ (if $k_{i}$ is odd). ∎

Lemma 4.4.

Suppose that $U$ is a finite-dimensional $\mathfrak{sl}_{2}$ -module. Suppose that $z\in U^{e}$ is an $h$ -eigenvector of eigenvalue $w$ . Then $z$ is contained in $\bigoplus_{i}(W_{i})^{e}$ where the $W_{i}$ are the irreducible components of $U$ that are isomorphic to $U_{w}$ . Furthermore, the $\mathfrak{sl}_{2}$ -submodule generated by $z$ is irreducible and is isomorphic to $U_{w}$ .

Proof.

The fact that $z\in\bigoplus_{i}(W_{i})^{e}$ follows from the weight space decomposition (4) for $U$ . One can show by induction that $f^{i}(z)$ is an $h$ -eigenvector with eigenvalue $w-2i$ for all $i\in\{0,\ldots,n\}$ and these vectors form the basis of an $\mathfrak{sl}_{2}$ -submodule isomorphic to $U_{w}$ . ∎

Theorem 4.5.

Suppose that $U_{m}$ and $U_{n}$ are $\mathfrak{sl}_{2}$ -modules as defined above. Then the following decomposition of $\mathfrak{sl}_{2}$ -modules is valid:

U_{m}\otimes U_{n}\cong\bigoplus_{i=0}^{\min\{m,n\}}U_{m+n-2i}.

Suppose that $u_{0},\ldots,u_{m}$ and $w_{0},\ldots,w_{n}$ are bases for $U_{m}$ and $U_{n}$ consisting of $h$ -eigenvectors, respectively, such that $e(u_{0})=e(w_{0})=0$ and $e(u_{i})=u_{i-1}$ and $e(w_{j})=w_{j-1}$ for $i\in\{1,\ldots,m\}$ and $j\in\{1,\ldots,n\}$ . Define, for $i=0,1,\ldots,\min\{m,n\}$ ,

(5)

z_{i}=\sum_{d=0}^{i}(-1)^{d}{u_{d}\otimes w_{i-d}}.

Then $e(z_{i})=0$ and the irreducible component of $U_{m}\otimes U_{n}$ isomorphic to $U_{m+n-2i}$ is generated by $z_{i}$ .

Proof.

The decomposition of $U_{m}\otimes U_{n}$ is well known as the Clebsch–Gordan formula for $\mathfrak{sl}_{2}$ [Kow14, Theorem 2.6.3]. The fact that $e(z_{i})=0$ is true since

	$\displaystyle e(z_{i})$	$\displaystyle=e\left(\sum_{d=0}^{i}(-1)^{d}{u_{d}\otimes w_{i-d}}\right)=\sum_{d=0}^{i}(-1)^{d}{e(u_{d})\otimes w_{i-d}}+\sum_{d=0}^{i}(-1)^{d}{u_{d}\otimes e(w_{i-d})}$
		$\displaystyle=\sum_{d=1}^{i}(-1)^{d}{u_{d-1}\otimes w_{i-d}}+\sum_{d=0}^{i-1}(-1)^{d}{u_{d}\otimes w_{i-d-1}}=0.$

Further, it also follows that $z_{i}$ is an $h$ -eigenvector of eigenvalue $n+m-2i$ . By Lemma 4.4, $z_{i}$ is contained in the direct sum of the $\mathfrak{sl}_{2}$ -submodules that are isomorphic to $U_{n+m-2i}$ , but the Clebsch–Gordan decomposition implies that there is a unique such submodule and this must be generated by $z_{i}$ . ∎

The element $z_{i}$ defined in (5) can be viewed as a Casimir element for the operator $e$ on the tensor product $U_{m}\otimes U_{n}$ .

Lemma 4.6.

Let $z\in U_{n}$ be an $h$ -eigenvector of weight $n-2i$ with some $i\in\{0,\ldots,n\}$ . Then

e(f(z))=(n-i)(i+1)z.

Proof.

We can assume without loss of generality that $U_{n}=\left<x_{1}^{n},x_{1}^{n-1}x_{2},\ldots,x_{1}x_{2}^{n-1},x_{2}^{n}\right>$ and $z=x_{1}^{n-i}x_{2}^{i}$ . Then

e(f(z))=e(f(x_{1}^{n-i}x_{2}^{i}))=(n-i)(i+1)x_{1}^{n-i}x_{2}^{i}.

∎

Corollary 4.7.

Let $U_{n}$ be an $(n+1)$ -dimensional irreducible $\mathfrak{sl}_{2}$ -module and consider the polynomial algebra $\mathbb{F}[U_{n}]$ as an $\mathfrak{sl}_{2}$ -module. For $i\geqslant 0$ , let $U_{n,i}$ denote the space of homogeneous polynomials of $\mathbb{F}[U_{n}]$ of degree $i$ . Let $z_{1}\in U_{n,k_{1}}^{e}$ and $z_{2}\in U_{n,k_{2}}^{e}$ that are also $h$ -eigenvectors of weight $w_{1}$ and $w_{2}$ , respectively. Suppose that $d\leqslant\min\{w_{1},w_{2}\}$ and define for $i\in\{1,2\}$ and for $k\in\{0,\ldots,d\}$ ,

z_{i,0}=z_{i}\quad\mbox{and}\quad z_{i,k}=\frac{1}{k(w_{i}-k+1)}f(z_{i,k-1})\quad\mbox{for}\quad k\geqslant 1.

Set

z_{1}\circ_{d}z_{2}=\sum_{i=0}^{d}(-1)^{i}z_{1,i}z_{2,d-i}.

Then $z_{1}\circ_{d}z_{2}\in U_{n,k_{1}+k_{2}}^{e}$ and it is an $h$ -eigenvector of weight $w_{1}+w_{2}-2d$ . Furthermore, letting $y\in U_{n,1}^{e}\setminus\{0\}$ be fixed and $z_{1},\ldots,z_{m}$ be a basis of $U_{n,k-1}^{e}$ formed by $h$ -eigenvectors, the elements $z_{i}\circ_{d}y$ , with $i\in\{1,\ldots,m\}$ and $d\geqslant 0$ generate $U_{n,k}^{e}$ and are $h$ -eigenvectors.

Proof.

Suppose that $W_{1}$ and $W_{2}$ are the irreducible components of $U_{n,k_{1}}$ and $U_{n,k_{2}}$ that contain $z_{1}$ and $z_{2}$ , respectively (they exist by Lemma 4.4). Then $W_{1}\cong U_{w_{1}}$ and $W_{2}\cong U_{w_{2}}$ . The fact that $e(z_{i,0})=0$ , for $i=1,2$ , is clear. Suppose that $k\geqslant 1$ note that $z_{i,k-1}$ is an $h$ -eigenvector of weight $w_{i}-2(k-1)$ . Thus, Lemma 4.6 implies that

e(z_{i,k})=\frac{1}{k(w_{i}-k+1)}e(f(z_{i,k-1}))=\frac{k(w_{i}-k+1)}{k(w_{i}-k+1)}z_{i,k-1}=z_{i,k-1}.

The product space $W_{1}W_{2}$ is an epimorphic image of the tensor product $W_{1}\otimes W_{2}$ under the $\mathfrak{sl}_{2}$ -homomorphism $\psi:W_{1}\otimes W_{2}\to W_{1}W_{2}$ induced by multiplication. Observe that $z_{1}\circ_{d}z_{2}$ is the image of

\sum_{i=0}^{d}(-1)^{i}z_{1,i}\otimes z_{2,d-i}

under $\psi$ , and so Theorem 4.5 implies that $e(z_{1}\circ_{d}z_{2})=0$ and also that $(W_{1}W_{2})^{e}$ is generated by such elements. The final assertion follows from the fact that $U_{n,k}=U_{n,k-1}U_{n,1}$ and that $U_{n,1}$ is irreducible, and in particular $U_{n,1}^{e}=\left<y\right>$ . ∎

5. Explicit generators of $Z_{n}$

In this section we return to our investigation of the invariant algebra $Z_{n}=\mathbb{F}[\mathbf{y}_{n}]^{x}$ as seen in Sections 2–3. Assume throughout this section that $\mathbb{F}$ is a field of characteristic zero. Note that $x$ acts on the vector space $V_{n}=\left<y_{0},y_{1},\ldots,y_{n}\right>$ and Lemma 4.2 shows that its action is equivalent to the action of $e\in\mathfrak{sl}_{2}$ . Thus, there exists $\hat{f}_{n}\in\mbox{End}(V_{n})$ such that the map $x\mapsto e,\ \hat{f}_{n}\mapsto f,\ [x,\hat{f}_{n}]\mapsto h$ can be extended to an isomorphism of Lie algebras $\langle x,\hat{f}_{n},[x,\hat{f}_{n}]\rangle\to\mathfrak{sl}_{2}$ . In fact, $\hat{f}_{n}$ is determined in [Bed10, Theorem 3.1] and in [Bed05, Theorem 3.1] as

(6)

\hat{f}_{n}(y_{i})=(i+1)(n-i)y_{i+1}\quad\mbox{for all}\quad i\in\{0,\ldots,n\}.

It is easy to compute, for all $y_{i}$ , that $[x,\hat{f}_{n}](y_{i})=(n-2i)y_{i}$ and that $[x,[x,\hat{f}_{n}]]=-2x$ , while $[\hat{f}_{n},[x,\hat{f}_{n}]]=2\hat{f}_{n}$ which verifies the isomorphism $\langle x,\hat{f}_{n},h\rangle\to\mathfrak{sl}_{2}$ . Because of the isomorphism with $\mathfrak{sl}_{2}$ , $\hat{h}=[x,\hat{f}_{n}]$ is an operator on $V_{n}$ with eigenvectors $y_{0},y_{1},\ldots,y_{n}$ corresponding to eigenvalues $n,n-2,\ldots,-n+2,-n$ . For $k\geqslant 0$ , let $Z_{n,k}$ denote the degree- $k$ homogeneous component of $Z_{n}$ . The algebras $\mathbb{F}[\mathbf{y}_{n}]$ and $\mathbb{F}[U_{n}]$ in Corollary 4.7 are isomorphic as $\mathfrak{sl}_{2}$ -modules. Furthermore, under the obvious isomorphism between $\mathbb{F}[\mathbf{y}_{n}]$ and $\mathbb{F}[U_{n}]$ , $Z_{n}$ corresponds to $\mathbb{F}[U_{n}]^{e}$ , while $Z_{n,k}$ corresponds to $U_{n,k}^{e}$ . Thus, the following result is an immediate consequence of Corollary 4.7.

Proposition 5.1.

Suppose that $z_{1}\in Z_{n,k_{1}}$ and $z_{2}\in Z_{n,k_{2}}$ are $\hat{h}$ -eigenvectors with eigenvalues $w_{1}$ and $w_{2}$ respectively. Suppose that $d\leqslant\min\{w_{1},w_{2}\}$ and define for $i\in\{1,2\}$ and for $k\in\{0,\ldots,d\}$ ,

z_{i,0}=z_{i}\quad\mbox{and}\quad z_{i,k}=\frac{1}{k(w_{i}-k+1)}\hat{f}_{n}(z_{i,k-1})\quad\mbox{for}\quad k\geqslant 1.

Define

(7)

z_{1}\circ_{d}z_{2}=\sum_{i=0}^{d}(-1)^{i}z_{1,i}z_{2,d-i}.

Then $z_{1}\circ_{d}z_{2}\in Z_{n,k_{1}+k_{2}}$ and this element is an $\hat{h}$ -eigenvector of weight $w_{1}+w_{2}-2d$ . Furthermore, letting $z_{1},\ldots,z_{m}$ be a basis of $Z_{n,k-1}$ formed by $\hat{h}$ -eigenvectors, the elements $z_{i}\circ_{d}y_{0}$ , with $i\in\{1,\ldots,m\}$ and $d\geqslant 0$ generate $Z_{n,k}$ and are $\hat{h}$ -eigenvectors.

The element $z\circ_{d}z$ appears as $\tau_{d}(z)$ in [Bed05, Bed10]. Note that using equation (7), we obtain that

y_{0}\circ_{d}y_{0}=\sum_{i=0}^{d}(-1)^{i}y_{i}y_{d-i}.

Applying Proposition 5.1 recursively for the homogeneous components $Z_{n,1}$ , $Z_{n,2}$ , etc., we obtain the following.

Corollary 5.2.

For $k\geqslant 1$ , the homogeneous component $Z_{n,k}$ is generated as a vector space by elements of the form

((y_{0}\circ_{d_{1}}y_{0})\circ_{d_{2}}y_{0})\cdots\circ_{d_{k-1}}y_{0}

where the number of factors is $k$ and the $d_{i}$ are chosen in such a way that $\circ_{d_{i}}$ be defined for each $i$ . In particular, the ring $Z_{n}$ admits a minimal generating set formed by polynomials of this form.

In some cases, it is possible to obtain explicit generators for the invariant ring $\mathbb{F}[\mathbf{y}_{n}]^{x}$ as shown in the following example.

Example 5.3.

As noted in Example 2.2, $Z_{1}=\mathbb{F}[y_{0}]$ , and it is well-known that

Z_{2}=\mathbb{F}[y_{0},y_{0}\circ_{2}y_{0}]=\mathbb{F}[y_{1},2y_{0}y_{2}-y_{1}^{2}]

(see [Dix58]). The generators for $Z_{3}$ were determined by Dixmier [Dix58], and we can write them as

Z_{3}=\mathbb{F}[z_{1},z_{2},z_{3},\zeta_{4}]

where

	$\displaystyle z_{1}$	$\displaystyle=y_{0};$
	$\displaystyle z_{2}$	$\displaystyle=2y_{0}y_{2}-y_{1}^{2}=y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{3}$	$\displaystyle=3y_{0}^{2}y_{3}-3y_{0}y_{1}y_{2}+y_{1}^{3}=-y_{0}\circ_{2}y_{0}\circ_{1}y_{0};$
	$\displaystyle\zeta_{4}$	$\displaystyle=\frac{z_{2}^{3}+z_{3}^{2}}{y_{0}^{2}}=-3y_{1}^{2}y_{2}^{2}+8y_{0}y_{2}^{3}+6y_{1}^{3}y_{3}-18y_{0}y_{1}y_{2}y_{3}+9y_{0}^{2}y_{3}^{2}$
		$\displaystyle=(-3/2)y_{0}\circ_{2}y_{0}\circ_{1}y_{0}\circ_{3}y_{0}.$

The expressions in $\circ_{d}$ are left normed; that is, $a\circ_{d_{1}}b\circ_{d_{2}}c=(a\circ_{d_{1}}b)\circ_{d_{2}}c$ . According to [Oom12] The algebra $\mathbb{F}[\mathbf{y}_{4}]^{x}$ is generated by

	$\displaystyle z_{1}$	$\displaystyle=y_{0};$
	$\displaystyle z_{2}$	$\displaystyle=2y_{0}y_{2}-y_{1}^{2}=y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{3}$	$\displaystyle=3y_{0}^{2}y_{3}-3y_{0}y_{1}y_{2}+y_{1}^{3}=-y_{0}\circ_{2}y_{0}\circ_{1}y_{0};$
	$\displaystyle z_{4}$	$\displaystyle=2y_{0}y_{4}-2y_{1}y_{3}+y_{2}^{2}=y_{0}\circ_{4}y_{0};$
	$\displaystyle\zeta_{5}$	$\displaystyle=\frac{z_{2}^{3}+z_{3}^{2}-3y_{0}^{2}z_{2}z_{4}}{y_{0}^{3}}=2y_{2}^{3}-6y_{1}y_{2}y_{3}+9y_{0}y_{3}^{2}+6y_{1}^{2}y_{4}-12y_{0}y_{2}y_{4}$
		$\displaystyle=-2y_{0}\circ_{2}y_{0}\circ_{4}y_{0}.$

Since $\mathbb{F}[\mathbf{y}_{n}]^{x}$ is a noetherian ring, by the Mauer–Weitzenböck Theorem [Fre17, Theorem 6.1], one may want to describe its Noether normalization. The algebra $\mathbb{F}[\mathbf{y}_{2}]^{x}$ is freely generated by $z_{1}$ and $z_{2}$ and so there is nothing to do. On the other hand, $\mathbb{F}[\mathbf{y}_{3}]^{x}$ is an integral extension of $\mathbb{F}[z_{1},z_{2},\zeta_{4}]$ , or of $\mathbb{F}[z_{1},z_{3},\zeta_{4}]$ , but not of $\mathbb{F}[z_{1},z_{2},z_{3}]$ . This is because $z_{3}$ satisfies the integral equation $t^{2}+z_{2}^{3}-y_{0}^{2}\zeta_{4}=0$ . Similarly, we obtain that $\mathbb{F}[\mathbf{y}_{4}]$ is integral over $\mathbb{F}[z_{1},z_{2},z_{4},\zeta_{5}]$ or over $\mathbb{F}[z_{1},z_{3},z_{4},\zeta_{5}]$ , but not over $\mathbb{F}[z_{1},z_{2},z_{3},z_{4}]$ .

Problem. Determine generators for the Noether normalization of $\mathbb{F}[\mathbf{y}_{n}]^{x}$ for $n\geqslant 5$ .

The algebra $\mathbb{F}[\mathbf{y}_{5}]$ is known to be minimally generated by 23 generators that satisfy 168 relations; see the discussion after Example 27 in [Oom09]. Using the computational algebra system Magma [BCP97], we determined explicit expressions in terms of the $\circ_{d}$ operation for $Z_{5}$ , $Z_{6}$ and $Z_{8}$ . See Appendix A.

5.1. The first generator sequence

We use Proposition 4.7 to obtain an explicit generating set for the field of fractions of $\mathbb{F}[\mathbf{y}_{n}]^{x}$ . Set $z_{1}=y_{0}$ , and, for $i\geqslant 2$ , define

z_{i}=\left\{\begin{array}[]{ll}y_{0}\circ_{i}y_{0}&\mbox{if $i$ is even}\\ y_{0}\circ_{1}(y_{0}\circ_{i-1}y_{0})&\mbox{if $i$ is odd.}\end{array}\right.

The explicit expressions for the first couple of values of the $z_{i}$ are as follows:

	$\displaystyle z_{1}$	$\displaystyle=y_{0};$
	$\displaystyle z_{2}$	$\displaystyle=2y_{0}y_{2}-y_{1}^{2};$
	$\displaystyle z_{3}$	$\displaystyle=3y_{0}^{2}y_{3}-3y_{0}y_{1}y_{2}+y_{1}^{3};$
	$\displaystyle z_{4}$	$\displaystyle=2y_{0}y_{4}-2y_{1}y_{3}+y_{2}^{2};$
	$\displaystyle z_{5}$	$\displaystyle=5y_{0}^{2}y_{5}-5y_{0}y_{1}y_{4}+y_{0}y_{2}y_{3}+2y_{1}^{2}y_{3}-y_{1}y_{2}^{2};$
	$\displaystyle z_{6}$	$\displaystyle=2y_{0}y_{6}-2y_{1}y_{5}+2y_{2}y_{4}-y_{3}^{2};$
	$\displaystyle z_{7}$	$\displaystyle=7y_{0}^{2}y_{7}-7y_{0}y_{1}y_{6}+3y_{0}y_{2}y_{5}-y_{0}y_{3}y_{4}+2y_{1}^{2}y_{5}-2y_{1}y_{2}y_{4}+y_{1}y_{3}^{2}.$

Recall that the lexicographic monomial order on $\mathbb{F}[\mathbf{y}_{n}]$ is taken ordering the variables in increasing order as $y_{n},y_{n-1},\ldots,y_{0}$ .

Lemma 5.4.

The element $z_{i}\in\mathbb{F}[\mathbf{y}_{n}]^{x}$ for all $i\leqslant n$ . Furthermore, the leading term of $z_{i}$ in the lexicographic monomial order is $2y_{i}y_{0}$ if $i$ is even, while the leading term is $i\cdot y_{i}y_{0}^{2}$ when $i$ is odd.

Proof.

The fact that $z_{i}\in\mathbb{F}[\mathbf{y}_{n}]^{x}$ follows from Corollary 4.7. Let us show the statement concerning the leading term. For $i\geqslant 2$ even, we have that

z_{i}=\sum_{j=0}^{i}(-1)^{j}y_{j}y_{i-j}.

The leading term of $z_{i}$ is the term that contains $y_{i}$ which is $2y_{0}y_{i}$ as claimed. Now for $i$ odd, the definition of $\circ_{d}$ in (7) and the definition of $\hat{f}$ in (6) imply that

	$\displaystyle z_{i}$	$\displaystyle=\frac{1}{2n-2i+2}y_{0}\hat{f}_{n}\left(\sum_{j=0}^{i-1}(-1)^{j}y_{j}y_{i-1-j}\right)-y_{1}\sum_{j=0}^{i-1}(-1)^{j}y_{j}y_{i-1-j}$
		$\displaystyle=\frac{1}{2n-2i-2}y_{0}\left(\sum_{j=0}^{i-1}(-1)^{j}(j+1)(n-j)y_{j+1}y_{i-1-j}+\right.$
		$\displaystyle\left.\sum_{j=0}^{i-1}(-1)^{j}(i-j)(n-i+j+1)y_{j}y_{i-j}\right)$
		$\displaystyle-y_{1}\sum_{j=0}^{i-1}(-1)^{j}y_{j}y_{i-1-j}.$

The leading term of this expression is the only term containing $y_{i}$ , which is $iy_{0}^{2}y_{i}$ as claimed. ∎

5.2. The second sequence

We define another sequence of elements in $\mathbb{F}[\mathbf{y}]^{x}$ which appears in [ŠW14]. (Recall we assume that the characteristic of $\mathbb{F}$ is zero.) Set, $w_{1}=y_{0}$ and for $n\geqslant 1$ ,

(8)

w_{n}=\frac{(-1)^{n}}{n!}y_{1}^{n}+\sum_{i=0}^{n-1}\frac{(-1)^{i}}{j!}y_{0}^{n-1-i}y_{1}^{i}y_{n-i}.

The first couple of values of the sequence $w_{i}$ are as follows.

	$\displaystyle w_{1}$	$\displaystyle=y_{0}$
	$\displaystyle w_{2}$	$\displaystyle=y_{0}y_{2}-\frac{1}{2}y_{1}^{2}$
	$\displaystyle w_{3}$	$\displaystyle=y_{0}^{2}y_{3}-y_{0}y_{1}y_{2}+\frac{1}{3}y_{1}^{3}$
	$\displaystyle w_{4}$	$\displaystyle=y_{0}^{3}y_{4}-y_{0}^{2}y_{1}y_{3}+\frac{1}{2}y_{0}y_{1}^{2}y_{2}-\frac{1}{8}y_{1}^{4}$
	$\displaystyle w_{5}$	$\displaystyle=y_{0}^{4}y_{5}-y_{0}^{3}y_{1}y_{4}+\frac{1}{2}y_{0}^{2}y_{1}^{2}y_{3}-\frac{1}{6}y_{0}y_{1}^{3}y_{2}+\frac{1}{30}y_{1}^{5}$
	$\displaystyle w_{6}$	$\displaystyle=y_{0}^{5}y_{6}-y_{0}^{4}y_{1}y_{5}+\frac{1}{2}y_{0}^{3}y_{1}^{2}y_{4}-\frac{1}{6}y_{0}^{2}y_{1}^{3}y_{3}+\frac{1}{24}y_{0}y_{1}^{4}y_{2}-\frac{1}{144}y_{1}^{6}$
	$\displaystyle w_{7}$	$\displaystyle=y_{0}^{6}y_{7}-y_{0}^{5}y_{1}y_{6}+\frac{1}{2}y_{0}^{4}y_{1}^{2}y_{5}-\frac{1}{6}y_{0}^{3}y_{1}^{3}y_{4}+\frac{1}{24}y_{0}^{2}y_{1}^{4}y_{3}-\frac{1}{120}y_{0}y_{1}^{5}y_{2}+\frac{1}{840}y_{1}^{7}$
	$\displaystyle w_{8}$	$\displaystyle=y_{0}^{7}y_{8}-y_{0}^{6}y_{1}y_{7}+\frac{1}{2}y_{0}^{5}y_{1}^{2}y_{6}-\frac{1}{6}y_{0}^{4}y_{1}^{3}y_{5}+\frac{1}{24}y_{0}^{3}y_{1}^{4}y_{4}-\frac{1}{120}y_{0}^{2}y_{1}^{5}y_{3}+\frac{1}{720}y_{0}y_{1}^{6}y_{2}-\frac{1}{5760}y_{1}^{8}.$

Lemma 5.5.

We have that $w_{n}\in\mathbb{F}[\mathbf{y}]^{x}$ for all $m\geqslant 1$ . Furthermore, the leading term of $w_{n}$ is $y_{0}^{i-1}y_{n}$ .

Proof.

The claim that $w_{n}\in\mathbb{F}[\mathbf{y}]^{x}$ follows from [ŠW14, Lemma 3.1], while the claim concerning the leading term is easily verified by inspection of (8). ∎

Theorem 5.6.

If the characteristic of $\mathbb{F}$ is zero, then the following are valid.

(1)

The sets $\{z_{1},z_{2},\ldots,\}$ and $\{w_{1},w_{2},\ldots\}$ are algebraically independent in $\mathbb{F}[\mathbf{y}]$ .
(2)

The fraction field of the algebra $Z_{n}$ is generated by $z_{1},\ldots,z_{n}$ and also by $w_{1},\ldots,w_{n}$ .
(3)

$Z_{n}$ lies in $\mathbb{F}[y_{0}^{-1},z_{1},z_{2},\ldots,z_{n}]$ and also in $\mathbb{F}[y_{0}^{-1},w_{1},w_{2},\ldots,w_{n}]$ .
(4)

The Krull dimension of $Z_{n}$ is $n$ .

Proof.

This follows from Theorem 3.4, Lemma 5.4 and from Lemma 5.5. ∎

Theorem 5.6 implies Theorem 1.1.

6. The Hilbert series of $Z_{n}$

In this section $\mathbb{F}$ is a field of characteristic zero. Suppose, as in Section 2, that $V_{n}=\left<y_{0},\ldots,y_{n}\right>$ is an $\mathbb{F}$ -vector space of dimension $n+1$ and consider $V_{n}$ as an $x$ -module. The action of $x$ on $V_{n}$ is equivalent to the action of $e\in\mathfrak{sl}_{2}$ on the $(n+1)$ -dimensional irreducible $\mathfrak{sl}_{2}$ -module $U_{n}$ by Lemma 4.2. Thus, the invariant ring $Z_{n}=\mathbb{F}[\mathbf{y}_{n}]^{x}$ is isomorphic to the algebra of polynomial invariants $\mathbb{F}[U_{n}]^{e}$ where $\mathbb{F}[U_{n}]$ is the polynomial algebra generated by $U_{n}$ . Furthermore, this isomorphism is an isomorphism of graded algebras with respect to the grading by degree on $\mathbb{F}[\mathbf{y}_{n}]$ and $\mathbb{F}[U_{n}]$ . For $n\geqslant 1$ , and $d\geqslant 0$ , let $\delta_{n,d}$ denote the dimension of the degree- $d$ homogeneous component of $Z_{n}$ (which is equal to the dimension of the degree- $d$ homogeneous component of $\mathbb{F}[U_{n}]^{e}$ ). Set

H_{n}(t)=\sum_{d=0}^{\infty}\delta_{n,d}t^{d}.

Consider the basis of $U_{n}$ formed by $u_{i}=x_{1}^{i}x_{2}^{n-i}$ for $i\in\{0,\ldots,n\}$ . Then

h(u_{i})=(-n+2i)u_{i}

and each monomial $m=u_{0}^{\alpha_{0}}\cdots u_{n}^{\alpha_{n}}$ in the $u_{i}$ is an eigenvector of $h$ with eigenvalue $w(m)$ (which can be calculated explicitly if needed). The eigenvalue $w(m)$ is also called the weight of $m$ . The degree of such a monomial is $\sum_{i}\alpha_{i}$ . Noting that $U_{n,d}$ is isomorphic, as an $\mathfrak{sl}_{2}$ -module, to the $d$ -th symmetric power $S^{d}(U_{n})$ of the irreducible $\mathfrak{sl}_{2}$ -module $U_{n}$ , the following result follows at once from Corollary 4.3.

Theorem 6.1.

$\delta_{n,d}$ is equal to the number of irreducible components of the symmetric power $S^{d}(U_{n})$ considered as an $\mathfrak{sl}_{2}$ -module. Moreover, if either $n$ or $d$ is even then $\delta_{n,d}$ is equal to the number of monomials in the $u_{i}$ with degree $d$ and weight zero; otherwise $\delta_{n,d}$ is equal to the number of such monomials with degree $d$ and weight one.

Example 6.2.

Consider for example the case of $n=1$ . Then, for each $d\geqslant 0$ ,

\delta_{1,d}=1

since, for even $d$ , the only weight zero monomial of degree $d$ is $u_{0}^{d/2}u_{1}^{d/2}$ , while for odd $d$ , the only weight one monomial of degree $d$ is $u_{0}^{(d+1)/2}u_{1}^{(d-1)/2}$ . This corresponds to the fact that $Z_{1}$ is generated by $y_{0}$ and the homogeneous component of degree $d$ is generated by $y_{0}^{d}$ (see Example 2.2). The Hilbert series is

H_{1}(t)=\sum_{d=0}^{\infty}t^{d}=\frac{1}{1-t}.

This shows also that $Z_{1}$ is generated by $y_{0}$ which is a generator of degree $1$ and $Z_{1}$ is isomorphic to the polynomial algebra $\mathbb{F}[t]$ in one variable $t$ .

Example 6.3.

Suppose that $n=2$ . Then the weight zero monomials of degree $d$ are $u_{0}^{i}u_{1}^{d-2i}u_{2}^{i}$ with $i\in\{0,\ldots\lfloor d/2\rfloor\}$ . Thus, $\delta_{2,d}=\lfloor d/2\rfloor+1$ and the sequence $\delta_{2,d}$ is

1,1,2,2,3,3,4,4,5,5,\ldots.

Easy computation (distinguishing between $d$ odd and $d$ even) shows that

\delta_{2,d}=\delta_{2,d-1}+\delta_{2,d-2}-\delta_{2,d-3}

for $d\geqslant 3$ with initial values $\delta_{2,0}=\delta_{2,1}=1$ and $\delta_{2,2}=2$ . Hence, the characteristic polynomial of $\delta_{2,d}$ considered as a recursive sequence is

t^{3}-t^{2}-t+1=(1-t)(1-t^{2}).

In fact, the Hilbert series is

H_{2}(t)=\sum_{d=0}^{\infty}\delta_{2,d}t^{d}=\frac{1}{(1-t)(1-t^{2})}

reflecting the fact that $Z_{2}=\mathbb{F}[z_{1},z_{2}]=\mathbb{F}[y_{0},2y_{0}y_{2}-y_{1}^{2}]$ generated by a generator of degree one and a generator of degree two and these generators are algebraically independent.

For $k,d,n\in\mathbb{N}_{0}$ , let $p(k,d,n)$ denote the number of partitions of $k$ with $d$ parts each of which of size at most $n$ allowing parts of size zero.

Theorem 6.4.

If $n,d\geqslant 0$ , then $\delta_{n,d}=p(\lfloor dn/2\rfloor,d,n)=p(\lfloor dn/2\rfloor,n,d)$ . Consequently, $\delta_{n,d}=\delta_{d,n}$

Proof.

Suppose that $m=u_{0}^{\alpha_{0}}u_{1}^{\alpha_{1}}\cdots u_{n}^{\alpha_{n}}$ is a monomial in the $u_{i}$ of degree $d$ ; that is $\sum_{i\geqslant 0}\alpha_{i}=d$ . Define the function $w^{\prime}$ on the generators $u_{0},\ldots,u_{n}$ as $w^{\prime}(u_{i})=i$ and extend it multiplicatively to the set of monomials in $u_{0},\ldots,u_{n}$ . Then $w^{\prime}(m)=(w(m)+dn)/2$ where $w(m)$ is the weight (that is, the $h$ -eigenvalue) of the monomial $m$ . If either $n$ is even or $d$ is even, then $\delta_{n,d}$ is equal to the number of monomials of degree $d$ and weight zero, and if $m$ has degree $d$ , then $w(m)=0$ if and only if $w^{\prime}(m)=dn/2$ . If both $n$ and $d$ are odd, then $\delta_{n,d}$ is equal to the number of monomials of degree $d$ and weight minus one (which is equal to the number of monomials of degree $d$ and weight one) and if the degree of $m$ is $d$ , then $w(m)=-1$ if and only if $w^{\prime}(m)=(dn-1)/2=\lfloor dn/2\rfloor$ . So in both cases, $\delta_{n,d}$ is equal to the number of monomials $m$ of degree $d$ and $w^{\prime}(m)=\lfloor dn/2\rfloor$ . That is, we need to count monomials $m=u_{0}^{\alpha_{0}}u_{1}^{\alpha_{1}}\cdots u_{n}^{\alpha_{n}}$ such that

\sum_{i=0}^{n}\alpha_{i}=d\quad\mbox{and}\quad\sum_{i=0}^{n}i\cdot\alpha_{i}=\lfloor dn/2\rfloor.

Now the sequence $\alpha_{0},\alpha_{1},\ldots,\alpha_{n}$ can be identified with the partition of $\lfloor dn/2\rfloor$ in which there are $\alpha_{i}$ parts of size $i$ for all $i\in\{0,\ldots,n\}$ . Omitting the parts of size zero, we obtain a correspondence between the set of such monomials $m$ and partitions of $\lfloor dn/2\rfloor$ with at most $d$ parts each of which has size at most $n$ .

Transposing the partitions in question, we obtain that $p(\lfloor dn/2\rfloor,d,n)=p(\lfloor dn/2\rfloor,n,d)$ , and this implies the final statement of the theorem. ∎

Example 6.5.

Consider the case $n=3$ . Then $\delta_{3,d}$ is equal to the number of partitions of $\lfloor 3d/2\rfloor$ with at most $d$ parts of size at most $3$ . The first elements of this sequence are $1$ , $1$ , $2$ , $3$ , $5$ , $6$ , $8$ , $10$ , $13$ , $15$ , $18$ , $21$ , $25$ , $28$ , $32$ , $36$ , etc. According to [SI20, A001971], this sequence $a(d)=\delta_{3,d}$ satisfies the recurrence relation

a(d)=2a(d-1)-a(d-2)+a(d-4)-2a(d-5)+a(d-6).

Furthermore, its generating function is

H_{3}(t)=\frac{1-t^{6}}{(1-t)(1-t^{2})(1-t^{3})(1-t^{4})}.

It is known that $\mathbb{F}[\mathbf{y}_{3}]^{x}=Z_{3}$ is generated by $y_{0}$ , $2y_{0}y_{2}-y_{1}^{2}$ , $3y_{0}^{2}y_{3}-3y_{0}y_{1}y_{2}+y_{1}^{3}$ , and $-3y_{1}^{2}y_{2}^{2}+8y_{0}y_{2}^{3}+6y_{1}^{3}y_{3}-18y_{0}y_{1}y_{2}y_{3}+9y_{0}^{2}y_{3}^{2}$ ; that is, a generator of degree one, one of degree 2, one of degree three, and one of degree 4 (see Example 5.3).

Example 6.6.

Consider the case $n=4$ . Then $\delta_{4,d}$ is equal to the number of partitions of $\lfloor 4d/2\rfloor=2d$ with at most $d$ parts of size at most $4$ . The first elements of this sequence are $1$ , $1$ , $3$ , $5$ , $8$ , $12$ , $18$ , $24$ , $33$ , $43$ , $55$ , $69$ , $86$ , $104$ , $126$ , $150$ , etc. According to [SI20, A001973], this sequence $a(d)=\delta_{4,d}$ satisfies the recurrence relation

a(d)=2a(d-1)-a(d-3)-a(d-4)+2a(d-6)-a(d-7).

Furthermore, its generating function is

H_{4}(t)=\frac{1+t^{3}}{(1-t)(1-t^{2})^{2}(1-t^{3})}

It is known that $\mathbb{F}[\mathbf{y}_{4}]^{x}=Z_{4}$ is generated by $z_{1}$ , $z_{2}$ , $z_{3}$ , $z_{4}$ (as defined in Section 5.1), and

\zeta_{5}=2y_{2}^{3}-6y_{1}y_{2}y_{3}+9y_{0}y_{3}^{2}+6y_{1}^{2}y_{4}-12y_{0}y_{2}y_{4};

that is a generator of degree one, two of degree two and two of degree three (see Example 5.3).

Example 6.7.

Consider the case $n=5$ . Then $\delta_{5,d}$ is equal to the number of partitions of $\lfloor 5d/2\rfloor$ with at most $d$ parts of size at most $5$ . The first elements of this sequence are $1$ , $1$ , $3$ , $6$ , $12$ , $20$ , $32$ , $49$ , $73$ , $102$ , $141$ , $190$ , $252$ , $325$ , $414$ , $521$ , $649$ , $795$ , $967$ , $1165$ , $1394$ , etc. According to [SI20, A001975], this sequence $a(d)=\delta_{5,d}$ satisfies the recurrence relation

	$\displaystyle a(d)$	$\displaystyle=2a(d-1)-a(d-2)+a(d-4)-2a(d-5)+2a(d-6)-2a(d-7)$
		$\displaystyle+2a(d-8)-2a(d-9)+2a(d-11)-2a(d-12)+2a(d-13)$
		$\displaystyle-2a(d-14)+2a(d-15)-a(d-16)+a(d-18)-2a(d-19)+a(d-20)$

Furthermore, its generating function is

H_{5}(t)=\frac{-(t^{14}-t^{13}+2t^{12}+t^{11}+2t^{10}+3t^{9}+t^{8}+5t^{7}+t^{6}+3t^{5}+2t^{4}+t^{3}+2t^{2}-t+1)}{(t^{4}+1)(t^{2}+t+1)(t^{2}-t+1)(t^{2}+1)^{2}(t+1)^{3}(t-1)^{5}}.

It is claimed in [Oom09, Oom12] that $Z_{5}$ is generated by 23 generators.

Theorem 1.3 stated in the Introduction gives a closed formula for the Hilbert series $H_{n}(t)$ for all $n$ . The formula originates from [Alm80a] where it is derived in a different context with proof in [AF78]. We include a proof for completeness and easy reference.

The proof of Theorem 1.3.

Suppose that $u_{-n},u_{-n+2},\ldots,u_{n-2},u_{n}$ are variables such that the weight of $u_{i}$ is $i$ . Suppose that $I_{n}$ denotes the index set $\{-n,-n+2,\ldots,n-2,n\}$ . We denote by $\boldsymbol{\alpha}$ a vector $(\alpha_{-n},\alpha_{-n+2},\ldots,\alpha_{n})$ and by $A_{d}$ the set of $\boldsymbol{\alpha}$ such that $0\leqslant\alpha_{i}\leqslant d$ and $\sum_{j\in I_{n}}\alpha_{j}=d$ . The weight of a monomial $\prod_{j\in I_{n}}u_{j}^{\alpha_{j}}$ is $\sum_{j\in I_{n}}\alpha_{j}\cdot j$ and its degree is $\sum_{j\in I_{n}}\alpha_{j}$ . Recall, for $n\geqslant 0$ and $d\geqslant 0$ , that $\delta_{n,d}$ is the number of monomials of degree $d$ and weight zero plus the number of monomials of degree $d$ and weight one in the $u_{i}$ . We are required to show that the right-hand side of (1) can be written as

\sum_{d=0}^{\infty}\delta_{n,d}t^{d}.

Let us first work the denominator of the function inside the integral. More specifically, we compute that

\displaystyle\prod_{k=0}^{n}\frac{1}{1-t\exp(i(n-2k)\varphi)}=\prod_{k\in I_{n}}\frac{1}{1-t\exp(ik\varphi)}=\prod_{k\in I_{n}}\sum_{j\geqslant 0}{\exp(ijk\varphi)t^{j}}.

The product of the power series in the last expression is a power series $\sum_{d\geqslant 0}c_{d}t^{d}$ whose $d$ -th coefficient is

c_{d}=\sum_{\boldsymbol{\alpha}\in A_{d}}\exp\left(i\varphi\sum_{k\in I_{n}}k\alpha_{k}\right).

The integral in (1) can be split into two parts:

(9)

\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{\prod_{k=0}^{n}(1-t\exp(i(n-2k)\varphi))}\,d\varphi+\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\exp(i\varphi)}{\prod_{k=0}^{n}(1-t\exp(i(n-2k)\varphi))}\,d\varphi.

By the calculations above, the first summand in (9) can be written as

(10)

\frac{1}{2\pi}\left(\int_{-\pi}^{\pi}\sum_{d\geqslant 0}c_{d}\,d\varphi\right)t^{d}=\frac{1}{2\pi}\sum_{d\geqslant 0}\left(\sum_{\boldsymbol{\alpha}\in A_{d}}\int_{-\pi}^{\pi}\exp\left(i\varphi\sum_{k\in I_{n}}k\alpha_{k}\right)\,d\varphi\right)t^{d}.

We have, for $k\in\mathbb{Z}$ , that

\int_{-\pi}^{\pi}\exp(ik\varphi)\,d\varphi=\left\{\begin{array}[]{ll}0&\mbox{if $k\neq 0$}\\ 2\pi&\mbox{if $k=0$.}\end{array}\right.

Hence, among the integrals in the coefficient of $t^{d}$ in (10) only those are different from zero which are taken for $\boldsymbol{\alpha}\in A_{d}$ with $\sum_{k\in I_{n}}k\alpha_{k}=0$ . Furthermore, for such an $\boldsymbol{\alpha}$ , the value of the integral is $2\pi$ and hence, since there is a $1/(2\pi)$ factor, each such $\boldsymbol{\alpha}$ contributes with one to the coefficient of $t^{d}$ . Now each such choice of $\boldsymbol{\alpha}$ corresponds to a monomial $u_{-n}^{\alpha_{-n}}\cdots u_{n}^{\alpha_{n}}$ with degree $d$ and weight zero. Thus, the first term in (9) can be written as $\sum_{d\geqslant 0}a_{d}t^{d}$ where $a_{d}$ is the number of monomials in the $u_{i}$ of degree $d$ and weight zero.

Now consider the second summand in (9) which is equal

(11)

\frac{1}{2\pi}\left(\int_{-\pi}^{\pi}\exp(i\varphi)\sum_{d\geqslant 0}c_{d}\,d\varphi\right)t^{d}=\frac{1}{2\pi}\sum_{d\geqslant 0}\left(\sum_{\boldsymbol{\alpha}\in A_{d}}\int_{-\pi}^{\pi}\exp\left(i\varphi\left(1+\sum_{k\in I_{n}}ka_{k}\right)\right)\,d\varphi\right)t^{d}.

For reasons similar to the ones in the previous paragraph, among the integrals in the coefficient of $t^{d}$ the ones that are different from zero are taken for $\boldsymbol{\alpha}\in A_{d}$ with $1+\sum_{k\in I_{n}}ka_{k}=0$ . Now each such choice of $\boldsymbol{\alpha}$ corresponds to a monomial $u_{-n}^{a_{-n}}\cdots u_{n}^{a_{n}}$ with degree $d$ and weight $-1$ . Thus, the second term in (9) can be written as $\sum_{d\geqslant 0}b_{d}t^{d}$ where $b_{d}$ is the number of monomials in the $u_{i}$ of degree $d$ and weight $-1$ which is the same as the number of monomials of degree $d$ and weight one. ∎

The first summand in (9) can be viewed as the Molien series for the invariant ring of the compact Lie group $S^{1}=\{\exp(i\varphi)\mid\varphi\in[-\pi,\pi)\}$ inside $\mathbb{F}[\mathbf{y}_{n}]$ where an element $\exp(i\varphi)\in S^{1}$ acts on the variables $y_{0},\ldots,y_{n}$ by multiplication with the scalars

\exp(in\varphi),\ \exp(i(n-2)\varphi),\ldots,\exp(i(-n+2)\varphi),\ \exp(-in\varphi).

(See [DK15, Theorem 3.4.2] for Molien series for the invariant ring of finite groups.) The second term is a translate of the same Molien series; taking such a translate corresponds to considering monomials of weight one instead of zero.

6.1. Computation of $\boldsymbol{H_{n}(t)}$ for higher values of $\boldsymbol{n}$

Let $V_{n,d}$ be the space of degree- $d$ polynomials in the variables $y_{0},\ldots,y_{n}$ considered as an $\mathfrak{sl}_{2}$ -module. That is, $V_{n,1}$ is considered as an irreducible $\mathfrak{sl}_{2}$ -module of dimension $n+1$ with weights $n,n-2,\ldots,-n$ and $V_{n,d}$ is isomorphic to the symmetric power $S^{d}(V_{n,1})$ . Then $V_{n,d}$ is spanned by monomials of the form $y_{0}^{\alpha_{0}}\cdots y_{n}^{\alpha_{n}}$ with $\alpha_{0}+\cdots+\alpha_{n}=d$ . As above, the weight of such a monomial is $n\alpha_{0}+(n-2)\alpha_{1}+\cdots-n\alpha_{n}$ . Set

M_{n}(d,w)=\mbox{the number of monomials in $V_{n,d}$ of weight $w$}

and define the generating function $\mu_{n}(t,z)$ as

\mu_{n}(t,z)=\sum_{d\geqslant 0,w}^{\infty}M_{n}(d,w)t^{d}z^{w}.

Then standard computation with generating functions shows that the following theorem is valid.

Theorem 6.8.

For $n\geqslant 1$ , we have

(12)

\mu_{n}(t,z)=\prod_{k\in\{-n,-n+2,\ldots,n-2,n\}}\frac{1}{1-tz^{k}}.

Example 6.9.

If $n=1$ , then

\mu_{1}(t,z)=1+t(z^{-1}+z)+t^{2}(z^{-2}+1+z^{2})+t^{3}(z^{-3}+z^{-1}+z+z^{3})+\cdots

In this case we have to count monomials of even degree and weight zero and monomials of odd degree and weight one. These monomials are counted by the terms

1+tz+t^{2}+t^{3}z+\cdots

showing that $\delta_{1,d}=1$ for all $d\geqslant 0$ . This agrees with the values of $\delta_{1,d}$ computed in Example 6.2.

Example 6.10.

If $n=2$ , then

	$\displaystyle\mu_{2}(t,z)$	$\displaystyle=1+t(z^{2}+z^{-2}+1)+t^{2}(z^{4}+z^{-4}+z^{2}+z^{-2}+2)$
		$\displaystyle+t^{3}(z^{6}+z^{-6}+z^{4}+z^{-4}+2z^{2}+2z^{-2}+2)$
		$\displaystyle+t^{4}(z^{8}+z^{-8}+z^{6}+z^{-6}+2z^{4}+2z^{-4}+2z^{2}+2z^{-2}+3)$
		$\displaystyle+t^{5}(z^{10}+z^{-10}+z^{8}+z^{-8}+2z^{6}+2z^{-6}+2z^{4}+2z^{-4}+3z^{2}+3z^{-2}+3)+O(t^{6})$

The part of this series expansion containing only the terms with $z^{0}$ is

1+t+2t^{2}+2t^{3}+3t^{4}+3t^{5}+\cdots

and the coefficients agree with the values for $\delta_{2,d}$ computed in Example 6.3.

Example 6.11.

Let us compute $H_{3}(t)$ using the method outlined by Ekhad and Zeilberger. By Theorem 6.8 we have that

\mu_{3}(t,z)=\frac{1}{(1-z^{-3}t)(1-z^{-1}t)(1-zt)(1-z^{3}t)}=z^{4}\frac{1}{(z^{3}-t)(z-t)(1-zt)(1-z^{3}t)}.

Using that the factors $z^{3}-t$ , $z-t$ , $1-zt$ , $1-z^{3}t$ are coprime polynomials in the variable $z$ over the rational function field $\mathbb{Q}(t)$ , we have that there are polynomials $Q_{1},Q_{2},Q_{3},Q_{4}\in\mathbb{Q}(t)[z]$ such that

	$\displaystyle\mu_{3}(t,z)$	$\displaystyle=z^{4}\left(\frac{Q_{1}}{z^{3}-t}+\frac{Q_{2}}{z+t}+\frac{Q_{3}}{1-zt}+\frac{Q_{4}}{1-z^{3}t}\right)$
		$\displaystyle=z\frac{Q_{1}}{1-z^{-3}t}+z^{3}\frac{Q_{2}}{1-z^{-1}t}+z^{4}\frac{Q_{3}}{1-zt}+z^{4}\frac{Q_{4}}{1-z^{3}t}.$

The expressions $Q_{i}$ in the last displayed equation can be determined in a computational algebra system, such as Magma [BCP97], using the Extended Euclidean Algorithm. Writing $\mu_{3}(t,z)=\sum_{k\in\mathbb{Z}}q_{k}z^{k}$ where $q_{k}\in\mathbb{Q}(t)$ , remembering that $n=3$ is odd, the Hilbert series $H_{3}(t)$ is determined by the coefficients $q_{1}$ and $q_{0}$ and they depend only on $Q_{1}$ and $Q_{2}$ . This way one obtains the same Hilbert series as in Example 6.5. The web page [EZ22] contains a report on computations that determined the rational expressions for $H_{n}(t)$ for $n\leqslant 16$ using partial fraction decomposition of $\mu_{n}(t,z)$ as given in equation (12); see also the note [EZ19] for details. This procedure was also implemented by the authors of the present paper and the implementation was used to verify the rational expressions for $H_{n}(t)$ in Examples 6.3, and 6.5–6.7. Using our Magma implementation of the procedure, we also computed the rational expression for $H_{n}(t)$ for $n\leqslant 18$ . The computation for $n=18$ took about 10 minutes on an Intel(R) Core(TM) i5-10210U CPU running at 1.6 GHz.

7. Prime characteristic

Throughout this final section, we let $n\geqslant 1$ fixed, $\mathbb{F}$ denotes a field of characteristic $p$ with $p\geqslant n+1$ , and $\mathfrak{g}=\mathfrak{g}(n+2)$ is the $(n+2)$ -dimensional standard filiform Lie algebra defined over $\mathbb{F}$ . Note that $U(\mathfrak{g})$ can be viewed as $U_{\mathbb{Z}}(\mathfrak{g})\otimes\mathbb{F}$ where $U_{\mathbb{Z}}(\mathfrak{g})$ is the universal enveloping algebra of the standard filiform Lie algebra $\mathfrak{g}_{\mathbb{Z}}(n+2)$ defined over $\mathbb{Z}$ . Since the elements $z_{1},\ldots,z_{n}$ in Section 5.1 are central in $U_{\mathbb{Z}}(\mathfrak{g})$ , they are also central in $U(\mathfrak{g})=U_{\mathbb{Z}}(\mathfrak{g})\otimes\mathbb{F}$ . In characteristic $p$ , however, there are other central elements that do not occur in characteristic zero. Let $Z_{p}(\mathfrak{g})$ denote the $p$ -center of $U(\mathfrak{g})$ ; that is

Z_{p}(\mathfrak{g})=\mathbb{F}[x^{p},y_{0},y_{1}^{p},\ldots,y_{n}^{p}].

The condition that $p\geqslant n+1$ implies that $Z_{p}(\mathfrak{g})$ is isomorphic to the polynomial algebra $\mathbb{F}[\mathbf{t}_{n+2}]$ and also that $Z_{p}(\mathfrak{g})\subseteq Z(\mathfrak{g})$ . Let $K(\mathfrak{g})$ and $K_{p}(\mathfrak{g})$ denote the fraction fields of $Z(\mathfrak{g})$ and $Z_{p}(\mathfrak{g})$ respectively. We also let $D(\mathfrak{g})$ denote the division algebra of fractions of $U(\mathfrak{g})$ . Suppose that $z_{i}$ is the sequence of elements defined in Section 5. The leading coefficient of $z_{i}$ is either $2$ or $i$ by Lemma 5.4, which is nonzero by the assumption that $p\geqslant n+1$ .

In the rest of this section, the elements $z_{1},\ldots,z_{n}$ are the same as defined in Section 5 and recall that $z_{1}=y_{0}\in Z_{p}(\mathfrak{g})$ .

Theorem 7.1.

We have that $K(\mathfrak{g})=K_{p}(\mathfrak{g})(z_{2},\ldots,z_{n})$ and $\dim_{K(\mathfrak{g})}D(\mathfrak{g})=p^{2}$ .

Proof.

Consider the chain

K_{p}(\mathfrak{g})\subset K(\mathfrak{g})\subset D(\mathfrak{g})

of division algebras (where the first two terms are fields). We have that

\dim_{K_{p}(\mathfrak{g})}D(\mathfrak{g})=p^{\dim\mathfrak{g}-\dim C(\mathfrak{g})}=p^{n+1}\quad\mbox{and}\quad\dim_{K(\mathfrak{g})}D(\mathfrak{g})\geqslant p^{2}

(see [dJS22, Theorem 3.2]). Furthermore, since $a^{p}\in K_{p}(\mathfrak{g})$ for all $a\in K(\mathfrak{g})$ , $K(\mathfrak{g})$ is a purely inseparable extension of $K_{p}(\mathfrak{g})$ of dimension $p^{k}$ where $k\leqslant n-1$ . Since $z_{i}\not\in K_{p}(\mathfrak{g})(z_{2},\ldots,z_{i-1})$ but $z_{i}^{p}\in K_{p}(\mathfrak{g})$ for all $i\geqslant 2$ and $K_{p}(\mathfrak{g})(z_{2},\ldots,z_{i})$ is a purely inseparable extension of $K_{p}(\mathfrak{g})(z_{2},\ldots,z_{i-1})$ , it follows that

\dim_{K_{p}(\mathfrak{g})(z_{2},\ldots,z_{i-1})}K_{p}(\mathfrak{g})(z_{2},\ldots,z_{i-1},z_{i})=p.

This implies that $K(\mathfrak{g})=K_{p}(z_{2},\ldots,z_{n})$ and $\dim_{K(\mathfrak{g})}D(\mathfrak{g})=p^{2}$ . ∎

Proposition 7.2.

Suppose that $R_{n}=Z_{p}(\mathfrak{g})[y_{0}^{-1},z_{2},\ldots,z_{n}]$ and, for $i\in\{2,\ldots,n\}$ , set $\alpha_{i}=z_{i}^{p}$ . Then $\alpha_{i}\in Z_{p}(\mathfrak{g})$ and following are valid.

(1)

$R_{n}\cong Z_{p}(\mathfrak{g})[y_{0}^{-1},t_{2},\ldots,t_{n}]/(t_{2}^{p}-\alpha_{2},\ldots,t_{n}^{p}-\alpha_{n})$ .
(2)

$R_{n}$ is a regular ring.

Consequently, $R_{n}$ is a normal domain (that is, integrally closed in its field of fractions).

Proof.

The fact that $\alpha_{i}\in Z_{p}(\mathfrak{g})$ follows from the fact that $\alpha_{i}$ is an expression in the commuting variables $y_{0},y_{1},\ldots,y_{n}$ and that $y_{0}\in Z_{p}(\mathfrak{g})$ and, for all $i\geqslant 1$ , $y_{i}^{p}\in Z_{p}(\mathfrak{g})$ .

We will prove the assertions (1)–(2) of the lemma simultaneously by induction on $n$ . The basis of the induction is the case when $n=1$ and the assertions in this case are valid since $R_{1}=Z_{p}(\mathfrak{g})[y_{0}^{-1}]\cong\mathbb{F}[x^{p},y_{0},y_{0}^{-1}]$ ( $x^{p}$ and $y_{0}$ are algebraically independent) is a regular domain. Since a regular domain is also normal, we obtain that $R_{1}$ is a normal domain.

Suppose that assertions (1) and (2) hold for the algebra $R_{n-1}$ that corresponds to the standard filiform Lie algebra $\mathfrak{g}(n+1)$ . That is,

	$\displaystyle R_{n-1}$	$\displaystyle=\mathbb{F}[y_{0}^{-1},y_{0},y_{1}^{p},\ldots,y_{n}^{p},z_{2},\ldots,z_{n-1}]$
		$\displaystyle\cong Z_{p}(\mathfrak{g}(n+1))[y_{0}^{-1},z_{2},\ldots,z_{n-1}]$
		$\displaystyle\cong Z_{p}(\mathfrak{g}(n+1))[y_{0}^{-1},t_{2},\ldots,t_{n-1}]/(t_{2}^{p}-\alpha_{2},\ldots,t_{n-1}^{p}-\alpha_{n-1}).$

Further, the induction hypothesis also states that $R_{n-1}$ is regular and hence it is normal. Since, $y_{n}^{p}$ is a free variable over $R_{n-1}$ , $R_{n-1}[y_{n}^{p}]$ is normal. Also note that $\alpha_{i}\in R_{n-1}[y_{n}^{p}]\setminus(R_{n-1}[y_{n}^{p}])^{p}$ and $t_{n}^{p}-\alpha_{n}$ is prime in $R_{n-1}[y_{n}^{p}][t_{n}]$ (see [dJS22, Lemma 2.5]). Now since $t_{n}^{p}-\alpha_{n}$ is prime, we obtain that

	$\displaystyle R_{n}$	$\displaystyle=Z_{p}(\mathfrak{g})[y_{0}^{-1},z_{2},\ldots,z_{n}]=R_{n-1}[y_{n}^{p},z_{n}]$
		$\displaystyle\cong R_{n-1}[y_{n}^{p}][t_{n}]/(t_{n}^{p}-\alpha_{n})$
		$\displaystyle\cong Z_{p}(\mathfrak{g})[y_{0}^{-1},t_{2},\ldots,t_{n}]/(t_{2}^{p}-\alpha_{2},\ldots,t_{n}^{p}-\alpha_{n}).$

This proves claim (1).

Let us turn to claim (2). We identify $Z_{p}(\mathfrak{g})[y_{0}^{-1},t_{2},\ldots,t_{n}]$ with the localization $\mathbb{F}[\mathbf{u}_{2n+1}]_{u_{2}}$ at the multiplicative set generated by $u_{2}$ of the polynomial ring $\mathbb{F}[\mathbf{u}_{2n+1}]$ in $2n+1$ variables where $x,y_{0},y_{1}^{p},\ldots,y_{n}^{p}$ are identified with the variables $u_{1},u_{2},\ldots,u_{n+2}$ and $t_{2},\ldots,t_{n}$ are identified with the variables $u_{n+3},\ldots,u_{2n+1}$ , respectively. In particular, $y_{0}$ corresponds to the variable $u_{2}$ and this is why the polynomial ring is localized at $u_{2}$ . Suppose, for $i\in\{2,\ldots,n\}$ that $f_{i}$ is the image of the polynomial $t_{i}^{p}-\alpha_{i}$ in $\mathbb{F}[\mathbf{u}_{2n-2}]_{u_{2}}$ . The argument in the previous paragraph implies that

R_{n}\cong\mathbb{F}[\mathbf{u}_{2n+1}]_{u_{2}}/(f_{2},\ldots,f_{n}).

For $i=\{2,\ldots,n\}$ , let $D_{i}$ denote the derivation $\partial/\partial u_{i+2}$ of $\mathbb{F}[\mathbf{u}_{2n+1}]_{u_{2}}$ . Then we have from Lemma 5.4 that $D_{i}(f_{j})=0$ for all $j<i$ and $D_{i}(f_{i})=c_{i}u_{2}^{k_{i}}$ where $k_{i}$ and $c_{i}$ are positive integers and $c_{i}<p$ . Hence the Jacobian matrix $(D_{i}f_{j})$ is upper triangular and $\det(D_{i}f_{j})=cu_{2}^{k}$ where $c,k\in\mathbb{N}$ and $p\nmid c$ .

Suppose that $\bar{P}\in\mbox{Spec}(R)$ . Then $\bar{P}=P/\left(f_{2},\ldots,f_{n}\right)$ for some prime ideal $P\in\mbox{Spec}(\mathbb{F}[\mathbf{u}_{2n+1}]_{u_{2}})$ such that $u_{2}\not\in P$ and $(f_{2},\ldots,f_{n})\subseteq P$ . Thus $\det(D_{i}f_{j})\not\in P$ , otherwise $u_{2}\in P$ would be true. The Jacobian Criterion for Regularity [Mat89, Theorem 30.4] implies that $\mathbb{F}[\mathbf{u}_{2n+1}]_{P}/\left(f_{2},\ldots,f_{n}\right)_{P}$ is a regular ring; that is $(R_{n})_{\bar{P}}$ is a regular ring. This shows that $R_{n}$ is regular. Since a regular ring is normal, $R_{n}$ is also a normal domain. ∎

The proof of Theorem 1.4.

Set $R=Z_{p}(\mathfrak{g})[z_{2},\ldots,z_{n}]$ and $R[y_{0}^{-1}]=Z_{p}(\mathfrak{g})[y_{0}^{-1},z_{2},\ldots,z_{n}]$ . Theorem 7.1 implies that

K(\mathfrak{g})=K_{p}(\mathfrak{g})(z_{2},\ldots,z_{n})

and so the fraction fields of $R$ , $R[y_{0}^{-1}]$ and $Z(\mathfrak{g})$ coincide. Let us denote this field by $K$ . For domains $A\subseteq B$ , let $A^{B}$ denote the integral closure of $A$ in $B$ . Since $R[y_{0}^{-1}]$ is integrally closed,

R^{K}\subseteq R[y_{0}^{-1}]^{K}=R[y_{0}^{-1}].

This means that the integral closure $R^{K}$ must be contained in $R[y_{0}^{-1}]$ and hence $R^{K}=R^{R[y_{0}^{-1}]}$ . For the other statement, recall that $Z(\mathfrak{g})$ is an integral extension of $R$ in $K$ and $Z(\mathfrak{g})$ is integrally closed in $K$ (by Zassenhaus’ Theorem [Zas54]). Thus

Z(\mathfrak{g})\subseteq R^{K}\subseteq Z(\mathfrak{g})^{K}=Z(\mathfrak{g}).

It follows that equality must hold in each step of the previous chain which implies that $Z(\mathfrak{g})=R^{K}$ as claimed. ∎

References

[AF78] Gert Almkvist and Robert Fossum. Decomposition of exterior and symmetric powers of indecomposable ${\bf Z}/p{\bf Z}$ -modules in characteristic $p$ and relations to invariants. In Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), volume 641 of Lecture Notes in Math., pages 1–111. Springer, Berlin, 1978.
[Alm80a] Gert Almkvist. Invariants, mostly old ones. Pacific J. Math., 86(1):1–13, 1980.
[Alm80b] Gert Almkvist. Reciprocity theorems for representations in characteristic $p$ . In Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, 32ème année (Paris, 1979), volume 795 of Lecture Notes in Math., pages 1–9. Springer, Berlin, 1980.
[BCP97] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).
[Bed05] L. Bedratyuk. Casimir elements and kernel of Weitzenbök derivation. 2005. DOI: 10.48550/arxiv.math/0512520; Preprint: arxiv.org/abs/math/0512520.
[Bed07] Leonid Bedratyuk. On complete system of invariants for the binary form of degree 7. J. Symbolic Comput., 42(10):935–947, 2007.
[Bed08] L. P. Bedratyuk. The kernel of the generalized Weitzenböck derivation of a polynomial ring. Mat. Stud., 29(2):115–120, 2008.
[Bed09] Leonid Bedratyuk. A complete minimal system of covariants for the binary form of degree 7. J. Symbolic Comput., 44(2):211–220, 2009.
[Bed10] L. P. Bedratyuk. Kernels of derivations of polynomial rings and Casimir elements. Ukraïn. Mat. Zh., 62(4):435–452, 2010.
[Bed11a] Leonid Bedratyuk. Bivariate Poincaré series for the algebra of covariants of a binary form. ISRN Algebra, pages Art. ID 312789, 11, 2011.
[Bed11b] Leonid Bedratyuk. Weitzenböck derivations and classical invariant theory, II: The symbolic method. Serdica Math. J., 37(2):87–106, 2011.
[BI15] Leonid Bedratyuk and Nadia Ilash. The degree of the algebra of covariants of a binary form. J. Commut. Algebra, 7(4):459–472, 2015.
[DK15] Harm Derksen and Gregor Kemper. Computational invariant theory, volume 130 of Encyclopaedia of Mathematical Sciences. Springer, Heidelberg, enlarged edition, 2015. Invariant Theory and Algebraic Transformation Groups, VIII.
[Dix58] Jacques Dixmier. Sur les représentations unitaires des groupes de Lie nilpotents. III. Canadian J. Math., 10:321–348, 1958.
[dJS22] Vanderlei Lopes de Jesus and Csaba Schneider. The center of the universal enveloping algebras of small-dimensional nilpotent Lie algebras in prime characteristic. To appear in Beitr. Algebra Geom. DOI: doi.org/10.1007/s13366-022-00631-5; Preprint: arxiv.org/abs/2111.13432, 2022.
[EZ19] Shalosh B. Ekhad and Doron Zeilberger. In how many ways can I carry a total of $n$ coins in my two pockets, and have the same amount in both pockets?, 2019. DOI: 10.48550/arxiv.1901.08172; Preprint: arxiv.org/abs/1901.08172.
[EZ22] Shalosh B. Ekhad and Doron Zeilberger. In how many ways can I carry a total of $n$ coins in my two pockets, and have the same amount in both pockets? Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org; URL: sites.math.rutgers.edu/ zeilberg/mamarim/mamarimhtml/change.html, accessed in August of 2022.
[Fre13] Gene Freudenburg. Foundations of invariant theory for the down operator. J. Symbolic Comput., 57:19–47, 2013.
[Fre17] Gene Freudenburg. Algebraic theory of locally nilpotent derivations, volume 136 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, second edition, 2017. Invariant Theory and Algebraic Transformation Groups, VII.
[Jac79] Nathan Jacobson. Lie algebras. Dover Publications, Inc., New York, 1979. Republication of the 1962 original.
[Kow14] Emmanuel Kowalski. An introduction to the representation theory of groups, volume 155 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2014.
[Mat89] Hideyuki Matsumura. Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid.
[Oom09] Alfons I. Ooms. Computing invariants and semi-invariants by means of Frobenius Lie algebras. J. Algebra, 321(4):1293–1312, 2009.
[Oom12] Alfons I. Ooms. The Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven. J. Algebra, 365:83–113, 2012.
[SI20] Neil J. A. Sloane and The OEIS Foundation Inc. The on-line encyclopedia of integer sequences, 2020. URL: oeis.org.
[ŠW14] Libor Šnobl and Pavel Winternitz. Classification and identification of Lie algebras, volume 33 of CRM Monograph Series. American Mathematical Society, Providence, RI, 2014.
[Zas54] Hans Zassenhaus. The representations of Lie algebras of prime characteristic. Proc. Glasgow Math. Assoc., 2:1–36, 1954.

Appendix A The generating sets of $Z_{5}$ , $Z_{6}$ , $Z_{8}$

We computed a minimal generating set for $Z_{5}$ of this form consisting of the following 23 polynomials.

	$\displaystyle\mbox{degree 1: }z_{1}$	$\displaystyle=y_{0};$
	$\displaystyle\mbox{degree 2: }z_{2}$	$\displaystyle=y_{0}\circ_{2}y_{0};z_{3}=y_{0}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 3: }z_{4}$	$\displaystyle=y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{5}=y_{0}\circ_{4}y_{0}\circ_{1}y_{0};z_{6}=y_{0}\circ_{4}y_{0}\circ_{2}y_{0};$
	$\displaystyle\mbox{degree 4: }z_{7}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{8}=y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{2}y_{0};z_{9}=y_{0}\circ_{4}y_{0}\circ_{1}y_{0}\circ_{5}y_{0};$
	$\displaystyle\mbox{degree 5: }z_{10}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{11}=z_{3}^{2}\circ_{3}y_{0};z_{12}=z_{3}^{2}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 6: }z_{13}$	$\displaystyle=z_{3}^{2}\circ_{4}y_{0}\circ_{1}y_{0};z_{14}=z_{3}^{2}\circ_{3}y_{0}\circ_{3}y_{0};$
	$\displaystyle\mbox{degree 7: }z_{15}$	$\displaystyle=z_{3}^{2}\circ_{3}y_{0}\circ_{3}y_{0}\circ_{1}y_{0};z_{16}=z_{3}^{2}\circ_{4}y_{0}\circ_{1}y_{0}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 8: }z_{17}$	$\displaystyle=z_{3}z_{12}\circ_{3}y_{0};z_{18}=z_{3}z_{11}\circ_{5}y_{0};$
	$\displaystyle\mbox{degree 9: }z_{19}$	$\displaystyle=z_{3}z_{12}\circ_{3}y_{0}\circ_{2}y_{0};$
	$\displaystyle\mbox{degree 11: }z_{20}$	$\displaystyle=z_{3}z_{17}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 12: }z_{21}$	$\displaystyle=z_{3}^{2}z_{16}\circ_{5}y_{0};$
	$\displaystyle\mbox{degree 13: }z_{22}$	$\displaystyle=z_{3}z_{12}^{2}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 18: }z_{23}$	$\displaystyle=z_{3}^{2}z_{22}\circ_{5}y_{0};$

The minimal generating system for $Z_{6}$ consists of the following 26 polynomials:

	$\displaystyle\mbox{degree 1: }z_{1}$	$\displaystyle=y_{0};$
	$\displaystyle\mbox{degree 2: }z_{2}$	$\displaystyle=y_{0}\circ_{2}y_{0};z_{3}=y_{0}\circ_{4}y_{0};z_{4}=y_{0}\circ_{6}y_{0};$
	$\displaystyle\mbox{degree 3: }z_{5}$	$\displaystyle=y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{6}=y_{0}\circ_{4}y_{0}\circ_{1}y_{0};z_{7}=y_{0}\circ_{4}y_{0}\circ_{2}y_{0};z_{8}=y_{0}\circ_{4}y_{0}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 4: }z_{9}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{10}=y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{1}y_{0};z_{11}=y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{12}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{6}y_{0};$
	$\displaystyle\mbox{degree 5: }z_{13}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{1}y_{0}\ z_{14}=y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{3}y_{0};$
	$\displaystyle z_{15}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 6: }z_{16}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{4}y_{0}\circ_{1}y_{0};z_{17}=y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{3}y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{18}$	$\displaystyle=z_{3}z_{8}\circ_{6}y_{0};$
	$\displaystyle\mbox{degree 7: }z_{19}$	$\displaystyle=z_{8}^{2}\circ_{3}y_{0};z_{20}=z_{8}^{2}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 8: }z_{21}$	$\displaystyle=z_{8}^{2}\circ_{3}y_{0}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 9: }z_{22}$	$\displaystyle=z_{8}^{2}\circ_{3}y_{0}\circ_{4}y_{0}\circ_{2}y_{0};$
	$\displaystyle\mbox{degree 10: }z_{23}$	$\displaystyle=z_{8}^{2}\circ_{3}y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{4}y_{0};z_{24}=z_{8}^{3}\circ_{6}y_{0};$
	$\displaystyle\mbox{degree 12: }z_{25}$	$\displaystyle=z_{8}z_{21}\circ_{4}y_{0};$
	$\displaystyle\mbox{degree 15: }z_{26}$	$\displaystyle=z_{8}^{2}z_{21}\circ_{6}y_{0};$

The generators of $Z_{8}$ are the following polynomials.

	$\displaystyle\mbox{Degree 1: }z_{1}$	$\displaystyle=y_{0};$
	$\displaystyle\mbox{Degree 2 :}z_{2}$	$\displaystyle=y_{0}\circ_{2}y_{0};z_{3}=y_{0}\circ_{4}y_{0};z_{4}=y_{0}\circ_{6}y_{0};z_{5}=y_{0}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 3 :}z_{6}$	$\displaystyle=y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{7}=y_{0}\circ_{4}y_{0}\circ_{1}y_{0};z_{8}=y_{0}\circ_{6}y_{0}\circ_{1}y_{0};z_{9}=y_{0}\circ_{4}y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{10}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{2}y_{0};z_{11}=y_{0}\circ_{6}y_{0}\circ_{3}y_{0};z_{12}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0};z_{13}=y_{0}\circ_{4}y_{0}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 4 :}z_{14}$	$\displaystyle=y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{15}=y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{16}=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{1}y_{0};$
	$\displaystyle z_{17}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{1}y_{0};z_{18}=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{2}y_{0};z_{19}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{20}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0};z_{21}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0};z_{22}=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0};$
	$\displaystyle z_{23}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 5 :}z_{24}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{2}y_{0}\circ_{1}y_{0};z_{25}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{1}y_{0};$
	$\displaystyle z_{26}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{1}y_{0};z_{27}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{28}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{2}y_{0};z_{29}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{3}y_{0};$
	$\displaystyle z_{30}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{3}y_{0};z_{31}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0};$
	$\displaystyle z_{32}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0};z_{33}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0};$
	$\displaystyle z_{34}$	$\displaystyle=z_{4}^{2}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 6 :}z_{35}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{1}y_{0};z_{36}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{1}y_{0};$
	$\displaystyle z_{37}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0};z_{38}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{3}y_{0};$
	$\displaystyle z_{39}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{3}y_{0};z_{40}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0};$
	$\displaystyle z_{41}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{4}y_{0};z_{42}=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{3}y_{0}\circ_{6}y_{0};$
	$\displaystyle z_{43}$	$\displaystyle=z_{4}z_{12}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 7 :}z_{44}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0};z_{45}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{3}y_{0};$
	$\displaystyle z_{46}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{3}y_{0};z_{47}=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0};$
	$\displaystyle z_{48}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{5}y_{0};z_{49}=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{6}y_{0};$
	$\displaystyle z_{50}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0};z_{51}=z_{12}^{2}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 8 :}z_{52}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{53}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{6}y_{0}\circ_{2}y_{0};$
	$\displaystyle z_{54}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{5}y_{0}\circ_{4}y_{0};$
	$\displaystyle z_{55}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0}\circ_{4}y_{0};$
	$\displaystyle z_{56}$	$\displaystyle=z_{4}z_{33}\circ_{6}y_{0};z_{57}=y_{0}\circ_{6}y_{0}\circ_{3}y_{0}\circ_{5}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{6}y_{0};z_{58}=z_{4}z_{32}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 9 :}z_{59}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{5}y_{0}\circ_{4}y_{0}\circ_{4}y_{0};$
	$\displaystyle z_{60}$	$\displaystyle=z_{4}z_{42}\circ_{6}y_{0};z_{61}=z_{12}z_{33}\circ_{6}y_{0};$
	$\displaystyle z_{62}$	$\displaystyle=y_{0}\circ_{6}y_{0}\circ_{4}y_{0}\circ_{3}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{6}y_{0}\circ_{2}y_{0}\circ_{6}y_{0};z_{63}=z_{4}z_{41}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 10 :}z_{64}$	$\displaystyle=z_{4}z_{50}\circ_{6}y_{0};z_{65}=z_{4}z_{49}\circ_{6}y_{0};z_{66}=z_{4}z_{48}\circ_{8}y_{0};$
	$\displaystyle\mbox{Degree 11 :}z_{67}$	$\displaystyle=z_{4}z_{56}\circ_{6}y_{0};z_{68}=z_{4}z_{57}\circ_{6}y_{0};$
	$\displaystyle\mbox{Degree 12 :}z_{69}$	$\displaystyle=z_{4}z_{60}\circ_{6}y_{0};$

In all of these computations, we used the information given in [Fre13] concerning the maximum degree of the polynomials in a minimal generating set. The degree-15 polynomial in the generating set for $Z_{6}$ has $1370$ nonzero terms, while the degree-12 polynomial in the generating set of $Z_{8}$ has $3651$ nonzero terms.

The center and invariants of standard filiform Lie algebras

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

2. The standard filiform Lie algebra

Lemma 2.1.

Proof.

Example 2.2.

3. The elements of 𝔽​[𝐲n]x\mathbb{F}[\mathbf{y}_{n}]^{x}

Lemma 3.1.

Proof.

Corollary 3.2.

Lemma 3.3.

Proof.

Theorem 3.4.

Proof.

4. Some facts concerning the representations of 𝔰​𝔩2\mathfrak{sl}_{2}

Theorem 4.1.

Lemma 4.2.

Proof.

Corollary 4.3.

Proof.

Lemma 4.4.

Proof.

Theorem 4.5.

Proof.

Lemma 4.6.

Proof.

Corollary 4.7.

Proof.

5. Explicit generators of ZnZ_{n}

Proposition 5.1.

Corollary 5.2.

Example 5.3.

5.1. The first generator sequence

Lemma 5.4.

Proof.

5.2. The second sequence

Lemma 5.5.

Proof.

Theorem 5.6.

Proof.

6. The Hilbert series of ZnZ_{n}

Theorem 6.1.

Example 6.2.

Example 6.3.

Theorem 6.4.

Proof.

Example 6.5.

Example 6.6.

Example 6.7.

The proof of Theorem 1.3.

6.1. Computation of 𝑯𝒏​(𝒕)\boldsymbol{H_{n}(t)} for higher values of 𝒏\boldsymbol{n}

Theorem 6.8.

Example 6.9.

Example 6.10.

Example 6.11.

7. Prime characteristic

Theorem 7.1.

Proof.

Proposition 7.2.

Proof.

The proof of Theorem 1.4.

References

Appendix A The generating sets of Z5Z_{5}, Z6Z_{6}, Z8Z_{8}

3. The elements of $\mathbb{F}[\mathbf{y}_{n}]^{x}$

4. Some facts concerning the representations of $\mathfrak{sl}_{2}$

5. Explicit generators of $Z_{n}$

6. The Hilbert series of $Z_{n}$

6.1. Computation of $\boldsymbol{H_{n}(t)}$ for higher values of $\boldsymbol{n}$

Appendix A The generating sets of $Z_{5}$ , $Z_{6}$ , $Z_{8}$