# Syzygies for the vector invariants of the dihedral group

Mátyás Domokos  Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15,
1053 Budapest, Hungary, ORCID iD: https://orcid.org/0000-0002-0189-8831
<EMAIL_ADDRESS>
###### Abstract.

The problem of finding generators of the $GL$-ideal of the relations between
the generators of the algebra of invariants of the dihedral group acting on
$m$-tuples of vectors from its defining $2$-dimensional representation is
studied. It is shown that this $GL$-ideal is generated by relations depending
on no more than $3$ vector variables. A minimal $GL$-ideal generating system
is found for the case when $m=2$, and for the case of the dihedral group of
order $8$ and arbitrary $m$.

###### Key words and phrases:

$GL$-ideal, vector invariants, ideal of relations, dihedral group

###### 2010 Mathematics Subject Classification:

Primary 13A50; Secondary 14L30, 20G05

Partially supported by the Hungarian National Research, Development and
Innovation Office, NKFIH K 138828, K 132002.

##  1\. Introduction

Given a group $G$ of linear transformations on a finite dimensional complex
vector space $V$ and a positive integer $m$, consider the diagonal action of
$G$ on the space $V^{m}$ of $m$-tuples of vectors from $V$. We call the
algebra $R(m):=\mathbb{C}[V^{m}]^{G}$ of polynomial functions on $V^{m}$
constant along the $G$-orbits the _algebra of vector invariants of_ $G$.
Following Weyl [17] who gave a systematic study of the case when $G$ is one of
the classical subgroups of the general linear group $GL(V)$, a description of
the generators of $R(m)$ is referred to as a _First Fundamental Theorem_ ,
whereas a description of the generators of the ideal of relations between the
generators of $R(m)$ is referred to as a _Second Fundamental Theorem_ for the
vector invariants of $G$. An indispensable tool to establish such theorems is
to take into account a natural right action of the group $GL_{m}(\mathbb{C})$
of invertible $m\times m$ matrices on $V^{m}$ that commutes with the
$G$-action: for $g=(g_{ij})_{i,j=1}^{m}\in GL_{m}(\mathbb{C})$ and
$v=(v_{1},\dots,v_{m})\in V^{m}$ we set

$v\cdot g=(\sum_{i=1}^{m}g_{i1}v_{i},\dots,\sum_{i=1}^{m}g_{im}v_{i}).$

This induces a left action of $GL_{m}(\mathbb{C})$ on $\mathbb{C}[V^{m}]$ via
$\mathbb{C}$-algebra automorphisms. Namely, for $f\in\mathbb{C}[V^{m}]$, $v\in
V^{m}$ and $g\in GL_{m}(\mathbb{C})$ we have $(g\cdot f)(v)=f(v\cdot g)$. The
subalgebra $R(m)$ is a $GL_{m}(\mathbb{C})$-invariant subspace. Moreover,
$R(m)$ is a graded subalgebra of $\mathbb{C}[V^{m}]$, where the latter is
endowed with the standard grading. Write $R(m)_{+}$ for the maximal ideal of
$R(m)$ spanned by its homogeneous elements of positive degree. It is clearly a
$GL_{m}(\mathbb{C})$-submodule, just like $(R(m)_{+})^{2}$. Let $W(m)$ be a
$GL_{m}(\mathbb{C})$-module direct complement of $(R(m)_{+})^{2}$ in
$R(m)_{+}$. Then the subspace $W(m)$ minimally generates the algebra $R(m)$.
For example, for the case when $G=GL_{3}(\mathbb{C})$ acting on the space
$V=\mathbb{C}^{3\times 3}$ of $3\times 3$ matrices by conjugation, an explicit
description of a minimal homogeneous generating system of $R(m)$ for general
$m$ is rather complicated. This was made transparent in [1] by determining the
$GL_{m}(\mathbb{C})$-module structure of $W(m)$. Let us turn next to the
relations between the generators of $R(m)$. The identity map $W(m)\to W(m)$
induces a $\mathbb{C}$-algebra surjection

(1) $\varphi(m):S(W(m))\to R(m)$

from the symmetric tensor algebra $S(W(m))$ of $W(m)$ onto $R(m)$, and
$\ker(\varphi(m))$ is the _ideal of relations for the minimal generating
subspace_ $W(m)$ of $R(m)$. Clearly, $\ker(\varphi(m))$ is a _$GL$ -ideal_,
i.e. a $GL_{m}(\mathbb{C})$-stable ideal in $S(W(m))$. A sensible way to
describe $\ker(\varphi(m))$ is to find elements in it which generate
irreducible $GL_{m}(\mathbb{C})$-submodules of $S(W(m))$, such that the sum of
these $GL_{m}(\mathbb{C})$-submodules is direct and forms a minimal generating
subspace of the ideal $\ker(\varphi(m))$ (so in particular, these elements
constitute a minimal generating system of $\ker(\varphi(m))$ as a $GL$-ideal).
For example, in the special case $G=GL_{3}(\mathbb{C})$ and
$V=\mathbb{C}^{3\times 3}$, the $GL_{m}(\mathbb{C})$-module structure of the
minimal degree non-zero homogeneous component of $\ker(\varphi(m))$ was
computed in [3].

In this paper we study the problem of finding generators of the $GL$-ideal
$\ker(\varphi(m))$ for the defining $2$-dimensional representation of the
dihedral group $D_{2n}$ of order $2n$ ($n\geq 3$ is a positive integer). For
the case $n=3$ the result can be easily deduced from [10], see Theorem 6.9.
For arbitrary $n$ and $m=2$ the solution will be given in Theorem 5.1.
Moreover, for arbitrary $n$ we reduce the problem to the case $m=3$ (see
Theorem 4.8). As an application of this reduction we give a complete solution
of the problem when $n=4$ in Theorem 6.8.

##  2\. First Fundamental Theorem for the dihedral group

Let $n\geq 3$ be a positive integer and $\omega$ a complex primitive $n$th
root of $1$. We take for $G$ the dihedral group $D_{2n}$, the subgroup of
$GL_{2}(\mathbb{C})$ generated by the matrices
$\left(\begin{array}[]{cc}\omega&0\\\ 0&\omega^{-1}\end{array}\right)$ and
$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$. The given
representation of $D_{2n}$ is the complexification of the defining real
representation of $D_{2n}$ as the group of isometries of the Euclidean plane
mapping a given regular $n$-gon into itself. Denoting by $x,y$ the coordinate
functions on $V:=\mathbb{C}^{2}$, the algebra $\mathbb{C}[x,y]^{D_{2n}}$ of
polynomial invariants is generated by the algebraically independent invariants

$q:=xy,\qquad p:=x^{n}+y^{n}.$

Consider now the diagonal action of $D_{2n}$ on the space
$V^{m}:=V\oplus\cdots\oplus V$ ($m$ direct summands) of $m$-tuples of vectors.
Denote by $x_{i}$ (respectively $y_{i}$) the function mapping an $m$-tuple of
vectors to the first (respectively second) coordinate of its $i$th vector
component, $i=1,\dots,m$. The coordinate ring $\mathbb{C}[V^{m}]$ is the
$2m$-variable polynomial ring $\mathbb{C}[x_{1},y_{1},\dots,x_{m},y_{m}]$. For
a degree $d$ homogeneous element $g\in\mathbb{C}[x,y]$ and
$\alpha=(\alpha_{1},\dots,\alpha_{m})$ with $\sum_{j}\alpha_{j}=d$ denote by
$g_{\alpha}$ the multihomogeneous component of
$\frac{1}{\binom{d}{\alpha}}g(x_{1}+\cdots+x_{m},y_{1}+\cdots+y_{m})$ having
total degree $\alpha_{j}$ in $x_{j},y_{j}$ for $j=1,\dots,m$; so

$g(x_{1}+\cdots+x_{m},y_{1}+\cdots+y_{m})=\sum_{\alpha}\binom{d}{\alpha}g_{\alpha},$

where $\binom{d}{\alpha}=\frac{d!}{\alpha_{1}!\cdots\alpha_{m}!}$. The
elements $g_{\alpha}\in\mathbb{C}[V^{m}]$ are called the _polarizations_ of
$g$, and they form a $\mathbb{C}$-vector space basis in the
$GL_{m}(\mathbb{C})$-submodule $\langle g\rangle_{GL_{m}(\mathbb{C})}$ of
$\mathbb{C}[V^{m}]$ generated by $g$. Here we treat $\mathbb{C}[V]$ as the
subalgebra of $\mathbb{C}[V^{m}]$ consisting of the polynomial functions that
depend only on the first vector component of the $m$-tuples in $V^{m}$, and
throughout the paper we write $\langle A\rangle_{GL_{m}(\mathbb{C})}$ for the
$GL_{m}(\mathbb{C})$-submodule generated by the element or subset $A$ of a
given $GL_{m}(\mathbb{C})$-module.

For example, the polarizations of $p\in\mathbb{C}[x,y]^{D_{2n}}$ are

$p_{\alpha}=x_{1}^{\alpha_{1}}\cdots
x_{m}^{\alpha_{m}}+y_{1}^{\alpha_{1}}\cdots y_{m}^{\alpha_{m}}\text{ for
}\alpha\in\mathbb{N}_{0}^{m}\text{ with }\sum\alpha_{i}=n$

and the polarizations of $q\in\mathbb{C}[x,y]^{D_{2n}}$ for $m=2$ are

$q_{2,0}=x_{1}y_{1},\quad q_{11}=\frac{1}{2}(x_{1}y_{2}+y_{1}x_{2}),\quad
q_{0,2}=x_{2}y_{2}.$

Our starting point is the following known description of the generators of
$\mathbb{C}[V^{m}]^{D_{2n}}$ due to Hunziker:

######  Theorem 2.1.

[12, Theorem 4.1] The polarizations of $q$ and $p$ form a minimal homogeneous
generating system of the $\mathbb{C}$-algebra $\mathbb{C}[V^{m}]^{D_{2n}}$.

######  Remark 2.2.

In fact [12] works over the real field, but by well known basic principles
[12, Theorem 4.1] implies the variant Theorem 2.1 above.

Therefore in this case $W(m)$ (cf. Section 1) is

$W(m)=\langle q\rangle_{GL_{m}(\mathbb{C})}\oplus\langle
p\rangle_{GL_{m}(\mathbb{C})}=\mathrm{Span}_{\mathbb{C}}\displaystyle\\{q_{\alpha},p_{\beta}\mid\alpha,\beta\in\mathbb{N}_{0}^{m},\
\sum_{i=1}^{m}\alpha_{i}=2,\ \sum_{j=1}^{m}\beta_{j}=n\\}.$

We have the $GL_{m}(\mathbb{C})$-module isomorphisms

$\langle q\rangle_{GL_{m}(\mathbb{C})}\cong S^{2}(\mathbb{C}^{m})\quad\text{
and }\quad\langle p\rangle_{GL_{m}(\mathbb{C})}\cong S^{n}(\mathbb{C}^{m})$

where $S^{k}(U)$ stands for the $k$th symmetric tensor power of the
$GL(U)$-module $U$.

Write $\mathcal{F}(n,m):=S(W(m))=\sum_{d=0}^{\infty}S^{d}(W(m))$ for the
symmetric tensor algebra of $W(m)$ (endowed with the natural
$GL_{m}(\mathbb{C})$-module structure induced by the
$GL_{m}(\mathbb{C})$-action on $W(m)$). Although
$S^{1}(W(m))\subset\mathcal{F}(n,m)$ is just $W(m)$, to avoid confusion later
we need to distinguish in the notation the elements of $W(m)$ when they are
considered as elements of $\mathcal{F}(n,m)$. We shall write
$\rho_{\alpha}\in\mathcal{F}(n,m)$ for the element corresponding to
$q_{\alpha}$, and $\pi_{\beta}\in\mathcal{F}(n,m)$ for the element
corresponding to $p_{\beta}$. So $\mathcal{F}(n,m)$ is the polynomial ring

$\mathcal{F}(n,m)=\mathbb{C}[\rho_{\alpha},\pi_{\beta}\mid\sum\alpha_{i}=2,\
\sum\beta_{j}=n].$

Thus in our case $\varphi(m)$ (cf. (1) in Section 1) is the
$\mathbb{C}$-algebra (as well as $GL_{m}(\mathbb{C})$-module) surjection

$\varphi(n,m):\mathcal{F}(n,m)\to\mathbb{C}[V^{m}]^{D_{2n}},\qquad\rho_{\alpha}\mapsto
q_{\alpha},\ \pi_{\beta}\mapsto p_{\beta}.$

Throughout the paper for $l\leq m$ we shall treat $\mathbb{C}[V^{l}]$ and
$\mathcal{F}(n,l)$ as a subalgebra of $\mathbb{C}[V^{m}]$ and
$\mathcal{F}(n,m)$ in the obvious way. Our aim is to describe the kernel
$\ker(\varphi(n,m))$ of $\varphi(n,m)$ as a $GL$-ideal.

##  3\. Preliminaries on $GL_{m}(\mathbb{C})$-modules

As a general reference to the material in this section, see for example the
book [14]. Recall that the isomorphism classes of the irreducible polynomial
$GL_{m}(\mathbb{C})$-modules of degree $d$ are labeled by partitions of $d$
with $m$ parts, where by a partition $\lambda$ of $d$ with $m$ parts
(notation: $\lambda\in\mathrm{Par}_{m}(d)$) we mean a sequence
$(\lambda_{1},\dots,\lambda_{m})$ of non-negative integers with
$\lambda_{1}\geq\cdots\geq\lambda_{m}$ and $\lambda_{1}+\cdots+\lambda_{m}=d$.
We denote by $\mathrm{ht}(\lambda)$ the number of non-zero elements in the
sequence $(\lambda_{1},\dots,\lambda_{m})$. Moreover, for $l<m$ and non-
negative integers $\lambda_{1}\geq\cdots\geq\lambda_{l}$ with
$\sum\lambda_{i}=d$ we identify the sequence $(\lambda_{1},\dots,\lambda_{l})$
with $(\lambda_{1},\dots,\lambda_{l},0,\dots,0)\in\mathrm{Par}_{m}(d)$.

For an $\ell$-dimensional vector space $U$ write $S^{\lambda}(U)$ for the
$GL(U)$-module associated to $U$ by the Schur functor $S^{\lambda}(-)$. This
is an irreducible polynomial $GL(U)$-module if $\mathrm{ht}(\lambda)\leq\ell$
and $S^{\lambda}(U)$ is the zero module when $\mathrm{ht}(\lambda)>\ell$. For
example, for the partition $\lambda=(n,0,\dots)$ with only one non-zero part
we have $S^{\lambda}(U)=S^{n}(U)$, the $n$th symmetric tensor power of $U$.
Moreover, for $\lambda\in\mathrm{Par}_{m}(d)$, $S^{\lambda}(\mathbb{C}^{m})$
is the irreducible polynomial $GL_{m}(\mathbb{C})$-module labeled by
$\lambda$. Note that when $U$ has a $GL_{m}(\mathbb{C})$-module structure ($m$
may differ from $\ell$), then $S^{\lambda}(U)$ becomes naturally a (typically
not irreducible) $GL_{m}(\mathbb{C})$-module.

$\mathcal{F}(n,m)$ and $\mathbb{C}[V^{m}]$ are polynomial
$GL_{m}(\mathbb{C})$-modules. An element $v$ of a polynomial
$GL_{m}(\mathbb{C})$-module generates a submodule isomorphic to
$S^{\lambda}(\mathbb{C}^{m})$ if

  1. (1)

it is fixed by $UT_{m}(\mathbb{C})$, the subgroup of upper triangular
unipotent matrices;

  2. (2)

for a diagonal element $\mathrm{diag}(z_{1},\dots,z_{m})\in
GL_{m}(\mathbb{C})$ we have $\mathrm{diag}(z_{1},\dots,z_{m})\cdot
v=z_{1}^{\lambda_{1}}\cdots z_{m}^{\lambda_{m}}v$.

Such an element is called a _highest weight vector_. An irreducible polynomial
$GL_{m}(\mathbb{C})$-module contains a unique (up to non-zero scalar
multiples) highest weight vector.

The action of $GL_{m}(\mathbb{C})$ on $\mathcal{F}(n,m)$ and on
$\mathbb{C}[V^{m}]$ induces a representation of its Lie algebra
$\mathfrak{gl}_{m}(\mathbb{C})$ on $\mathcal{F}(n,m)$ and on
$\mathbb{C}[V^{m}]$, such that $\varphi(n,m)$ is a homomorphism of
$\mathfrak{gl}_{m}(\mathbb{C})$-modules. In particular, $\ker(\varphi(n,m))$
is preserved by $\mathfrak{gl}_{m}(\mathbb{C})$. To detect explicit highest
weight vectors it is convenient to pass to the Lie algebra action. Therefore
for later use we shall record the formulae determining this
$\mathfrak{gl}_{m}(\mathbb{C})$-representation on $\mathcal{F}(n,m)$ in the
case $m=3$. Denote by $E_{i,j}$ the matrix unit having entry $1$ in the
$(i,j)$ position and the entry $0$ in all other positions. The Lie algebra
$\mathfrak{gl}_{m}(\mathbb{C})$ of $GL_{m}(\mathbb{C})$ has basis
$\\{E_{i,j}\mid 1\leq i,j\leq m\\}$. We have

$\displaystyle E_{1,2}.\pi_{i,j,k}=j\pi_{i+1,j-1,k},\quad
E_{2,3}.\pi_{i,j,k}=k\pi_{i,j+1,k-1},\quad
E_{ss}.\pi_{\alpha_{1},\alpha_{2},\alpha_{3}}=\alpha_{s}\pi_{\alpha}$
$\displaystyle E_{2,1}.\pi_{i,j,k}=i\pi_{i-1,j+1,k},\qquad
E_{3,2}.\pi_{i,j,k}=j\pi_{i,j-1,k+1}$ $\displaystyle
E_{1,2}.\rho_{i,j,k}=j\rho_{i+1,j-1,k},\quad
E_{2,3}.\rho_{i,j,k}=k\rho_{i,j+1,k-1},\quad
E_{ss}.\rho_{\alpha_{1},\alpha_{2},\alpha_{3}}=\alpha_{s}\rho_{\alpha}$
$\displaystyle E_{2,1}.\rho_{i,j,k}=i\rho_{i-1,j+1,k},\qquad
E_{3,2}.\rho_{i,j,k}=j\rho_{i,j-1,k+1}$

where in the above formulae $\pi_{a,b,c}$ or $\rho_{a,b,c}$ is interpreted as
zero unless $a,b,c$ are all non-negative.

##  4\. Reduction to the case $m=3$

The algebra $\mathbb{C}[V^{m}]^{D_{2n}}$ is a graded subalgebra of
$\mathbb{C}[V^{m}]$, where the latter is endowed with the standard grading. We
introduce a grading on $\mathcal{F}(n,m)$ by making the homomorphism
$\varphi(n,m)$ degree preserving; that is, the degree of the variables
$\rho_{\alpha}$ is $2$, and the degree of the variables $\pi_{\beta}$ is $n$.
We shall denote by $\mathcal{F}(n,m)_{d}$ and by $\ker(\varphi(n,m))_{d}$ the
degree $d$ homogeneous component of $\mathcal{F}(n,m)$ and its graded subspace
$\ker(\varphi(n,m))$, and we write $\mathcal{F}(n,m)_{\leq d}$ and
$\ker(\varphi(n,m))_{\leq d}$ for the sum of the homogeneous components of
degree at most $d$ in $\mathcal{F}(n,m)$ and $\ker(\varphi(n,m))$.

######  Proposition 4.1.

The ideal $\ker(\varphi(n,m))$ is generated by its elements of degree at most
$2n+2$.

###### Proof.

By a general result of Derksen [8, Theorem 2] the ideal $\ker(\varphi(n,m))$
is generated by its elements of degree at most $2\tau_{D_{2n}}(V^{m})$, where
$\tau_{D_{2n}}(V^{m})$ denotes the minimal positive integer $k$ such that all
homogeneous polynomials in $\mathbb{C}[V^{m}]$ of degree $k$ are contained in
the ideal $\mathbb{C}[V^{m}]^{D_{2n}}_{+}\mathbb{C}[V^{m}]$ of
$\mathbb{C}[V^{m}]$ generated by the homogeneous $D_{2n}$-invariants of
positive degree (called the Hilbert ideal). The number $\tau_{D_{2n}}(V^{m})$
is bounded by the Noether number of $D_{2n}$ by [7, Lemma 1.6]. The Noether
number of $D_{2n}$ is $n+1$ by [15] (see also [6, Corollary 5.6] for a
stronger statement). This implies $\tau_{D_{2n}}(V^{m})\leq n+1$. Note finally
that obviously $\tau_{D_{2n}}(V^{m})\geq\tau_{D_{2n}}(V)$, and the latter
number is known to be $n+1$: indeed, $\mathbb{C}[V]^{D_{2n}}$ is generated by
algebraically independent homogeneous invariants of degree $2$ and $n$, and
thus the Hilbert series of the corresponding coinvariant algebra
$\mathbb{C}[V^{m}]/\mathbb{C}[V^{m}]^{D_{2n}}_{+}\mathbb{C}[V^{m}]$ equals
$(1+t)(1+t+\cdots+t^{n-1})$ (see [5]). ∎

Note that $\mathcal{F}(n,m)$ has the tensor product decomposition
$\mathcal{F}(n,m)=\mathcal{D}(m)\otimes\mathcal{E}(n,m)$ where

$\mathcal{D}(m)=S(\langle
q\rangle_{GL_{m}(\mathbb{C})})=\mathbb{C}[\rho_{\alpha}\mid\alpha\in\mathbb{N}_{0}^{m},\
\sum\alpha_{i}=2]$

and

$\mathcal{E}(n,m)=S(\langle
p\rangle_{GL_{m}(\mathbb{C})})=\mathbb{C}[\pi_{\beta}\mid\beta\in\mathbb{N}_{0}^{m},\
\sum\beta_{j}=n]$

For $m\geq 3$ set

$\mathcal{R}_{2,2,2}:=\det\left(\begin{array}[]{ccc}\rho_{2,0,0}&\rho_{1,1,0}&\rho_{1,0,1}\\\
\rho_{1,1,0}&\rho_{0,2,0}&\rho_{0,1,1}\\\
\rho_{1,0,1}&\rho_{0,1,1}&\rho_{0,0,2}\end{array}\right)\in\mathcal{D}(m)\subset\mathcal{F}(n,m).$

######  Lemma 4.2.

For $m\geq 3$ the element $\mathcal{R}_{2,2,2}$ belongs to
$\ker(\varphi(n,m))$, and
$\langle\mathcal{R}_{2,2,2}\rangle_{GL_{m}(\mathbb{C})}\cong
S^{(2,2,2)}(\mathbb{C}^{m})$ as $GL_{m}(\mathbb{C})$-modules.

###### Proof.

The matrix

$\left(\begin{array}[]{ccc}q_{2,0,0}&q_{1,1,0}&q_{1,0,1}\\\
q_{1,1,0}&q_{0,2,0}&q_{0,1,1}\\\
q_{1,0,1}&q_{0,1,1}&q_{0,0,2}\end{array}\right)=\left(\begin{array}[]{cc}x_{1}&y_{1}\\\
x_{2}&y_{2}\\\
x_{3}&y_{3}\end{array}\right)\cdot\left(\begin{array}[]{cc}0&1/2\\\
1/2&0\end{array}\right)\cdot\left(\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\\\
y_{1}&y_{2}&y_{3}\end{array}\right)$

has rank $2$, implying that its determinant is zero. On the other hand, this
determinant is $\varphi(n,m)(\mathcal{R}_{2,2,2})$. Thus
$\mathcal{R}_{2,2,2}\in\ker(\varphi(n,m))$. A straightforward calculation
yields $E_{1,2}.\mathcal{R}_{2,2,2}=0$, $E_{2,3}.\mathcal{R}_{2,2,2}=0$, and
$E_{i,i}.\mathcal{R}_{2,2,2}=2\mathcal{R}_{2,2,2}$ ($i=1,2,3)$, hence
$\mathcal{R}_{2,2,2}$ is a highest weigh vector in $\mathcal{F}(n,m)$ with
weight $(2,2,2)$. ∎

######  Remark 4.3.

In fact it is known that $\mathcal{R}_{2,2,2}$ generates as a $GL$-ideal the
kernel of the restriction of $\varphi(n,m)$ to $\mathcal{D}(m)$ (see the
Second Fundamental Theorem for the orthogonal group in [17, Theorem 2.17.A]).

Setting
$\bar{\mathcal{D}}(m):=\mathcal{D}(m)/\langle\mathcal{R}_{2,2,2}\rangle_{GL_{m}(\mathbb{C})}\mathcal{D}(m)$,
by Lemma 4.2 we conclude that $\varphi(n,m)$ factors through the natural
surjection $\mathcal{F}(n,m)\to\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$,
so we get the graded $GL_{m}(\mathbb{C})$-module algebra surjection

$\bar{\varphi}(n,m):\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)\to\mathbb{C}[V^{m}]^{D_{2n}}.$

######  Lemma 4.4.

For $m\geq 3$ the $GL$-ideal $\ker(\varphi(n,m))$ is generated by
$\mathcal{R}_{2,2,2}$ and any subset of $\ker(\varphi(n,m))$ whose image under
the natural surjection
$\mathcal{F}(n,m)\to\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$ generates
$\ker(\bar{\varphi}(n,m))$ as a $GL$-ideal.

###### Proof.

This is an immediate consequence of the construction of $\bar{\varphi}(n,m)$.
∎

Our next aim is to compute the $GL_{m}(\mathbb{C})$-module structure of the
homogeneous components of $\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$ up to
degree $2n+2$.

######  Proposition 4.5.

We have the following isomorphisms of $GL_{m}(\mathbb{C})$-modules:

  * (i)

$\mathcal{D}(m)\cong\sum_{d=0}^{\infty}\sum_{\lambda\in\mathrm{Par}_{m}(d)}S^{2\lambda}(\mathbb{C}^{m})$
where $2\lambda$ stands for the partition $(2\lambda_{1},\dots,2\lambda_{m})$
of $2d$.

  * (ii)

$\bar{\mathcal{D}}(m)\cong\sum_{d=0}^{\infty}\sum_{\lambda\in\mathrm{Par}_{m}(d),\
\mathrm{ht}(\lambda)\leq 2}S^{2\lambda}(\mathbb{C}^{m})$.

  * (iii)

$\mathcal{E}(n,m)\cong
S(S^{n}(\mathbb{C}^{m}))=\bigoplus_{d=0}^{\infty}S^{d}(S^{n}(\mathbb{C}^{m}))$
(the symmetric tensor algebra of $S^{n}(\mathbb{C}^{m})$.

  * (iv)

$S^{2}(S^{n}(\mathbb{C}^{m}))\cong\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}S^{(2n-2j,2j)}(\mathbb{C}^{m})$.

###### Proof.

(i) We have $\langle q\rangle_{GL_{m}(\mathbb{C})}\cong S^{2}(\mathbb{C}^{m})$
and therefore (i) follows from the well-known decomposition

$S^{d}(S^{2}(\mathbb{C}^{m}))\cong\sum_{\lambda\in\mathrm{Par}_{m}(d)}S^{2\lambda}(\mathbb{C}^{m})$

(see for example [14, Section 11.4.5, Theorem]).

(ii) $\bar{\mathcal{D}}(m)$ can be identified with the coordinate ring of the
variety of $m\times m$ symmetric matrices of rank at most $2$ endowed with the
natural $GL_{m}(\mathbb{C})$-action. For the well-known decomposition of this
$GL_{m}(\mathbb{C})$-module see for example [14, Section 11.5.1, Second
Fundamental Theorem].

(iii) follows from the isomorphism $\langle p\rangle_{GL_{m}(\mathbb{C})}\cong
S^{n}(\mathbb{C}^{m})$.

(iv) can be found for example in [13, Example I.8.9] (see also Remark 5.3 and
the proof of Theorem 5.1 (i)). ∎

Write $\bar{\mathcal{D}}(m)_{d}$, $\mathcal{E}(n,m)_{d}$ for the degree $d$
homogeneous component of $\bar{\mathcal{D}}(m)$, $\mathcal{E}(n,m)$, and write
$(\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m))_{\leq d}$ for the sum of the
homogeneous components of $\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$ of
degree at most $d$.

######  Corollary 4.6.

As a $GL_{m}(\mathbb{C})$-module,
$(\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m))_{\leq 2n+2}$ is isomorphic to

$\displaystyle\sum_{d=0}^{n+1}\sum_{\lambda\in\mathrm{Par}_{m}(d),\
\mathrm{ht}(\lambda)\leq
2}S^{2\lambda}(\mathbb{C}^{m})+\sum_{d=0}^{\lfloor\frac{n+2}{2}\rfloor}\sum_{\lambda\in\mathrm{Par}_{m}(d),\
\mathrm{ht}(\lambda)\leq 2}S^{n}(\mathbb{C}^{m})\otimes
S^{2\lambda}(\mathbb{C}^{m})$
$\displaystyle+\sum_{j=0}^{\lfloor\frac{n}{2}\rfloor}(S^{(2n-2j,2j)}(\mathbb{C}^{m})+S^{(2n-2j,2j)}(\mathbb{C}^{m})\otimes
S^{2}(\mathbb{C}^{m})).$

###### Proof.

As $\mathcal{E}(n,m)_{d}$ is non-zero only if $d$ is a multiple of $n$, the
sum of the homogeneous components of
$\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$ of degree at most $2n+2$ is

$\sum_{d=0}^{2n+2}\bar{\mathcal{D}}(m)_{d}\otimes\mathcal{E}(n,m)_{0}+\sum_{d=0}^{n+2}\bar{\mathcal{D}}(m)_{d}\otimes\mathcal{E}(n,m)_{n}+\sum_{d=0}^{2}\bar{\mathcal{D}}(m)_{d}\otimes\mathcal{E}(n,m)_{2n}.$

Now the statement follows from the $GL_{m}(\mathbb{C})$-module isomorphisms
given in Proposition 4.5. ∎

Recall that for $l\leq m$ we view $\mathcal{F}(n,l)$ as a subspace of
$\mathcal{F}(n,m)$ in the obvious way. Similarly, $\mathcal{D}(l)$,
$\bar{\mathcal{D}}(l)$, $\mathcal{E}(n,l)$, are viewed as subspaces of
$\mathcal{D}(m)$, $\bar{\mathcal{D}}(m)$, $\mathcal{E}(n,m)$, and so
$\bar{\mathcal{D}}(l)\otimes\mathcal{E}(n,l)$ is viewed as a subspace of
$\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$.

######  Proposition 4.7.

Let $U$ be a $GL_{m}(\mathbb{C})$-invariant subspace of
$(\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m))_{\leq 2n+2}$. Then

$U=\langle
U\cap\bar{\mathcal{D}}(3)\otimes\mathcal{E}(n,3)\rangle_{GL_{m}(\mathbb{C})}.$

###### Proof.

By complete reducibility of $GL_{m}(\mathbb{C})$-modules, it is sufficient to
prove the statement in the special case when $U$ is a _minimal_
$GL_{m}(\mathbb{C})$-invariant subspace of
$(\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m))_{\leq 2n+2}$. So assume that
$U$ is a minimal $GL_{m}(\mathbb{C})$-invariant subspace. Pieri’s Formula (cf.
[13, I.5.16]) implies that if $S^{\nu}(\mathbb{C}^{m})$ occurs as a summand in
$S^{\lambda}(\mathbb{C}^{m})\otimes S^{\mu}(\mathbb{C}^{m})$ then
$\mathrm{ht}(\nu)\leq\mathrm{ht}(\lambda)+\mathrm{ht}(\mu)$. Thus by Corollary
4.6 we conclude that $U\cong S^{\lambda}(\mathbb{C}^{m})$ for some partition
$\lambda$ with $\mathrm{ht}(\lambda)\leq 3$. It follows that $U=\langle
w\rangle_{GL_{m}(\mathbb{C})}$ for some highest weight vector $w$ with weight
$\lambda$. That is, $w$ is a non-zero element of $U$ fixed by the subgroup of
unipotent upper triangular matrices in $GL_{m}(\mathbb{C})$, and for a
diagonal element $\mathrm{diag}(z_{1},\dots,z_{m})\in GL_{m}(\mathbb{C})$ we
have $\mathrm{diag}(z_{1},\dots,z_{m})\cdot w=z_{1}^{\lambda_{1}}\cdots
z_{m}^{\lambda_{m}}w$. In particular, $\mathrm{ht}(\lambda)\leq 3$ implies
that $w$ belongs to $\bar{\mathcal{D}}(3)\otimes\mathcal{E}(n,3)$. ∎

We arrived at the main result of this Section:

######  Theorem 4.8.

For $m\geq 3$ the ideal $\ker(\varphi(n,m))$ of $\mathcal{F}(n,m)$ is
generated by the $GL_{m}(\mathbb{C})$-submodule
$\langle\ker(\varphi(n,3))_{\leq 2n+2}\rangle_{GL_{m}(\mathbb{C})}$ of
$\mathcal{F}(n,m)$. That is, $\ker(\varphi(n,m))$ is generated as a $GL$-ideal
by elements of degree $\leq 2n+2$ contained in the kernel of $\varphi(n,3)$.

###### Proof.

The natural surjection
$\mathcal{F}(n,m)\to\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$ maps
$\ker(\varphi(n,m))$ onto $\ker(\bar{\varphi}(n,m))$, therefore by Proposition
4.1 the latter is generated as an ideal by $\ker(\bar{\varphi}(n,m))_{\leq
2n+2}$. Now $\ker(\bar{\varphi}(n,m))_{\leq 2n+2}$ is a
$GL_{m}(\mathbb{C})$-invariant subspace of
$(\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m))_{\leq 2n+2}$, hence by
Proposition 4.7 we have

$\ker(\bar{\varphi}(n,m))_{\leq 2n+2}=\langle\ker(\bar{\varphi}(n,m))_{\leq
2n+2}\cap\bar{\mathcal{D}}(3)\otimes\mathcal{E}(n,3)\rangle_{GL_{m}(\mathbb{C})}=\langle\ker(\bar{\varphi}(n,3))_{\leq
2n+2}\rangle_{GL_{m}(\mathbb{C})}.$

So the image $\ker(\bar{\varphi}(n,3))_{\leq 2n+2}$ of
$\ker(\varphi(n,3))_{\leq 2n+2}$ under the natural surjection
$\mathcal{F}(n,m)\to\bar{\mathcal{D}}(m)\otimes\mathcal{E}(n,m)$ generates
$\ker(\bar{\varphi}(n,m))$ as a $GL$-ideal. Since the element
$\mathcal{R}_{2,2,2}$ belongs to $\ker(\varphi(n,3))_{\leq 2n+2}$, we can
conclude by Lemma 4.4 that $\ker(\varphi(n,3))_{\leq 2n+2}$ generates
$\ker(\varphi(n,m))$ as a $GL$-ideal. Equivalently, $\ker(\varphi(n,m))$ is
generated as an ideal of $\mathcal{F}(n,m)$ by its subspace
$\langle\ker(\varphi(n,3))_{\leq 2n+2}\rangle_{GL_{m}(\mathbb{C})}$. ∎

##  5\. The case $m=2$

Set

(2)
$\mathcal{R}(n)_{n,2}:=\pi_{n,0}\rho_{0,2}-2\pi_{n-1,1}\rho_{1,1}+\pi_{n-2,2}\rho_{2,0}.$

For $k=1,\dots,\lfloor\frac{n}{2}\rfloor$ set

(3)
$\displaystyle\mathcal{R}(n)_{2n-2k,2k}:=(-1)^{k}\frac{1}{2}\binom{2k}{k}\pi_{n-k,k}^{2}+\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j}\pi_{n-j,j}\pi_{n-2k+j,2k-j}$
$\displaystyle-4^{k}\rho_{2,0}^{n-2k}(\rho_{1,1}^{2}-\rho_{2,0}\rho_{0,2})^{k}.$

######  Theorem 5.1.

For
$\lambda\in\\{(n,2),(2n-2,2),(2n-4,4),\dots,(2n-2\lfloor\frac{n}{2}\rfloor,2\lfloor\frac{n}{2}\rfloor)\\}$
consider the element $\mathcal{R}(n)_{\lambda}$ in $\mathcal{F}(n,2)$
introduced above.

  * (i)

For $m\geq 2$ the element $\mathcal{R}(n)_{\lambda}$ generates a
$GL_{m}(\mathbb{C})$-submodule in $\ker(\varphi(n,m))$ isomorphic to
$S^{\lambda}(\mathbb{C}^{m})$.

  * (ii)

For $m=2$ the ideal, $\ker(\varphi(n,2))$ is minimally generated by the above
elements $\mathcal{R}(n)_{\lambda}$ as a $GL_{2}(\mathbb{C})$-ideal.

######  Remark 5.2.

A minimal presentation of the algebra $\mathbb{C}[V\oplus V^{*}]^{D_{2n}}$ by
generators and relations is given in [2] for $n=3,4,6$ and in [4, Theorem 2.1]
for arbitrary $n$. Since $V^{*}\cong V$ as $D_{2n}$-modules, these results can
be translated to an explicit minimal presentation by generators and relations
of $\mathbb{C}[V^{2}]^{D_{2n}}$. So the novel part of our Theorem 5.1 is the
nice explicit form of the relations $\mathcal{R}(n)_{\lambda}$ and the
understanding of the $GL_{2}(\mathbb{C})$-module structure of the minimal
syzygies.

Proof of Theorem 5.1 (i). We have $E_{1,2}.\mathcal{R}(n)_{n,2}=0$,
$E_{1,1}.\mathcal{R}(n)_{n,2}=n\mathcal{R}(n)_{n,2}$,
$E_{2,2}.\mathcal{R}(n)_{n,2}=2\mathcal{R}(n)_{n,2}$. Thus
$\mathcal{R}(n)_{n,2}$ is a highest weight vector with weight $(n,2)$, and
therefore it generates an irreducible $GL_{m}(\mathbb{C})$-submodule of
$\mathcal{F}(n,m)$ isomorphic to $S^{(n,2)}(\mathbb{C}^{m})$ for $m\geq 2$.
The elements $\rho_{2,0}$, $\rho_{1,1}^{2}-\rho_{2,0}\rho_{0,2}$ and
$(-1)^{k}\frac{1}{2}\binom{2k}{k}\pi_{n-k,k}^{2}+\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j}\pi_{n-j,j}\pi_{n-2k+j,2k-j}$
are annihilated by $E_{1,2}\in\mathfrak{gl}_{2}(\mathbb{C})$, so they are
highest weight vectors with weights $(2,0)$, $(2,2)$ and $(2n-2k,2k)$. It
follows that $\mathcal{R}(n)_{2n-2k,2k}$ is a highest weights vector of weight
$(2n-2k,2k)$, and thus it generates an irreducible
$GL_{m}(\mathbb{C})$-submodule of $\mathcal{F}(n,m)$ isomorphic to
$S^{(2n-2k,2k)}(\mathbb{C}^{m})$ for $m\geq 2$. The equality

$p_{n,0}q_{0,2}-2p_{n-1,1}q_{1,1}+p_{n-2,2}q_{2,0}=0$

can be verified by direct computation, showing that
$\varphi(n,2)(\mathcal{R}(n)_{n,2})=0$. To verify that
$\mathcal{R}(n)_{2n-2k,2k}$ belongs to $\ker(\varphi(n,2))$ for
$k=1,\dots,\lfloor\frac{n}{2}\rfloor$ we need to prove the equality

(4)
$(-1)^{k}\frac{1}{2}\binom{2k}{k}p_{n-k,k}^{2}+\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j}p_{n-j,j}p_{n-2k+j,2k-j}-4^{k}q_{2,0}^{n-2k}(q_{1,1}^{2}-q_{2,0}q_{0,2})^{k}=0.$

Since $\mathcal{R}(n)_{2n-2k,2k}$ is a highest weight vector, it is fixed by
the subgroup $UT_{2}(\mathbb{C})$ of unipotent upper triangular matrices in
$GL_{2}(\mathbb{C})$. Therefore $\varphi(n,2)(\mathcal{R}(n)_{2n-2k,2k})$ (the
left hand side of (4)) is a $UT_{2}(\mathbb{C})$-invariant in
$\mathbb{C}[V^{2}]$, and thus it is constant along the
$UT_{2}(\mathbb{C})$-orbits in $V^{2}$. The $UT_{2}(\mathbb{C})$-orbit of each
point from a Zariski dense open subset in $V^{2}$ has non-empty intersection
with the subset of lower triangular matrices in $\mathbb{C}^{2\times
2}=V^{2}$. Therefore it is sufficient to show that
$\sigma(\varphi(n,2)(\mathcal{R}(n)_{2n-2k,2k}))=0$, where $\sigma$ is the
homomorphism
$\mathbb{C}[x_{1},y_{1},x_{2},y_{2}]\to\mathbb{C}[x_{1},y_{1},y_{2}]$ given by
the specialization $x_{2}\mapsto 0$. Now
$\sigma(p_{n-j,j})=y_{1}^{n-j}y_{2}^{j}$ for $j>0$, $\sigma(q_{0,2})=0$,
$\sigma(q_{1,1})=\frac{1}{2}x_{1}y_{2}$, hence

(5)
$\sigma(4^{k}q_{2,0}^{n-2k}(q_{1,1}^{2}-q_{2,0}q_{0,2})^{k})=x_{1}^{n}y_{1}^{n-2k}y_{2}^{2k}.$

On the other hand, we have

(6)
$\displaystyle\sigma((-1)^{k}\frac{1}{2}\binom{2k}{k}p_{n-k,k}^{2}+\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j}p_{n-j,j}p_{n-2k+j,2k-j})$
$\displaystyle=((-1)^{k}\frac{1}{2}\binom{2k}{k}+\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j})y_{1}^{2n-2k}y_{2}^{2k}+x_{1}^{n}y_{1}^{n-2k}y_{2}^{2k}.$

Note that

(7)
$(-1)^{k}\frac{1}{2}\binom{2k}{k}+\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j}=\frac{1}{2}\sum_{j=0}^{2k}\binom{2k}{j}=\frac{1}{2}(1-1)^{2k}=0.$

The equalities (5),(6),(7) show
$\sigma(\varphi(n,2)(\mathcal{R}(n)_{2n-2k,2k}))=0$, implying in turn the
equality (4), and thus (i) is proved. $\square$

######  Remark 5.3.

The expression
$(-1)^{k}\frac{1}{2}\binom{2k}{k}\pi_{n-k,k}^{2}+\sum_{j=0}^{k-1}(-1)^{j}\binom{2k}{j}\pi_{n-j,j}\pi_{n-2k+j,2k-j}$
appears in the invariant theory of $m$-ary forms of degree $n$, see [11,
Section 114]. Note also that this gives an explicit formula for the highest
weight vector of the summand $S^{(2n-2k,2k)}(\mathbb{C}^{m})$ of
$S^{2}(S^{n}(\mathbb{C}^{m}))$in the decomposition given in Proposition 4.5
(iv).

###  5.1. Hironaka decomposition

The following statement is known, see for example [4, Theorem 2.1]. We shall
present an alternative proof.

######  Proposition 5.4.

The elements $p_{n,0}$, $q_{2,0}$, $p_{0,n}$, $q_{0,2}$ form a homogeneous
system of parameters in $\mathbb{C}[V^{2}]^{D_{2n}}$, and
$\mathbb{C}[V^{2}]^{D_{2n}}$ is a free module over the subalgebra
$P(n,2):=\mathbb{C}[p_{n,0},q_{2,0},p_{0,n},q_{0,2}]$ generated by

(8) $\\{q_{1,1}^{j},\ p_{n-i,i}\mid j=0,1,\dots,n;\quad i=1,\dots,n-1\\}.$

###### Proof.

The common zero locus in $V^{2}$ of $p_{n,0}$, $q_{2,0}$, $p_{0,n}$, $q_{0,2}$
is the zero element of $V^{2}$, hence these polynomials form a homogeneous
system of parameters in $\mathbb{C}[V^{2}]^{D_{2n}}$ and in
$\mathbb{C}[V^{2}]$ (so both of these algebras are finitely generated free
modules over their subalgebra $P(n,2)$, see for example [16, Section 2.3]).

Denote by $H$ the cyclic subgroup of $D_{2n}$ generated by
$\left(\begin{array}[]{cc}\omega&0\\\ 0&\omega^{-1}\end{array}\right)$. Then
$\mathbb{C}[V^{2}]^{H}$ is spanned as a $\mathbb{C}$-vector space by the
monomials
$x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}y_{1}^{\beta_{1}}y_{2}^{\beta_{2}}$,
where $\alpha_{1}+\alpha_{2}-\beta_{1}-\beta_{2}$ is divisible by $n$. One can
easily deduce (see for example the method of the proof of Proposition 6.3)
that $\mathbb{C}[V^{2}]^{H}$ is a free $P(n,2)$-module generated by

$\displaystyle\\{(x_{1}y_{2})^{j},\ (x_{2}y_{1})^{j},\ x_{1}^{n-i}x_{2}^{i},\
y_{1}^{n-i}y_{2}^{i},\ x_{1}^{n},\ x_{2}^{n},\ x_{1}^{n}x_{2}^{n}\mid
j=0,1,\dots,n-1,\quad i=1,\dots,n-1\\}.$

Denote by $(P(n,2)_{+})$ the ideal in $\mathbb{C}[V^{2}]$ generated by
$p_{n,0}$, $q_{2,0}$, $p_{0,n}$, $q_{0,2}$. The other generator
$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$ of $D_{2n}$ maps the
cosets of $x_{1}^{n}$ and $x_{2}^{n}$ in $\mathbb{C}[V^{2}]/(P(n,2)_{+})$ to
their negative, fixes the coset of $x_{1}^{n}x_{2}^{n}$ in
$\mathbb{C}[V^{2}]/(P(n,2)_{+})$, permutes the other elements in the above
$P(n,2)$-module generating system of $\mathbb{C}[V^{2}]^{H}$, and fixes all
elements of $P(n,2)$. It follows that $\mathbb{C}[V^{2}]^{D_{2n}}$ is a free
$P(n,2)$-module generated by

$\\{(x_{1}y_{2})^{j}+(y_{1}x_{2})^{j},\
x_{1}^{n-i}x_{2}^{i}+y_{1}^{n-i}y_{2}^{i},\ x_{1}^{n}x_{2}^{n}\mid
j=0,1,\dots,n-1,\quad i=1,\dots,n-1\\}.$

Note finally that $(x_{1}y_{2})^{j}+(y_{1}x_{2})^{j}$ is congruent to
$2^{j}q_{1,1}^{j}$ modulo $(P(n,2)_{+})$, and $x_{1}^{n}x_{2}^{n}$ is
congruent to $-\frac{1}{2}(2q_{1,1})^{n}$ modulo $(P(n,2)_{+})$. This shows
that $\mathbb{C}[V^{2}]^{D_{2n}}$ is a free module over $P(n,2)$ generated by
(8). ∎

###  5.2. Further relations

Define $\mathcal{R}(n)_{n,2}^{n-j,2+j}$ for $j=0,1,\dots,n-2$ recursively by
setting $\mathcal{R}(n)_{n,2}^{n,2}:=\mathcal{R}(n)_{n,2}$, and set

$\mathcal{R}(n)_{n,2}^{n-j-1,2+j+1}:=\frac{1}{n-2-j}E_{2,1}.\mathcal{R}(n)_{n,2}^{n-j,2+j}$

for $j=0,1,\dots,n-3$. Then $\\{\mathcal{R}(n)_{n,2}^{n-j,2+j}\mid
j=0,1,\dots,n-2\\}$ is a $\mathbb{C}$-vector space basis in
$\langle\mathcal{R}(n)_{n,2}\rangle_{GL_{2}(\mathbb{C})}$. Moreover, one shows
by induction on $j$ that

(9)
$\mathcal{R}(n)_{n,2}^{n-j,2+j}=\pi_{n-j,j}\rho_{0,2}-2\pi_{n-j-1,j+1}\rho_{1,1}+\pi_{n-j-2,j+2}\rho_{2,0}.$

Proof of Theorem 5.1 (ii). The elements $q_{2,0},q_{1,1},q_{0,2}$ in
$\mathbb{C}[V^{2}]^{D_{2n}}$ are algebraically independent. Therefore any non-
zero element in $\ker(\varphi(n,2))$ involves a variable $\pi_{n-j,j}$ for
some $j$. Moreover, a non-trivial linear combination of the elements
$p_{n-j,j}\in\mathbb{C}[V^{2}]^{D_{2n}}$ $(j=0,1,\dots,n)$ involves a term
$x_{1}^{i}x_{2}^{n-i}$ for some $i$, hence is not contained in the algebra
generated by $q_{2,0},q_{1,1},q_{0,2}$ (all of whose terms have the same
degree in the $x$-variables as in the $y$-variables). So every non-zero
element in $\ker(\varphi(n,2))$ involves a term divisible by a monomial of the
form $\pi_{n-j,j}\pi_{n-i,i}$ or $\pi_{n-j,j}\rho_{2-i,i}$. In particular, the
minimal possible degree of an element of $\ker(\varphi(n,2))$ is $n+2$, the
degree of $\mathcal{R}(n)_{n,2}$. It follows that a basis of
$\langle\mathcal{R}(n)_{n,2}\rangle_{GL_{2}(\mathbb{C})}$ is contained in any
minimal homogeneous generating system of $\ker(\varphi(n,2))$. Denote by
$\mathcal{I}$ the ideal of $\mathcal{F}(n,2)$ generated by
$\langle\mathcal{R}(n)_{n,2}\rangle_{GL_{2}(\mathbb{C})}$, and denote by
$\mathcal{P}$ the subalgebra of $\mathcal{F}(n,2)$ generated by
$\pi_{n,0},\rho_{2,0},\pi_{0,n},\rho_{0,2}$. The relations (9) show that
modulo $\mathcal{I}$, $\pi_{n-j,j}\rho_{1,1}$ is congruent to an element in
the ideal of $\mathcal{F}(n,2)$ generated by $\pi_{n,0}$, $\pi_{0,n}$,
$\rho_{2,0}$, $\rho_{0,2}$. Consequently, any element of degree less than $2n$
in $\mathcal{F}(n,2)$ is congruent modulo $\mathcal{I}$ to an element of
$\sum_{i=0}^{n-1}\mathcal{P}\rho_{1,1}^{i}+\sum_{j=1}^{n-1}\mathcal{P}\pi_{n-j,j}$.
It follows by Proposition 5.4 that a minimal homogeneous generating system of
$\ker(\varphi)$ has no elements of degree $n+3,n+4,\dots,2n-1$.

Set

$\mathcal{K}:=\bigoplus_{\lambda}\langle\mathcal{R}(n)_{\lambda}\rangle_{GL_{2}(\mathbb{C})}$

where
$\lambda\in\\{(2n-2,2),(2n-4,4),\dots,(2n-2\lfloor\frac{n}{2}\rfloor,2\lfloor\frac{n}{2}\rfloor)\\}$
(the above sum is direct, because by Theorem 5.1 (i) the summands are pairwise
non-isomorphic minimal $GL_{2}(\mathbb{C})$-invariant subspaces in
$\mathcal{F}(n,2)$). Observe next that any term of any element of the ideal
$\mathcal{I}$ of $\mathcal{F}(n,2)$ is divisible by a monomial of the form
$\pi_{n-j,j}\rho_{2-i,i}$, whereas no term of the elements in $\mathcal{K}$ is
divisible by a monomial of the form $\pi_{n-j,j}\rho_{2-i,i}$. It follows that
$\mathcal{R}(n)_{n,2}^{n-j,2+j}$ $(j=0,\dots,n-2)$ together with a basis of
$\mathcal{K}$ is part of some minimal homogeneous generating system of
$\ker(\varphi(n,2))$. The dimension of $\mathcal{K}$ is
$\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}(2n-4i+1)=\frac{n(n-1)}{2}$. On the
other hand, it follows from Proposition 5.4 that there must exist
$\binom{n}{2}$ elements of degree $2n$ in $\ker(\varphi(n,2))$ that allow to
rewrite the products $\pi_{n-j,j}\pi_{n-i,i}$ as an element of
$\sum_{i=0}^{n}\mathcal{P}\rho_{1,1}^{i}+\sum_{j=1}^{n-1}\mathcal{P}\pi_{n-j,j}$,
and these relations together with the earlier relations of degree $n+2$ are
sufficient to generate $\ker(\varphi(n,2))$ up to degree $2n$. Denote by
$\mathcal{J}$ the ideal generated by $I\cup\mathcal{K}$. Our observations so
far imply that any element of $\mathcal{F}(n,2)$ is congruent modulo
$\mathcal{J}$ to an element of
$\sum_{i=0}^{\infty}\mathcal{P}\rho_{1,1}^{i}+\sum_{j=1}^{n-1}\mathcal{P}\pi_{n-j,j}$.
It remains to show that $\rho_{1,1}^{n+1}$ is congruent modulo $\mathcal{J}$
to an element in
$\sum_{i=0}^{n}\mathcal{P}\rho_{1,1}^{i}+\sum_{j=1}^{n-1}\mathcal{P}\pi_{n-j,j}$.
When $n$ is even, consider the element $\rho_{1,1}\mathcal{R}_{n,n}$. It
belongs to $\mathcal{J}$ and has the term $-4^{n}\rho_{1,1}^{n+1}$. The other
terms of $\rho_{1,1}\mathcal{R}_{n,n}$ are congruent modulo $\mathcal{I}$ to
an element of the ideal of $\mathcal{F}(n,2)$ generated by $\pi_{n,0}$,
$\pi_{0,n}$, $\rho_{2,0}$, $\rho_{0,2}$ by (9). It follows that
$\rho_{1,1}^{n+1}$ is congruent modulo $\mathcal{J}$ to an element of
$\sum_{i=0}^{n}\mathcal{P}\rho_{1,1}^{i}+\sum_{j=1}^{n-1}\mathcal{P}\pi_{n-j,j}$.
So we are done when $n$ is even. For odd $n$ consider the element
$\rho_{1,1}(E_{2,1}.\mathcal{R}_{n+1,n-1})$ of $\mathcal{J}$. It has the term
$-2^{n}\rho_{1,1}^{n+1}$, and all the other terms are congruent modulo
$\mathcal{I}$ to elements in
$\sum_{i=0}^{n}\mathcal{P}\rho_{1,1}^{i}+\sum_{j=1}^{n-1}\mathcal{P}\pi_{n-j,j}$.
This finishes the proof also for the case of odd $n$. $\square$

##  6\. The case $n=4$

###  6.1. Secondary S-generating systems

In this subsection we return to the general setup of Section 1. The action of
the subgroup of diagonal matrices in $GL_{m}(\mathbb{C})$ induces an
$\mathbb{N}_{0}^{m}$-grading on $\mathbb{C}[V^{m}]$ and
$\mathbb{C}[V^{m}]^{G}$. Write $\mathbb{C}[V^{m}]_{\alpha}$ and
$\mathbb{C}[V^{m}]^{G}_{\alpha}$ for the multihomogeneous component of
$\mathbb{C}[V^{m}]$ and $\mathbb{C}[V^{m}]^{G}$ of multidegree
$\alpha=(\alpha_{1},\dots,\alpha_{m})$.

Now assume that $G$ is finite. Take a homogeneus system of parameters
$p_{1},\dots,p_{k}$ in $\mathbb{C}[V]^{G}$ (so $k=\dim(V)$). For $j=1,\dots,m$
and $i=1,\dots,k$ denote by $p_{i}^{(j)}$ the element of
$\mathbb{C}[V^{m}]^{G}$ that maps $(v_{1},\dots,v_{m})\in V^{m}$ to
$p_{i}(v_{j})$. Then

(10) $p_{1}^{(j)},\dots,p_{k}^{(j)}\qquad(j=1,\dots,m)$

is a homogeneous system of parameters for $\mathbb{C}[V^{m}]^{G}$. This means
that denoting by $P(m)$ the $\mathbb{C}$-subalgebra of $\mathbb{C}[V^{m}]^{G}$
generated by the (algebraically independent) elements (10), the space
$\mathbb{C}[V^{m}]^{G}$ is a finitely generated free $P(m)$-module. The
elements (10) are called _primary generators of_ $\mathbb{C}[V^{m}]^{G}$,
whereas a finite free $P(m)$-module generating system of
$\mathbb{C}[V^{m}]^{G}$ is called a _system of secondary generators_ of
$\mathbb{C}[V^{m}]^{G}$. By the Graded Nakayama Lemma a set of homogeneous
elements in $\mathbb{C}[V^{m}]^{G}$ forms a system of secondary generators if
and only if they form a basis in a vector space direct complement in
$\mathbb{C}[V^{m}]^{G}$ of $(P(m)^{+})$, the ideal in $\mathbb{C}[V^{m}]^{G}$
generated by the elements (10).

View the symmetric group $S_{m}$ as the subgroup of permutation matrices in
$GL_{m}(\mathbb{C})$. Then $s\in S_{m}$ maps $\mathbb{C}[V^{m}]_{\alpha}$ to
$\mathbb{C}[V^{m}]_{s\cdot\alpha}$, where
$s\cdot\alpha=(\alpha_{s^{-1}(1)},\dots,\alpha_{s^{-1}(m)})$.

######  Definition 6.1.

A subset $L$ of $\mathbb{C}[V^{m}]^{G}$ is called a _system of secondary
S-generators of_ $\mathbb{C}[V^{m}]^{G}$ if $L$ consists of multihomogeneous
elements of decreasing multidegrees and

$L^{*}:=\\{s\cdot f\mid f\in L,\quad s\in
S_{m}/\mathrm{Stab}(\underline{\deg}(f))\\},$

is a system of secondary generators of $\mathbb{C}[V^{m}]^{G}$ where
$\underline{\deg}(f)\in\mathbb{N}_{0}^{m}$ stands for the multidegree of $f$
and $S_{m}/\mathrm{Stab}(\underline{\deg}(f))$ stands for a chosen set of left
coset representatives in $S_{m}$ with respect to the stabilizer subgroup
$\mathrm{Stab}(\underline{\deg}(f))$ of the multidegree of $f$. (Note that
this notion depends on the choice $p_{1},\dots,p_{k}$ of homogeneous system of
parameters in $\mathbb{C}[V]^{G}$.)

######  Proposition 6.2.

A system of secondary S-generators of $\mathbb{C}[V^{m}]^{G}$ exists.

###### Proof.

Since the elements in (10) are multihomogeneous, the ideal $(P(m)^{+})$ of
$\mathbb{C}[V^{m}]^{G}$ is spanned by multihomogenous elements, and therefore
a system of secondary generators consisting of multihomogeneous elements
exist. Moreover, a set $M$ of multihomogeneous elements forms a system of
secondary generators if and only if for each multidegree
$\alpha\in\mathbb{N}_{0}^{m}$ the subset $M\cap\mathbb{C}[V^{m}]_{\alpha}$ is
a $\mathbb{C}$-vector space basis in a direct complement in
$\mathbb{C}[V^{m}]^{G}_{\alpha}$ of
$(P(m)^{+})\cap\mathbb{C}[V^{m}]_{\alpha}$.

The action of $S_{m}$ preserves the set (10), hence it preserves the algebra
$P(m)$ and the ideal $(P(m)^{+})$ in $\mathbb{C}[V^{m}]^{G}$. An element $s\in
S_{m}$ gives a vector space isomorphism between $\mathbb{C}[V^{m}]_{\alpha}$
and $\mathbb{C}[V^{m}]_{s\cdot\alpha}$, such that the subspace
$\mathbb{C}[V^{m}]^{G}_{\alpha}$ is mapped onto
$\mathbb{C}[V^{m}]^{G}_{s\cdot\alpha}$ and
$\mathbb{C}[V^{m}]_{\alpha}\cap(P(m)^{+})$ is mapped onto
$\mathbb{C}[V^{m}]_{s\cdot\alpha}\cap(P(m)^{+})$. Therefore to get a system of
secondary S-generators of $\mathbb{C}[V^{m}]^{G}$ we just need to take for
each decreasing multidegree $\alpha$ a basis in a direct complement in
$\mathbb{C}[V^{m}]^{G}_{\alpha}$ of
$\mathbb{C}[V^{m}]_{\alpha}\cap(P(m)^{+})$, and the union of these sets as
$\alpha$ ranges over all decreasing multidegrees will be a system of secondary
S-generators. ∎

###  6.2. Hironaka decomposition for $n=4$, $m=3$

The elements

(11) $p_{4,0,0},p_{0,4,0},p_{0,0,4},q_{2,0,0},q_{0,2,0},q_{0,0,2}$

constitute a _homogeneous system of parameters_ in the algebras
$\mathbb{C}[V^{3}]$ and $\mathbb{C}[V^{3}]^{D_{8}}$. We shall refer to the
elements (11) as the _primary generators of_ $\mathbb{C}[V^{3}]^{D_{8}}$, and
denote by $P(4,3)$ the subalgebra of $\mathbb{C}[V^{3}]$ generated by them. We
are looking for an explicit free $P(4,3)$-module generating system (called
_system of secondary generators_) of $\mathbb{C}[V^{3}]^{D_{8}}$.

Denote by $H$ the cyclic subgroup of $D_{8}$ generated by
$\left(\begin{array}[]{cc}\omega&0\\\ 0&\omega^{-1}\end{array}\right)$. The
elements (11) constitute a homogeneous system of parameters in
$\mathbb{C}[V^{3}]^{H}$ as well.

######  Proposition 6.3.

The following table gives a secondary S-generating system of
$\mathbb{C}[V^{3}]^{H}$ with respect to $P(4,3)$:

$\begin{array}[]{c||c}\text{multidegree}&\text{generator}\\\
\hline\cr\hline\cr(0,0,0)&1\\\ (1,1,0)&x_{1}y_{2},\ y_{1}x_{2}\\\
(2,1,1)&x_{1}^{2}x_{2}x_{3},\ y_{1}^{2}y_{2}y_{3},\ x_{1}^{2}y_{2}y_{3},\
y_{1}^{2}x_{2}x_{3}\\\ (2,2,0)&x_{1}^{2}x_{2}^{2},\ y_{1}^{2}y_{2}^{2},\
x_{1}^{2}y_{2}^{2},\ y_{1}^{2}x_{2}^{2}\\\ (3,1,0)&x_{1}^{3}x_{2},\
y_{1}^{3}y_{2}\\\ (4,0,0)&x_{1}^{4}\\\ (3,2,1)&x_{1}^{3}y_{2}^{2}y_{3},\
y_{1}^{3}x_{2}^{2}x_{3},\ x_{1}^{3}x_{2}^{2}y_{3},\ y_{1}^{3}y_{2}^{2}x_{3}\\\
(3,3,0)&x_{1}^{3}y_{2}^{3},\ y_{1}^{3}x_{2}^{3}\\\
(4,1,1)&x_{1}^{4}x_{2}y_{3},\ x_{1}^{4}y_{2}x_{3}\\\
(3,3,2)&x_{1}^{3}x_{2}^{3}x_{3}^{2},\ y_{1}^{3}y_{2}^{3}y_{3}^{2},\
x_{1}^{3}x_{2}^{3}y_{3}^{2},\ y_{1}^{3}y_{2}^{3}x_{3}^{2}\\\
(4,2,2)&x_{1}^{4}x_{2}^{2}x_{3}^{2},\ x_{1}^{4}y_{2}^{2}y_{3}^{2},\
x_{1}^{4}x_{2}^{2}y_{3}^{2},\ x_{1}^{4}y_{2}^{2}x_{3}^{2}\\\
(4,3,1)&x_{1}^{4}x_{2}^{3}x_{3},\ x_{1}^{4}y_{2}^{3}y_{3}\\\
(4,4,0)&x_{1}^{4}x_{2}^{4}\\\ (4,3,3)&x_{1}^{4}x_{2}^{3}y_{3}^{3},\
x_{1}^{4}y_{2}^{3}x_{3}^{3}\\\ (4,4,4)&x_{1}^{4}x_{2}^{4}x_{3}^{4}\end{array}$

###### Proof.

$\mathbb{C}[V^{3}]^{H}$ is spanned as a $\mathbb{C}$-vector space by the
monomials $x_{1}^{i}x_{2}^{j}x_{3}^{k}y_{1}^{l}y_{2}^{m}y_{3}^{r}$ such that
$i+j+k-l-m-r$ is divisible by $4$. Consider the lexicographic monomial order
in $\mathbb{C}[x,y]$ induced by the order $x<y$ of the variables. Then it is
easy to check that the ideal generated by $xy$ and $x^{4}+y^{4}$ has the
Gröbner basis $xy,x^{4}+y^{4},x^{5}=x(x^{4}+y^{4})-y^{3}(xy)$, hence the ideal
generated by the initial monomials in $(xy,x^{4}+y^{4})$ is
$(xy,y^{4},x^{5})$. It follows that the monomials not divisible by any of
$x_{1}y_{1}$, $x_{2}y_{2}$, $x_{3}y_{3}$, $y_{1}^{4}$, $y_{2}^{4}$,
$y_{3}^{4}$, $x_{1}^{5}$, $x_{2}^{5}$, $x_{3}^{5}$ form a basis in a vector
space direct complement of the ideal in $\mathbb{C}[V^{3}]$ generated by the
elements in (11). As each monomial spans an $H$-invariant subspace in
$\mathbb{C}[V^{3}]$, the $H$-invariant monomials among them form a system of
secondary generators for $\mathbb{C}[V^{3}]^{H}$, and the table above contains
all those with decreasing multidegree. ∎

######  Proposition 6.4.

The following table gives a secondary S-generating system of
$\mathbb{C}[V^{3}]^{D_{8}}$:

$\begin{array}[]{c||c}\text{multidegree}&\text{generator}\\\
\hline\cr\hline\cr(0,0,0)&1\\\ (1,1,0)&q_{1,1,0}\\\ (2,1,1)&p_{2,1,1},\
q_{1,1,0}q_{1,0,1}\\\ (2,2,0)&p_{2,2,0},\ q_{1,1,0}^{2}\\\
(3,1,0)&p_{3,1,0}\\\ (3,2,1)&p_{3,1,0}q_{0,1,1},\ q_{1,1,0}^{2}q_{1,0,1}\\\
(3,3,0)&q_{1,1,0}^{3}\\\ (4,1,1)&p_{3,1,0}q_{1,0,1}\\\
(3,3,2)&p_{2,1,1}p_{1,2,1},\ p_{3,1,0}q_{0,1,1}^{2}\\\
(4,2,2)&p_{3,1,0}q_{1,0,1}q_{0,1,1},\ q_{1,1,0}^{2}q_{1,0,1}^{2}\\\
(4,3,1)&q_{1,1,0}^{3}q_{1,0,1}\\\ (4,4,0)&q_{1,1,0}^{4}\\\
(4,3,3)&p_{3,1,0}q_{1,0,1}q_{0,1,1}^{2}\end{array}$

###### Proof.

Following the notation of Section 6.1, $(P(4,3)^{+})$ stands for the ideal in
$\mathbb{C}[V^{3}]^{D_{8}}$ generated by the elements (11). The element
$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$ of $D_{8}$ permutes up
to sign the cosets modulo $(P(4,3)^{+})$ of the monomials in the table in
Proposition 6.3 (note that $x^{4}$ is congruent to $-y^{4}$ modulo
$(P(4,3)^{+})$). It follows by Proposition 6.3 that a $D_{8}$-invariant direct
complement of $(P(4,3)^{+})$ in the sum of the homogeneous components of
$\mathbb{C}[V^{3}]^{H}$ with decreasing multidegree has the basis $C_{+}\cup
C_{-}$, where

$\displaystyle
C_{+}=\\{1,x_{1}y_{2}+y_{1}x_{2},x_{1}^{2}x_{2}x_{3}+y_{1}^{2}y_{2}y_{3},x_{1}^{2}y_{2}y_{3}+y_{1}^{2}x_{2}x_{3},x_{1}^{2}x_{2}^{2}+y_{1}^{2}y_{2}^{2},x_{1}^{2}y_{2}^{2}+y_{1}^{2}x_{2}^{2},$
$\displaystyle
x_{1}^{3}x_{2}+y_{1}^{3}y_{2},x_{1}^{3}y_{2}^{2}y_{3}+y_{1}^{3}x_{2}^{2}x_{3},x_{1}^{3}x_{2}^{2}y_{3}+y_{1}^{3}y_{2}^{2}x_{3},x_{1}^{3}y_{2}^{3}+y_{1}^{3}x_{2}^{3},x_{1}^{4}x_{2}y_{3}-x_{1}^{4}y_{2}x_{3},$
$\displaystyle
x_{1}^{3}x_{2}^{3}x_{3}^{2}+y_{1}^{3}y_{2}^{3}y_{3}^{2},x_{1}^{3}x_{2}^{3}y_{3}^{2}+y_{1}^{3}y_{2}^{3}x_{3}^{2},x_{1}^{4}x_{2}^{2}x_{3}^{2}-x_{1}^{4}y_{2}^{2}y_{3}^{2},x_{1}^{4}x_{2}^{2}y_{3}^{2}-x_{1}^{4}y_{2}^{2}x_{3}^{2},$
$\displaystyle
x_{1}^{4}x_{2}^{3}x_{3}-x_{1}^{4}y_{2}^{3}y_{3},x_{1}^{4}x_{2}^{4},x_{1}^{4}x_{2}^{3}y_{3}^{3}-x_{1}^{4}y_{2}^{3}x_{3}^{3}\\}$

and

$\displaystyle
C_{-}=\\{x_{1}y_{2}-y_{1}x_{2},x_{1}^{2}x_{2}x_{3}-y_{1}^{2}y_{2}y_{3},x_{1}^{2}y_{2}y_{3}-y_{1}^{2}x_{2}x_{3},x_{1}^{2}x_{2}^{2}-y_{1}^{2}y_{2}^{2},x_{1}^{2}y_{2}^{2}-y_{1}^{2}x_{2}^{2},$
$\displaystyle
x_{1}^{3}x_{2}-y_{1}^{3}y_{2},x_{1}^{3}y_{2}^{2}y_{3}-y_{1}^{3}x_{2}^{2}x_{3},x_{1}^{3}x_{2}^{2}y_{3}-y_{1}^{3}y_{2}^{2}x_{3},x_{1}^{3}y_{2}^{3}-y_{1}^{3}x_{2}^{3},x_{1}^{4}x_{2}y_{3}+x_{1}^{4}y_{2}x_{3},$
$\displaystyle
x_{1}^{3}x_{2}^{3}x_{3}^{2}-y_{1}^{3}y_{2}^{3}y_{3}^{2},x_{1}^{3}x_{2}^{3}y_{3}^{2}-y_{1}^{3}y_{2}^{3}x_{3}^{2},x_{1}^{4}x_{2}^{2}x_{3}^{2}+x_{1}^{4}y_{2}^{2}y_{3}^{2},x_{1}^{4}x_{2}^{2}y_{3}^{2}+x_{1}^{4}y_{2}^{2}x_{3}^{2},$
$\displaystyle
x_{1}^{4}x_{2}^{3}x_{3}+x_{1}^{4}y_{2}^{3}y_{3},x_{1}^{4}x_{2}^{3}y_{3}^{3}+x_{1}^{4}y_{2}^{3}x_{3}^{3},x_{1}^{4},x_{1}^{4}x_{2}^{4}x_{3}^{4}\\}.$

The elements in $C_{+}$ are $D_{8}$-invariant, whereas the elements in $C_{-}$
span a $1$-dimensional $D_{8}$-invariant subspace on which $D_{8}$ acts via
the determinant representation. It follows that $C_{+}$ is a system of
secondary S-generators for $\mathbb{C}[V^{3}]^{D_{8}}$. It is easy to see that
modulo the ideal $(P(4,3)^{+})$ the elements listed in the table in the
statement of our proposition agree with non-zero scalar multiples of the
elements in $C_{+}$. ∎

###  6.3. Relations for $n=4$ and $m=3$

######  Proposition 6.5.

The $GL_{3}(\mathbb{C})$-ideal $\ker(\varphi(4,3))$ is minimally generated by
$\mathcal{R}_{2,2,2}$, $\mathcal{R}(4)_{4,2}$, $\mathcal{R}(4)_{6,2}$, and
$\mathcal{R}(4)_{4,4}$.

######  Remark 6.6.

We have $\dim(S^{(4,2)}(\mathbb{C}^{3}))=27$,
$\dim(S^{(6,2)}(\mathbb{C}^{3}))=60$, $\dim(S^{(4,4)}(\mathbb{C}^{3}))=15$,
and $\dim(S^{(2,2,2)}(\mathbb{C}^{3}))=1$. Therefore Proposition 6.5 implies
that a minimal homogeneous generating system of the ideal $\ker(\varphi(4,3))$
consists of $103$ elements. For comparison we mention that a minimal
homogeneous generating system of $\ker(\varphi(4,2))$ consists of $3+5+1=9$
elements.

The logic of the proof of Proposition 6.5 is furnished by the following
general lemma:

######  Lemma 6.7.

Let $\varphi:\mathcal{F}\to R$ be a surjective homomorphism of graded
$\mathbb{C}$-algebras, where $R$ is a connected graded Cohen-Macaulay algebra
(cf. [16, Section 2.3]), so there exist homogeneous elements
$h_{1},\dots,h_{k}$ and $t_{1},\dots,t_{l}$ in $\mathcal{F}$ with the
following properties:

  1. (1)

$\varphi(h_{1}),\dots,\varphi(h_{k})$ is a homogeneous system of parameters in
$R$.

  2. (2)

$R=P\varphi(t_{1})\oplus\cdots\oplus P\varphi(t_{l})$ where
$P=\mathbb{C}[\varphi(h_{1}),\dots,\varphi(h_{l})]$.

Let $\mathcal{K}$ be a homogeneous ideal in $\mathcal{F}$ contained in
$\ker(\varphi)$, and assume that for some $d\in\mathbb{N}$ we have

$\mathcal{F}_{\leq d}=\mathrm{Span}_{\mathbb{C}}\mathcal{T}_{\leq
d}+\mathcal{H}_{\leq d}+\mathcal{K}_{\leq d}$

where $\mathcal{H}$ is the ideal in $\mathcal{F}$ generated by
$h_{1},\dots,h_{k}$, $\mathcal{T}:=\\{t_{1},\dots,t_{l}\\}$, $d\in\mathbb{N}$
and for a subset $A$ of homogeneous elements (respectively graded subspace) in
$\mathcal{F}$ we write $A_{\leq d}$ for the set of elements (respectively sum
of homogeneous components) of $A$ with degree $\leq d$.

Then we have $\mathcal{K}_{\leq d}=\ker(\varphi)_{\leq d}$.

###### Proof.

One can show by an induction on the degree that our assumptions imply the
equality

$\mathcal{F}_{\leq d}=\mathcal{K}_{\leq d}+\mathcal{P}_{\leq
d-\deg(t_{1})}t_{1}+\cdots+\mathcal{P}_{\leq d-\deg(t_{l})}t_{l},$

where $\mathcal{P}=\mathbb{C}[h_{1},\dots,h_{k}]$. Moreover, the restriction
of $\varphi$ to $\mathcal{P}$ and to
$\mathcal{P}t_{1}+\cdots+\mathcal{P}t_{l}$ is injective. This clearly implies
the desired equality $\mathcal{K}_{\leq d}=\ker(\varphi)_{\leq d}$. ∎

Proof of Proposition 6.5. For a decreasing multidegree
$\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ denote by $\mathcal{T}_{\alpha}$
the set of products of the variables $\pi_{i_{1},i_{2},i_{3}}$,
$\rho_{j_{1},j_{2},j_{3}}$ having multidegree $\alpha$ that correspond to the
elements of multidegree $\alpha$ in the secondary S-generating system of
$\mathbb{C}[V^{3}]^{D_{8}}$ given in Proposition 6.2. For example,
$\mathcal{T}_{1,1,1}=\emptyset$ and
$\mathcal{T}_{4,2,2}=\\{\pi_{3,1,0}\rho_{1,0,1}\rho_{0,1,1},\rho_{1,1,0}^{2}\rho_{1,0,1}^{2}\\}$.
Denote by $\mathcal{K}$ the $GL$-ideal in $\mathcal{F}_{3}$ generated by
$\mathcal{R}_{2,2,2}$, $\mathcal{R}(4)_{4,2}$, $\mathcal{R}(4)_{6,2}$, and
$\mathcal{R}(4)_{4,4}$, and write $\mathcal{H}$ for the ideal in
$\mathcal{F}(4,3)$ generated by
$\pi_{4,0,0},\pi_{0,4,0},\pi_{0,0,4},\rho_{2,0,0},\rho_{0,2,0},\rho_{0,0,2}$.
The subspaces $\mathcal{K}$ and $\mathcal{H}$ are spanned by multihomogeneous
elements of $\mathcal{F}(4,3)$, and we shall write $\mathcal{K}_{\alpha}$,
$\mathcal{H}_{\alpha}$, $\mathcal{F}(4,3)_{\alpha}$ for their components of
multidegree $\alpha$. We shall show that

(12)
$\mathrm{Span}_{\mathbb{C}}\\{\mathcal{T}_{\alpha}\\}+\mathcal{K}_{\alpha}+\mathcal{H}_{\alpha}=\mathcal{F}(4,3)_{\alpha}\text{
for all decreasing }\alpha\in\mathbb{N}_{0}^{3}\text{ with
}\alpha_{1}+\alpha_{2}+\alpha_{3}\leq 10.$

Note that $\mathcal{H}$ is not a $GL$-ideal in $\mathcal{F}(4,3)$, however, it
is preserved by the subgroup $S_{3}$ of $GL_{3}(\mathbb{C})$. For $s\in S_{3}$
we have $s\cdot\mathcal{H}_{\alpha}=\mathcal{H}_{s\cdot\alpha}$ and
$s\cdot\mathcal{K}_{\alpha}=\mathcal{K}_{s\cdot\alpha}$. Thus (12) implies
$\mathcal{F}_{s\cdot\alpha}=\mathrm{Span}_{\mathbb{C}}\\{s\cdot\mathcal{T}_{\alpha}\\}+\mathcal{H}_{s\cdot\alpha}+\mathcal{K}_{s\cdot\alpha}$.
Set

$\mathcal{T}:=\bigsqcup_{\alpha\in\mathbb{N}_{0}^{3},\
\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}}\bigsqcup_{s\in
S_{3}/\mathrm{Stab}_{S_{3}}(\alpha)}s\cdot\mathcal{T}_{\alpha}.$

Then the generators of $\mathcal{H}$ are mapped to a homogeneous system of
parameters in $R:=\mathbb{C}[V^{3}]^{D_{8}}$ and $\mathcal{T}$ is mapped by
$\varphi(4,3)$ to a system of secondary generators of $R$ (see Definition
6.1). Given (12) the above considerations show that the assumptions of Lemma
6.7 hold for $\mathcal{F}=\mathcal{F}(4,3)$, $\varphi=\varphi(4,3)$ and
$d=10$, and therefore by Lemma 6.7 we conclude that $\mathcal{K}_{\leq
10}=\ker(\varphi(4,3))_{\leq 10}$. Now $\ker(\varphi(4,3))$ is generated by
its elements of degree at most $10$ by Proposition 4.1, implying the equality
$\mathcal{K}=\ker(\varphi(4,3))$.

It remains to prove (12). For multidegrees $\alpha$ with $\alpha_{3}=0$ this
was done in the proof of Theorem 5.1. Note also that
$\mathcal{F}_{\alpha}=\\{0\\}$ if $\alpha_{1}+\alpha_{2}+\alpha_{3}$ is odd.

Denote by $\mathcal{M}_{\alpha}$ the products of multidegree $\alpha$ in
$\\{\pi_{\beta},\rho_{\gamma}\mid\beta_{i}<4,\gamma_{j}<2\\}$ (the generators
of $\mathcal{F}(4,3)$ different from
$\pi_{4,0,0},\pi_{0,4,0},\pi_{0,0,4},\rho_{2,0,0},\rho_{0,2,0},\rho_{0,0,2}$).
In the first table below we collect for all decreasing
$\alpha\in\mathbb{N}_{0}^{3}$ with $\alpha_{3}>0$ and
$\sum_{i=1}^{3}\alpha_{i}\leq 6$ even the elements of $\mathcal{M}_{\alpha}$.

$\begin{array}[]{c||c|c}\alpha&\mathcal{T}_{\alpha}&\mathcal{M}_{\alpha}\setminus\mathcal{T}_{\alpha}\\\
\hline\cr\hline\cr(2,1,1)&\pi_{2,1,1},\ \rho_{1,1,0}\rho_{1,0,1}&\\\
\hline\cr(4,1,1)&\pi_{3,1,0}\rho_{1,0,1}&\pi_{3,0,1}\rho_{1,1,0}\\\
\hline\cr(3,2,1)&\rho_{1,1,0}^{2}\rho_{1,0,1},\
\pi_{3,1,0}\rho_{0,1,1}&\pi_{2,1,1}\rho_{1,1,0},\ \pi_{2,2,0}\rho_{1,0,1}\\\
\hline\cr(2,2,2)&&\rho_{1,1,0}\rho_{1,0,1}\rho_{0,1,1},\
\pi_{2,1,1}\rho_{0,1,1},\ \pi_{1,2,1}\rho_{1,0,1},\ \pi_{1,1,2}\rho_{1,1,0}\\\
\hline\cr\end{array}$

Now going multidegree by multidegree we shall show that the products
$\mathcal{M}_{\alpha}\setminus\mathcal{T}_{\alpha}$ are congruent modulo
$\mathcal{H}+\mathcal{K}$ to a $\mathbb{C}$-linear combination of the elements
in $\mathcal{T}_{\alpha}$. Denote by $\mathcal{I}$ the $GL$-ideal of
$\mathcal{F}(4,3)$ generated by $\mathcal{R}_{2,2,2}$ and
$\mathcal{R}(4)_{4,2}$, and we shall write $a\equiv_{\mathcal{I}}b$ for some
$a,b\in\mathcal{F}(4,3)$ if $a-b\in\mathcal{H}+\mathcal{I}$. We shall write
$a\equiv b$ for $a,b\in\mathcal{F}(4,3)$ if $a-b\in\mathcal{H}+\mathcal{K}$.
Note that $\mathcal{I}\subseteq\mathcal{K}$ and therefore
$a\equiv_{\mathcal{I}}b$ implies $a\equiv b$.

There is nothing to do for the multidegree $(2,1,1)$. Taking into account
Theorem 5.1 this shows also that for $d<6$ we have
$\ker(\varphi(4,3))_{d}=\\{0\\}$.

$\mathbf{(4,1,1)}$: The relation (2) in the special case $n=4$, $m=3$ is

(13)
$\mathcal{R}(4)_{4,2}:=\pi_{4,0,0}\rho_{0,2,0}-2\pi_{3,1,0}\rho_{1,1,0}+\pi_{2,2,0}\rho_{2,0,0}.$

Applying $E_{3,2}\in\mathfrak{gl}_{3}(\mathbb{C})$ to $\mathcal{R}(4)_{4,2}$
we get the following element of
$\langle\mathcal{R}(4)_{4,2}\rangle_{GL_{3}(\mathbb{C})}$:

(14)
$\mathcal{R}(4)_{4,2}^{4,1,1}:=\frac{1}{2}E_{3,2}.\mathcal{R}_{4,2,0}=\pi_{4,0,0}\rho_{0,1,1}-\pi_{3,1,0}\rho_{1,0,1}-\pi_{3,0,1}\rho_{1,1,0}+\pi_{2,1,1}\rho_{2,0,0}$

This relation implies that

(15) $\pi_{3,0,1}\rho_{1,1,0}\equiv_{\mathcal{I}}-\pi_{3,1,0}\rho_{1,0,1}.$

$\mathbf{(3,2,1)}$: Consider the following elements of
$\langle\mathcal{R}(4)_{4,2}\rangle_{GL_{3}(\mathbb{C})}$:

(16)
$\displaystyle\mathcal{R}(4)_{4,2}^{3,2,1}(1):=\frac{1}{2}E_{3,1}.\mathcal{R}(4)_{4,2}=-\pi_{3,1,0}\rho_{0,1,1}-3\pi_{2,1,1}\rho_{1,1,0}+\pi_{2,2,0}\rho_{1,0,1}$
$\displaystyle+2\pi_{3,0,1}\rho_{0,2,0}+\pi_{1,2,1}\rho_{2,0,0}$

(17)
$\displaystyle\mathcal{R}(4)_{4,2}^{3,2,1}(2):=E_{2,1}.\mathcal{R}(4)_{4,2}^{4,1,1}=$
$\displaystyle
3\pi_{3,1,0}\rho_{0,1,1}-\pi_{2,1,1}\rho_{1,1,0}-3\pi_{2,2,0}\rho_{1,0,1}$
$\displaystyle+2\pi_{1,2,1}\rho_{2,0,0}-\pi_{3,0,1}\rho_{0,2,0}$

Now
$3\mathcal{R}(4)_{4,2}^{3,2,1}(1)+\mathcal{R}(4)_{4,2}^{3,2,1}(2)\in\mathcal{K}$
implies that

(18) $\pi_{2,1,1}\rho_{1,1,0}\equiv_{\mathcal{I}}0$

and (16) and (18) imply

(19) $\pi_{2,2,0}\rho_{1,0,1}\equiv_{\mathcal{I}}\pi_{3,1,0}\rho_{0,1,1}.$

$\mathbf{(2,2,2)}$: $\mathcal{R}_{2,2,2}\in\mathcal{I}$ implies

(20) $\rho_{1,1,0}\rho_{1,0,1}\rho_{0,1,1}\equiv_{\mathcal{I}}0.$

Set $U:=\langle\mathcal{R}(4)_{4,2}\rangle_{GL_{3}(\mathbb{C})}$. Then $\dim
U_{2,2,2}=3$ (since there are $3$ semistandard tableaux of shape $(4,2)$ and
content $1,1,2,2,3,3$). The table above shows that
$|\mathcal{M}_{2,2,2}\setminus\mathcal{T}_{2,2,2}|=4$, hence by Lemma 6.7 we
conclude that $\dim\ker(\varphi(4,3))_{2,2,2}=4$. It follows that

$\ker(\varphi(4,3))_{2,2,2}=\mathbb{C}\mathcal{R}_{2,2,2}\oplus U_{2,2,2}.$

Consequently, $\ker(\varphi(4,3))_{2,2,2}=\mathcal{I}_{2,2,2}$, and

(21)
$\pi_{2,1,1}\rho_{0,1,1}\equiv_{\mathcal{I}}0,\quad\pi_{1,2,1}\rho_{1,0,1}\equiv_{\mathcal{I}}0,\quad\pi_{1,1,2}\rho_{1,1,0}\equiv_{\mathcal{I}}0.$

The calculations so far show that

$\ker(\varphi(4,3))_{6}=\mathbb{C}\mathcal{R}_{2,2,2}\oplus\langle\mathcal{R}(4)_{4,2}\rangle_{GL_{3}(\mathbb{C})}.$

Moreover, for later reference we mention some consequences of the relations of
degree $6$. The relations (9) in the special case $n=4$ give that

$\pi_{3,1,0}\rho_{1,1,0}\equiv_{\mathcal{I}}0,\quad\pi_{2,2,0}\rho_{1,1,0}\equiv_{\mathcal{I}}0.$

Denote by $A$ the union of the $S_{3}$-orbits of $\pi_{2,1,1}\rho_{1,1,0}$,
$\pi_{2,1,1}\rho_{0,1,1}$, $\rho_{1,1,0}\rho_{1,0,1}\rho_{0,1,1}$,
$\pi_{3,1,0}\rho_{1,1,0}$ and $\pi_{2,2,0}\rho_{1,1,0}$. Each element of $A$
is congruent to zero modulo $\mathcal{H}+\mathcal{I}$ by (18), (20), (21) (and
taking into account that the ideal $\mathcal{H}+\mathcal{I}$ is preserved by
the action of $S_{3}$ on $\mathcal{F}(4,3)$). Denote by $B$ the $S_{3}$-orbit
of $\pi_{2,2,0}\rho_{1,0,1}$. By (19) and as $\mathcal{I}+\mathcal{H}$ is
$S_{3}$-stable, we conclude that each element of $B$ is congruent modulo
$\mathcal{H}+\mathcal{I}$ to an element of the $S_{3}$-orbit of
$\pi_{3,1,0}\rho_{0,1,1}$. Set $C:=\\{\pi_{3,0,1}\rho_{1,1,0}\\}$, by (15) the
element of $C$ is congruent modulo $\mathcal{H}+\mathcal{I}$ to
$-\pi_{3,1,0}\rho_{0,1,1}$. Summarizing, we have that the factor space

(22) $\displaystyle\mathcal{F}(4,3)/(\mathcal{I}+\mathcal{H})\text{ is spanned
over }\mathbb{C}\text{ by products of the variables }\pi_{i_{1},i_{2},i_{3}},\
\rho_{j_{1},j_{2},j_{3}}$ $\displaystyle\text{ not divisible by any element of
}A\cup B\cup C.$

Now we turn to the relations of degree $8$. Every non-zero term in any element
of the $GL$-ideal $\mathcal{I}$ of $\mathcal{F}(4,3)$ generated by
$\mathcal{R}_{2,2,2}$ and $\mathcal{R}(4)_{4,2}$ involves a variable
$\rho_{i,j,k}$, and every non-zero term of any element of $\mathcal{H}$
involves a variable from
$\\{\pi_{4,0,0},\pi_{0,4,0},\pi_{0,0,4},\rho_{2,0,0},\rho_{0,2,0},\rho_{0,0,2}\\}$.
Hence none of $\mathcal{R}(4)_{6,2}$ or $\mathcal{R}(4)_{4,4}$ is contained in
$\mathcal{I}+\mathcal{H}$. Consequently, by basic principles about semisimple
representations we have

(23) $(\mathcal{I}+\mathcal{H})\cap\langle\mathcal{R}(4)_{6,2},\
\mathcal{R}(4)_{4,4}\rangle_{GL_{3}(\mathbb{C})}=\\{0\\}.$

For the multidegrees $\alpha$ with $\sum\alpha_{i}=8$ we shall first show that
each element of $\mathcal{M}_{\alpha}\setminus\mathcal{T}_{\alpha}$ involving
a variable $\rho_{i,j,k}$ is congruent to an element of
$\mathrm{Span}_{\mathbb{C}}\mathcal{T}_{\alpha}$ modulo
$\mathcal{I}+\mathcal{H}$. In fact by (22) it is sufficient to do it for the
elements of the subset $\mathcal{U}_{\alpha}$ of
$\mathcal{M}_{\alpha}\setminus\mathcal{T}_{\alpha}$ not divisible by any
element of $A\cup B\cup C$. Denote by $\mathcal{N}_{\alpha}$ the set of
elements of $\mathcal{M}_{\alpha}\setminus\mathcal{T}_{\alpha}$ of the form
$\pi_{i_{1},i_{2},i_{3}}\pi_{j_{1},j_{2},j_{3}}$. In the table below we
collect for all decreasing $\alpha\in\mathbb{N}_{0}^{3}$ with $\alpha_{3}>0$
and $\sum_{i=1}^{3}\alpha_{i}=8$ the elements of $\mathcal{T}_{\alpha}$,
$\mathcal{U}_{\alpha}$, $\mathcal{N}_{\alpha}$:

$\begin{array}[]{c||c|c|c}\alpha&\mathcal{T}_{\alpha}&\mathcal{U}_{\alpha}\setminus\mathcal{N}_{\alpha}&\mathcal{N}_{\alpha}\\\
\hline\cr\hline\cr(6,1,1)&&&\pi_{3,1,0}\pi_{3,0,1}\\\
\hline\cr(5,2,1)&&&\pi_{3,1,0}\pi_{2,1,1},\ \pi_{3,0,1}\pi_{2,2,0}\\\
\hline\cr(4,3,1)&\rho_{1,1,0}^{3}\rho_{1,0,1}&&\pi_{3,1,0}\pi_{1,2,1},\
\pi_{2,2,0}\pi_{2,1,1},\ \pi_{3,0,1}\pi_{1,3,0}\\\
\hline\cr(4,2,2)&\rho_{1,1,0}^{2}\rho_{1,0,1}^{2},\
\pi_{3,1,0}\rho_{1,0,1}\rho_{0,1,1}&&\pi_{3,1,0}\pi_{1,1,2},\
\pi_{3,0,1}\pi_{1,2,1},\\\ &&&\pi_{2,2,0}\pi_{2,0,2},\ \pi_{2,1,1}^{2}\\\
\hline\cr(3,3,2)&\pi_{3,1,0}\rho_{0,1,1}^{2},\
\pi_{2,1,1}\pi_{1,2,1}&\pi_{1,3,0}\rho_{1,0,1}^{2}&\pi_{3,1,0}\pi_{0,2,2},\
\pi_{3,0,1}\pi_{0,3,1},\\\ &&&\pi_{1,3,0}\pi_{2,0,2},\
\pi_{2,2,0}\pi_{1,1,2}\\\ \hline\cr\end{array}$

We see that $\mathcal{U}_{\alpha}\neq\emptyset$ only for $\alpha=(3,3,2)$, and
then $\mathcal{U}_{3,3,2}=\\{\pi_{1,3,0}\rho_{1,0,1}^{2}\\}$. Consider

(24)
$\displaystyle\mathcal{R}(4)_{4,2}^{3,3,0}:=E_{2,1}.\mathcal{R}(4)_{4,2}^{4,2,0}=$
$\displaystyle-4\pi_{2,2,0}\rho_{1,1,0}$
$\displaystyle+2\pi_{3,1,0}\rho_{0,2,0}+2\pi_{1,3,0}\rho_{2,0,0}$

and

(25)
$\displaystyle\mathcal{R}(4)_{4,2}^{2,3,1}:=E_{3,1}.\mathcal{R}(4)_{4,2}^{3,3,0}=-8\pi_{1,2,1}\rho_{1,1,0}-4\pi_{2,2,0}\rho_{0,1,1}+4\pi_{1,3,0}\rho_{1,0,1}$
$\displaystyle+6\pi_{2,1,1}\rho_{0,2,0}+2\pi_{0,3,1}\rho_{2,0,0}$ (26)
$\displaystyle\equiv_{\mathcal{I}}-4\pi_{2,2,0}\rho_{0,1,1}+4\pi_{1,3,0}\rho_{1,0,1}$

(the congruence (26) follows from (18) and the fact that
$\mathcal{I}+\mathcal{H}$ is $S_{3}$-stable).

The element $\rho_{1,0,1}\mathcal{R}(4)_{4,2}^{2,3,1}\in\mathcal{I}$ shows
that

$\pi_{1,3,0}\rho_{1,0,1}^{2}\equiv_{\mathcal{I}}\pi_{2,2,0}\rho_{0,1,1}\rho_{1,0,1}\equiv_{\mathcal{I}}\pi_{3,1,0}\rho_{0,1,1}^{2}$

(we used also (19)).

At this stage by (23) to prove that each element of $\mathcal{N}_{\alpha}$ is
congruent to an element of $\mathrm{Span}_{\mathbb{C}}\mathcal{T}_{\alpha}$
modulo $\mathcal{K}+\mathcal{H}$ it is sufficient to verify that
$|\mathcal{N}_{\alpha}|$ equals the dimension of the component of multidegree
$\alpha$ of
$\langle\mathcal{R}(4)_{6,2}\rangle_{GL_{3}(\mathbb{C})}\oplus\langle\mathcal{R}(4)_{4,4}\rangle_{GL_{3}(\mathbb{C})}$.
Recall that $\langle\mathcal{R}(4)_{6,2}\rangle_{GL_{3}(\mathbb{C})}\cong
S^{(4,2)}(\mathbb{C}^{3})$ and
$\langle\mathcal{R}(4)_{4,4}\rangle_{GL_{3}(\mathbb{C})}\cong
S^{(4,4)}(\mathbb{C}^{3})$ as $GL_{3}(\mathbb{C})$-modules, and under these
isomorphisms the multihomogeneous component of multidegree $\alpha$
corresponds to the weight subspace with weight $\alpha$. The dimension of
$S^{\lambda}(\mathbb{C}^{3})_{\alpha}$ (the weight subspace with weight
$\alpha$) equals the number of semi-standard tableaux of shape $\lambda$ and
content $\alpha$. The dimensions relevant to us are given in the following
table:

$\begin{array}[]{c||c|c|c|c|c}\alpha&(6,1,1)&(5,2,1)&(4,3,1)&(4,2,2)&(3,3,2)\\\
\hline\cr\hline\cr\dim(S^{(6,2)}(\mathbb{C}^{3})_{\alpha})&1&2&2&3&3\\\
\hline\cr\dim(S^{(4,4)}(\mathbb{C}^{3})_{\alpha})&0&0&1&1&1\end{array}$

In the above table the sum of the dimensions in the column of $\alpha$ agrees
with $|\mathcal{N}_{\alpha}|$, hence we showed that

$\mathcal{K}_{\leq
8}=\langle\mathcal{R}(4)_{6,2}\rangle_{GL_{3}(\mathbb{C})}\oplus\langle\mathcal{R}(4)_{4,4}\rangle_{GL_{3}(\mathbb{C})}\oplus\mathcal{I}_{\leq
8}.$

Finally we turn to the relations of degree $10$. Our study of the degree $8$
relations implies that any product
$\pi_{i_{1},i_{2},i_{3}}\pi_{j_{1},j_{2},j_{3}}$ is congruent modulo
$\mathcal{K}+\mathcal{H}$ to a linear combination of elements from the
$S_{3}$-orbit of $\pi_{2,1,1}\pi_{1,2,1}$ and products of the variables of
$\mathcal{F}(4,3)$ involving at most one factor of the form
$\pi_{k_{1},k_{2},k_{3}}$. Moreover,

$\pi_{2,1,1}\pi_{1,2,1}\rho_{i,j,k}\equiv_{\mathcal{I}}0$

by (18) and (21). Taking into account (22) we conclude that

(27) $\displaystyle\mathcal{F}(4,3)_{10}\text{ is spanned modulo
}\mathcal{H}+\mathcal{K}\text{ by products involving at most one variable}$
$\displaystyle\text{ of the form }\pi_{i_{1},i_{2},i_{3}}\text{ and not
divisible by any element of }A\cup B\cup C.$

For a decreasing $\alpha$ with $\alpha_{1}+\alpha_{2}+\alpha_{3}=10$ denote by
$\mathcal{V}_{\alpha}$ the subset of
$\mathcal{M}_{\alpha}\setminus\mathcal{T}_{\alpha}$ obtained by removing the
products divisible by some $\pi_{i_{1},i_{2},i_{3}}\pi_{j_{1},j_{2},j_{3}}$ or
by any element of $A\cup B\cup C$. By (27) it is sufficient to prove that
every element of $\mathcal{V}_{\alpha}$ is congruent modulo
$\mathcal{K}+\mathcal{H}$ to an element in
$\mathrm{Span}_{\mathbb{C}}\mathcal{T}_{\alpha}$. The table below gives
$\mathcal{T}_{\alpha}$ and $\mathcal{V}_{\alpha}$ for all decreasing $\alpha$
with $\alpha_{3}\neq 0$ and $\sum\alpha_{i}=10$.

$\begin{array}[]{c||c|c}\alpha&\mathcal{T}_{\alpha}&\mathcal{V}_{\alpha}\\\
\hline\cr\hline\cr(8,1,1)&&\\\ \hline\cr(7,2,1)&&\\\ \hline\cr(6,3,1)&&\\\
\hline\cr(6,2,2)&&\\\ \hline\cr(5,4,1)&&\rho_{1,1,0}^{4}\rho_{1,0,1}\\\ &&\\\
\hline\cr(5,3,2)&&\rho_{1,1,0}^{3}\rho_{1,0,1}^{2}\\\ \hline\cr(4,4,2)&&\\\
\hline\cr(4,3,3)&\pi_{3,1,0}\rho_{1,0,1}\rho_{0,1,1}^{2}&\pi_{1,3,0}\rho_{1,0,1}^{3},\
\pi_{1,0,3}\rho_{1,1,0}^{3}\\\ \hline\cr\end{array}$

We see that $\mathcal{V}_{\alpha}$ is non-empty only for
$\alpha\in\\{(5,4,1),\ (5,3,2),\ (4,3,3)\\}$.

$\mathbf{(5,4,1)}$: The relation $\mathcal{R}_{2n-4,4}$ in the special case
$n=4$ is

$\mathcal{R}(4)_{4,4}:=\pi_{4,0,0}\pi_{0,4,0}-4\pi_{3,1,0}\pi_{1,3,0}+3\pi_{2,2,0}^{2}-16(\rho_{1,1,0}^{2}-\rho_{2,0,0}\rho_{0,2,0})^{2}.$

Apply $\frac{1}{4}E_{3,2}$ to $\mathcal{R}(4)_{4,4}$ we get

$\displaystyle\mathcal{R}(4)_{4,4}^{4,3,1}:=\frac{1}{4}E_{3,2}.\mathcal{R}(4)_{4,4}=\pi_{4,0,0}\pi_{0,3,1}-\pi_{3,0,1}\pi_{1,3,0}-3\pi_{3,1,0}\pi_{1,2,1}+3\pi_{2,2,0}\pi_{2,1,1}$
$\displaystyle-16(\rho_{2,0,0}\rho_{0,2,0}-\rho_{1,1,0}^{2})(\rho_{2,0,0}\rho_{0,1,1}-\rho_{1,1,0}\rho_{1,0,1}).$

We deduce from $\mathcal{R}(4)_{4,4}^{4,3,1}\in\mathcal{K}$ that

(28)
$\rho_{1,1,0}^{3}\rho_{1,0,1}\equiv\frac{1}{16}(-\pi_{3,0,1}\pi_{1,3,0}-3\pi_{3,1,0}\pi_{1,2,1}+3\pi_{2,2,0}\pi_{2,1,1}).$

Multiplying (28) by $\rho_{1,1,0}$ and using

$\pi_{1,3,0}\rho_{1,1,0}\equiv 0,\ \pi_{3,1,0}\rho_{1,1,0}\equiv 0,\
\pi_{2,2,0}\rho_{1,1,0}\equiv 0$

we get

$\rho_{1,1,0}^{4}\rho_{1,0,1}\equiv_{\mathcal{I}}0.$

$\mathbf{(5,3,2)}$: Multiplying (28) by $\rho_{1,0,1}$ and using

$\pi_{3,0,1}\rho_{1,0,1}\equiv_{\mathcal{I}}0,\
\pi_{1,2,1}\rho_{1,0,1}\equiv_{\mathcal{I}}0,\
\pi_{2,1,1}\pi_{1,0,1}\equiv_{\mathcal{I}}0$

we get

$\rho_{1,1,0}^{3}\rho_{1,0,1}^{2}\equiv_{\mathcal{I}}0.$

$\mathbf{(4,3,3)}$: Applying the transposition $(1,2)\in S_{3}$ to (19) we get

$\pi_{1,3,0}\rho_{1,0,1}\equiv_{\mathcal{I}}\pi_{2,2,0}\rho_{0,1,1}.$

Multiplying this by $\rho_{1,0,1}^{2}$ we get

$\pi_{1,3,0}\rho_{1,0,1}^{3}\equiv_{\mathcal{I}}\pi_{2,2,0}\rho_{1,0,1}^{2}\rho_{0,1,1}\equiv_{\mathcal{I}}\pi_{3,1,0}\rho_{1,0,1}\rho_{0,1,1}^{2}$

(for the second congruence see (19)). Finally, applying the transposition
$(2,3)\in S_{3}$ to the congruence
$\pi_{1,3,0}\rho_{1,0,1}^{3}\equiv_{\mathcal{I}}\pi_{3,1,0}\rho_{1,0,1}\rho_{0,1,1}^{2}$
and using (15) we obtain

$\pi_{1,0,3}\rho_{1,1,0}^{3}\equiv_{\mathcal{I}}-\pi_{3,1,0}\rho_{1,0,1}\rho_{0,1,1}^{2}.$

$\square$

######  Theorem 6.8.

For arbitrary $m\geq 3$ the kernel of
$\varphi(4,m):\mathcal{F}(4,m)\to\mathbb{C}[V^{m}]^{D_{8}}$ is minimally
generated as a $GL$-ideal by $\mathcal{R}_{2,2,2}$, $\mathcal{R}(4)_{4,2}$,
$\mathcal{R}(4)_{6,2}$, $\mathcal{R}(4)_{4,4}$.

###### Proof.

This is an immediate consequence of Proposition 6.5 and Theorem 4.8. ∎

The symmetric group $S_{3}$ is isomorphic to the dihedral group $D_{6}$ of
order $6$, and the natural $3$-dimensional permutation representation of
$S_{3}$ can be identified with the sum of the trivial representation and the
defining $2$-dimensional representation of $D_{6}$. Therefore [10, Theorem
3.1] dealing with multisymmetric polynomials can be restated in the notation
of the present paper as follows:

######  Theorem 6.9.

For arbitrary $m\geq 3$ the kernel of
$\varphi(3,m):\mathcal{F}(3,m)\to\mathbb{C}[V^{m}]^{D_{6}}$ is minimally
generated as a $GL$-ideal by $\mathcal{R}_{2,2,2}$, $\mathcal{R}(3)_{3,2}$,
$\mathcal{R}(3)_{4,2}$.

######  Remark 6.10.

  * (i)

The method of the present paper would yield a shorter proof of Theorem 6.9
than the proof in [10].

  * (ii)

For $n=2$ the group $D_{4}$ is isomorphic to Klein’s four group, so it is
abelian and its action on $V$ is diagonalizable. In an appropriate basis we
have $\mathbb{C}[V]^{D_{4}}=\mathbb{C}[x^{2},y^{2}]$, and
$\mathbb{C}[V^{m}]=\mathbb{C}[x_{i}x_{j},y_{i}y_{j}\mid 1\leq 1\leq j\leq m]$.
The corresponding $GL$-ideal of relations among the generators of
$\mathbb{C}[V^{m}]^{D_{4}}$ is generated by the two relations
$x_{1}^{2}x_{2}^{2}-(x_{1}x_{2})^{2}=0$ and
$y_{1}^{2}y_{2}^{2}-(y_{1}y_{2})^{2}=0$. For some results on presentations of
rings of invariants of abelian groups see for example [9] and the references
therein.

##  7\. Some computations

Next we determine the $GL_{m}(\mathbb{C})$-module structure of
$\mathbb{C}[V^{m}]^{D_{2n}}$. It turns out that the multiplicities of the
irreducible summands are conveniently expressed in terms of the coefficients
of the Hilbert series of $\mathbb{C}[V^{m}]^{D_{2n}}$. Denote by $h(d)$ the
dimension of the degree $d$ homogeneous component of
$\mathbb{C}[x,y]^{D_{2n}}$. Note that in the formal power series ring
$\mathbb{Z}[[t]]$ we have the equality

(29) $\sum_{d=0}^{\infty}h(d)t^{d}=\frac{1}{(1-t^{2})(1-t^{n})}.$

######  Proposition 7.1.

The multiplicity of $S^{\lambda}(\mathbb{C}^{m})$ as a summand in
$\mathbb{C}[V^{m}]^{D_{2n}}$ is non-zero only if $\mathrm{ht}(\lambda)\leq 2$,
and in this case the multiplicity is

$\begin{cases}h(\lambda_{1}-\lambda_{2})&\text{ if }2\mid\lambda_{2}\\\
h(\lambda_{1}-\lambda_{2}-n)&\text{ if }2\nmid\lambda_{2}.\end{cases}$

###### Proof.

Identify $V^{m}$ with the space $\mathbb{C}^{2\times m}$ of $2\times m$
matrices, endowed with the $GL_{2}(\mathbb{C})\times
GL_{m}(\mathbb{C})$-action $(g,h)\cdot A:=gAh^{-1}$ for $g\in
GL_{2}(\mathbb{C})$, $h\in GL_{m}(\mathbb{C})$, and $A\in\mathbb{C}^{2\times
m}$. This induces an action of $GL_{2}(\mathbb{C})\times GL_{m}(\mathbb{C})$
on $\mathbb{C}[V^{m}]$ via $\mathbb{C}$-algebra automorphisms in the standard
way: $((g,h)\cdot f)(A)=f(g^{-1}Ah)$ for $g\in GL_{2}(\mathbb{C})$, $h\in
GL_{m}(\mathbb{C})$, $A\in\mathbb{C}^{2\times m}$ and $f\in\mathbb{C}[V^{m}]$.
Note that the restriction of this action to the subgroup $GL_{m}(\mathbb{C})$
agrees with the $GL_{m}(\mathbb{C})$-action on $\mathbb{C}[V^{m}]$ considered
in the previous sections. By the Cauchy Formula (see for example [14, Section
9.6.3]) we have

$\mathbb{C}[V^{m}]_{d}\cong\sum_{\lambda\in\mathrm{Par}_{m}(d),\
\mathrm{ht}(\lambda)\leq 2}S^{\lambda}(V^{*})\otimes
S^{\lambda}(\mathbb{C}^{m})$

as $GL_{2}(\mathbb{C})\times GL_{m}(\mathbb{C})$-modules. It follows that we
have the following isomorphism of $GL_{m}(\mathbb{C})$-modules:

$\mathbb{C}[V^{m}]^{D_{2n}}_{d}\cong\sum_{\lambda\in\mathrm{Par}_{m}(d),\
\mathrm{ht}(\lambda)\leq 2}S^{\lambda}(V^{*})^{D_{2n}}\otimes
S^{\lambda}(\mathbb{C}^{m}).$

Note that for a partition $\lambda=(\lambda_{1},\lambda_{2})$, we have the
$D_{2n}$-module isomorphism $S^{\lambda}(V^{*})\cong
S^{\lambda_{1}-\lambda_{2}}(V^{*})$ when $\lambda_{2}$ is even, whereas for
$\lambda_{2}$ odd we have $S^{\lambda}(V^{*})\cong\det\otimes
S^{\lambda_{1}-\lambda_{2}}(V^{*})$, where $\det$ stands for the
$1$-dimensional representation of $D_{2n}$ obtained by composing the
determinant with the $2$-dimensional defining representation of $D_{2n}$. Now
the result follows from the well-known description of $\mathbb{C}[V]\cong
S(V^{*})$ as a $D_{2n}$-module. ∎

For a fixed $n$ Proposition 7.1 and Corollary 4.6 allow us to compute the
multiplicities of the simple $GL_{3}(\mathbb{C})$-module summands of the
homogeneous components of $\ker(\bar{\varphi}(n,3))$ up to degree $2n+2$. We
shall do this computation for the case $n=4$. To simplify notation write
$S(\lambda)$ for $S^{\lambda}(\mathbb{C}^{m})$. Corollary 4.6 yields

(30) $\displaystyle(\bar{\mathcal{D}}(m)\otimes\mathcal{E}(4,m))_{\leq 10}$
$\displaystyle\cong S(0)+S(2)+2S(4)+S(2,2)+2S(6)+S(5,1)$
$\displaystyle+2S(4,2)+3S(8)+S(7,1)+4S(6,2)+S(5,3)+3S(4,4)$
$\displaystyle+3S(10)+2S(9,1)+5S(8,2)+3S(7,3)+5S(6,4)$
$\displaystyle+S(5,2,1)+S(4,2,2)+2S(7,2,1)$
$\displaystyle+2S(6,3,1)+2S(5,4,1)+2S(6,2,2)$
$\displaystyle+S(5,3,2)+2S(4,4,2).$

By Proposition 7.1 and (29) we have

(31) $\displaystyle\mathbb{C}[V^{m}]^{D_{8}}_{\leq 10}$ $\displaystyle\cong
S(0)+S(2)+2S(4)+S(2,2)+2S(6)+S(5,1)$
$\displaystyle+S(4,2)+3S(8)+S(7,1)+2S(6,2)+S(4,4)+3S(10)$
$\displaystyle+2S(9,1)+2S(8,2)+S(7,3)+S(6,4).$

Combining (30) and (31) we get

(32) $\displaystyle\ker(\bar{\varphi}(4,m)_{\leq 10})$ $\displaystyle\cong
S(4,2)+2S(6,2)+2S(4,4)+3S(8,2)+4S(6,4)$
$\displaystyle+S(5,3)+2S(7,3)+S(5,2,1)+S(4,2,2)+2S(7,2,1)$
$\displaystyle+2S(6,3,1)+2S(6,2,2)+2S(5,4,1)+2S(4,4,2)+S(5,3,2).$

An alternative approach to Theorem 6.8 would be to find the highest weight
vectors in the $GL$-ideal generated by $\mathcal{R}_{2,2,2}$,
$\mathcal{R}(4)_{4,2}$, $\mathcal{R}(4)_{6,2}$, $\mathcal{R}(4)_{4,4}$
required by the decomposition (32). However, this seems to be more laborious
than the approach in Section 6 based on the Hironaka decomposition.

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