# standard monomials and invariant theory for arc spaces III: special linear
group

Andrew R. Linshaw  Department of Mathematics, University of Denver
<EMAIL_ADDRESS>and  Bailin Song  School of Mathematical Sciences,
University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

This is the third in a series of papers on standard monomial theory and
invariant theory of arc spaces. For any algebraically closed field $K$, we
prove the arc space analogue of the first and second fundamental theorems of
invariant theory for the special linear group. This is more subtle than the
results for the general linear and symplectic groups obtained in the first two
papers because the arc space of the corresponding affine quotients can be
nonreduced.

###### Key words and phrases:

standard monomial; invariant theory; arc space

A. Linshaw is supported by Simons Foundation Grant #635650 and NSF Grant
DMS-2001484.

B. Song is supported by NSFC No. 11771416

##  1\. Introduction

### Classical invariant theory for the special linear group

Let $SL_{h}(K)$ be the special linear group of degree $h$ over $K$. Let
$W=K^{\oplus h}$ be the standard representation of $SL_{h}(K)$ and
$V=W^{\oplus p}\bigoplus{W^{*}}^{\oplus q}$. The affine coordinate ring of $V$
is $B^{K}=K[a^{(0)}_{il},b^{(0)}_{jl}|\ 0\leq i\leq p,\ 1\leq j\leq p,\ 1\leq
l\leq h]$.

######  Theorem 1.1.

The invariant ring ${B^{K}}^{SL_{h}(K)}$ is generated by

(1.1)
$\begin{split}&X^{(0)}_{ij}=\sum_{l=1}^{h}a^{(0)}_{il}b^{(0)}_{jl},\quad\quad
1\leq i\leq p,\qquad 1\leq j\leq q;\\\
&Y^{(0)}_{u_{1},\dots,u_{h}}=\left|\begin{array}[]{cccc}a_{u_{1}1}^{(0)}&a_{u_{1}2}^{(0)}&\cdots&a_{u_{1}h}^{(0)}\\\
a_{u_{2}1}^{(0)}&a_{u_{2}2}^{(0)}&\cdots&a_{u_{2}h}^{(0)}\\\
\vdots&\vdots&\vdots&\vdots\\\
a_{u_{h}1}^{(0)}&a_{u_{h}2}^{(0)}&\cdots&a_{u_{h}h}^{(0)}\\\
\end{array}\right|,\quad\quad
Z^{(0)}_{v_{1},\dots,v_{h}}=\left|\begin{array}[]{cccc}b_{v_{1}1}^{(0)}&b_{v_{1}2}^{(0)}&\cdots&b_{v_{1}h}^{(0)}\\\
b_{v_{2}1}^{(0)}&b_{v_{2}2}^{(0)}&\cdots&b_{v_{2}h}^{(0)}\\\
\vdots&\vdots&\vdots&\vdots\\\
b_{v_{h}1}^{(0)}&b_{v_{h}2}^{(0)}&\cdots&b_{v_{h}h}^{(0)}\\\
\end{array}\right|,\\\ &1\leq u_{1}<u_{2}<\cdots<u_{h}\leq p,\quad\quad\quad
1\leq v_{1}<v_{2}<\cdots<v_{h}\leq q.\end{split}$

Let $Y^{(0)}_{u,u,u_{3},\dots,u_{h}}=0=Z^{(0)}_{v,v,v_{3}\dots,v_{h}}.$ For a
permutation $\sigma$ of $1,2,\dots,h$, let

$Y^{(0)}_{u_{\sigma(1)},\dots,u_{\sigma(h)}}=\operatorname{sign}(\sigma)Y^{(0)}_{u_{1},\dots,u_{h}},\quad
Z^{(0)}_{v_{\sigma(1)},\dots,v_{\sigma(h)}}=\operatorname{sign}(\sigma)Z^{(0)}_{v_{1},\dots,v_{h}};$

The ideal of relations among the generators is $I_{h}^{+}$ generated by

(1.2)
$\left|\begin{array}[]{cccc}X^{(0)}_{u_{1}v_{1}}&X^{(0)}_{u_{1}v_{2}}&\cdots&X^{(0)}_{u_{1}v_{h}}\\\
X^{(0)}_{u_{2}v_{1}}&X^{(0)}_{u_{2}v_{2}}&\cdots&X^{(0)}_{u_{2}v_{h}}\\\
\vdots&\vdots&\vdots&\vdots\\\
X^{(0)}_{u_{h}v_{1}}&X^{(0)}_{u_{h}v_{2}}&\cdots&X^{(0)}_{u_{h}v_{h}}\\\
\end{array}\right|-Y^{(0)}_{u_{1},\dots,u_{h}}Z^{(0)}_{v_{1},\dots,v_{h}},$
(1.3)
$\sum_{i=0}^{h}(-1)^{i}X^{(0)}_{u_{i}v}Y^{(0)}_{u_{0},\dots,u_{i-1},u_{i+1},\dots,u_{h}},\quad\sum_{j=0}^{h}(-1)^{j}X^{(0)}_{uv_{j}}Z^{(0)}_{v_{0},\dots,v_{j-1},v_{j+1},\dots,v_{h}},$
(1.4)
$\sum_{i=1}^{h}(-1)^{i}Y^{(0)}_{u^{\prime}_{h},\dots,u^{\prime}_{2},u_{i}}Y^{(0)}_{u_{1},\dots,u_{i-1},u^{\prime}_{1},u_{i+1},\dots,u_{h}},\quad\sum_{j=1}^{h}(-1)^{j}Z^{(0)}_{v^{\prime}_{h},\dots,v^{\prime}_{2},v_{j}}Z^{(0)}_{v_{1},\dots,v_{j-1},v^{\prime}_{1},v_{j+1},\dots,v_{h}}.$

The generating set (1.1) for ${B^{K}}^{SL_{h}(K)}$ is known as the first
fundamental theorem of invariant theory (FFT) for $SL_{h}(K)$. The generating
set (1.2)-(1.4) for the ideal of relations is known as the second fundamental
theorem of invariant theory (SFT). In the case $K=\mathbb{C}$, these theorems
are classical and were proven by Weyl [12]. In the general case, the FFT is
due to de Concini and Procesi [2], and the SFT is due to Lakshmibai and Shukla
[7]; see also the book [6] of Lakshmibai and Raghavan for a uniform approach
to these theorems for all the classical groups based on standard monomial
theory.

### Standard monomial theory

Let

$X^{(0)}_{u_{1},\dots,u_{r};v_{1},\dots,v_{r}}=\left|\begin{array}[]{cccc}X^{(0)}_{u_{1}v_{1}}&X^{(0)}_{u_{1}v_{2}}&\cdots&X^{(0)}_{u_{1}v_{r}}\\\
X^{(0)}_{u_{2}v_{1}}&X^{(0)}_{u_{2}v_{2}}&\cdots&X^{(0)}_{u_{2}v_{r}}\\\
\vdots&\vdots&\vdots&\vdots\\\
X^{(0)}_{u_{r}v_{1}}&X^{(0)}_{u_{r}v_{2}}&\cdots&X^{(0)}_{u_{r}v_{r}}\\\
\end{array}\right|.$

There is a partial order for the set

$\\{Y^{(0)}_{u_{1},\dots,u_{h}},Z^{(0)}_{v_{1},\dots,v_{h}},X^{(0)}_{u_{1},\dots,u_{r};v_{1},\dots,v_{r}}|r\leq
h,u_{i}<u_{i+1},v_{j}<v_{j+1}\\}$

given by

  1. (1)

$Y^{(0)}_{u_{1},\dots,u_{h}}\leq
Y^{(0)}_{u^{\prime}_{1},\dots,u^{\prime}_{h}}$ if $u_{i}\leq u^{\prime}_{i}$
for all $1\leq i\leq h$;

  2. (2)

$Z^{(0)}_{v_{1},\dots,v_{h}}\leq
Z^{(0)}_{v^{\prime}_{1},\dots,v^{\prime}_{h}}$ if $u_{i}\leq u^{\prime}_{i}$
for all $1\leq i\leq h$;

  3. (3)

$Y^{(0)}_{u_{1},\dots,u_{h}}\leq
X^{(0)}_{u^{\prime}_{1},\dots,u^{\prime}_{r};v^{\prime}_{1},\cdots,v^{\prime}_{r}}$
if $u_{i}\leq u^{\prime}_{i}$ for all $1\leq i\leq r$;

  4. (4)

$Z^{(0)}_{v_{1},\dots,v_{h}}\leq
X^{(0)}_{u^{\prime}_{1},\dots,u^{\prime}_{r};v^{\prime}_{1},\cdots,v^{\prime}_{r}}$
if $v_{i}\leq v^{\prime}_{i}$ for all $1\leq i\leq r$;

  5. (5)

$X^{(0)}_{u_{1},\dots,u_{r};v_{1},\cdots,v_{r}}\leq
X^{(0)}_{u^{\prime}_{1},\dots,u^{\prime}_{s};v^{\prime}_{1},\cdots,v^{\prime}_{s}}$
if $s\leq r$, $u_{i}\leq u^{\prime}_{i}$ and $v_{i}\leq v^{\prime}_{i}$ for
all $1\leq i\leq s$.

The invariant ring ${B^{K}}^{SL_{h}(K)}$ has a standard monomial basis
(chapter 11 in [6]) with respect to this partially ordered set: the ordered
products $A_{1}A_{2}\cdots A_{k}$ of of the elements $A_{i}$ in the partial
order set with $A_{i}\leq A_{i+1}$ form a basis of ${B^{K}}^{SL_{h}(K)}$. This
is needed in the proof of Theorem 1.1 in [2] and [7].

### Arc spaces

Our main goal is to prove the arc space analogue Theorem 1.1. Recall that for
a scheme $X$ of finite type over $K$, the arc space $J_{\infty}(X)$ is
determined by its functor of points. For every $K$-algebra $A$, we have a
bijection

$\operatorname{Hom}(\operatorname{Spec}A,J_{\infty}(X))\cong\operatorname{Hom}(\operatorname{Spec}A[[t]],X).$

If $f:X\to Y$ is a morphism of schemes, there is an induced morphism of
schemes $f_{\infty}:J_{\infty}(X)\to J_{\infty}(Y)$. If $f$ is a closed
immersion, then $f_{\infty}$ is also a closed immersion.

If $X=\operatorname{Spec}K[x_{1},\dots,x_{n}]$, then
$J_{\infty}(Y)=\operatorname{Spec}K[x^{(k)}_{i}|1\leq i\leq
n,k\in\mathbb{Z}_{\geq 0}]$. The identification is made as follows: for a
$K$-algebra $A$, a morphism $\phi:K[x_{1},\dots,x_{n}]\to A[[t]]$ determined
by $\phi(x_{i})=\sum_{k=0}^{\infty}a_{i}^{(k)}t^{k}$ corresponds to a morphism
$K[x_{i}^{(k)}]\to A$ determined by $x_{i}^{(k)}\to a_{i}^{(k)}$. Note that
$K[x_{1},\dots,x_{n}]$ can naturally be identified with the subalgebra
$K[x^{(0)}_{1},\dots,x^{(0)}_{n}]$ of $K[x_{i}^{(k)}]$, and from now on we use
$x_{i}^{(0)}$ instead of $x_{i}$.

The polynomial ring $K[x^{(k)}_{i}]$ has a derivation $\partial$ defined on
generators by

(1.5) $\partial x_{i}^{(k)}=(k+1)x_{i}^{(k+1)}.$

It will be more convenient to work with the normalized $k$-derivation
$\frac{1}{k!}\partial^{k}$, but this is a priori not well-defined on
$K[x_{i}^{(k)}]$ if $\text{char}\ K$ is positive. However, $\partial$ is well
defined on $\mathbb{Z}[x_{i}^{(k)}]$, and
${\bar{\partial}}^{k}=\frac{1}{k!}\partial^{k}$ maps $\mathbb{Z}[x_{i}^{(k)}]$
to itself. Therefore for any $K$, there is an induced $K$-linear map

(1.6) ${\bar{\partial}}^{k}:K[x_{i}^{(k)}]\to K[x_{i}^{(k)}],$

obtained by tensoring with $K$.

If $X$ is the affine scheme with affine coordinate ring
$K[X]=K[x^{(0)}_{1},\dots,x^{(0)}_{n}]/(f_{1},\dots,f_{r})$, then
$J_{\infty}(X)$ is the affine scheme with affine coordinate ring

$K[J_{\infty}(X)]=K[x_{i}^{(k)}|\ i=1,\dots,n,\ k\in\mathbb{Z}_{\geq
0}]/({\bar{\partial}}^{l}f_{j}|\ j=1,\dots,r,\ l\geq 0).$

For every $f\in K[x^{(0)}_{1},\dots,x^{(0)}_{n}]$, we have

$\phi(f)=\sum_{k=0}^{\infty}({\bar{\partial}}^{k}f)(a_{1}^{(0)},\cdots,a_{n}^{(k)})\,t^{k}.$

Therefore $\phi$ induces a morphism
$K[x^{(0)}_{1},\dots,x^{(0)}_{n}]/(f_{1},\cdots f_{r})\to A[[t]]$ if and only
if

$({\bar{\partial}}^{k}f_{i})(a_{1}^{(0)},\cdots,a_{n}^{(k)})=0,\ \text{for
all}\ i=1,\dots,r,\ k\geq 0.$

If $Y$ is the affine scheme with affine coordinate ring
$K[Y]=K[y^{(0)}_{1},\dots,y^{(0)}_{m}]/(g_{1},\dots,g_{s})$, a morphism
$P:X\to Y$ incudes a ring homomorphism $P^{*}:K[Y]\to K[X]$. Then the induced
homomorphism of arc spaces $P_{\infty}:J_{\infty}(X)\to J_{\infty}(Y)$ is
given by

$P^{*}_{\infty}(y_{i}^{(k)})={\bar{\partial}}^{k}P^{*}(y_{i}^{(0)}).$

In particular, $P^{*}_{\infty}$ commutes with ${\bar{\partial}}^{k}$ for all
$k\geq 0$.

By a theorem of Kolchin [4], $J_{\infty}(X)$ is irreducible whenever $X$ is
irreducible. However, even if $X$ is irreducible and reduced, $J_{\infty}(X)$
need not be reduced. An ideal $\mathcal{I}\subset K[J_{\infty}(X)]$ is said to
be differentially finitely generated if there exists a finite subset
$\\{f_{1},\dots,f_{m}\\}\subset\mathcal{I}$ such that all normalized
derivatives ${\bar{\partial}}^{i}f_{j}$ lie in $\mathcal{I}$, and
$\mathcal{I}$ is generated by $\\{{\bar{\partial}}^{i}f_{j}|\ j=1,\dots,m,\
i\geq 0\\}$. If $\text{char}\ K=0$, the nilradical $\mathcal{N}\subset
K[J_{\infty}(X)]$ is always a differential ideal, that is,
$\partial\mathcal{N}\subseteq\mathcal{N}$; although this fails in
characteristic $p$, the notion of differential finite generation is still
meaningful. The question of when $\mathcal{N}$ is differentially finitely
generated has been raised by several authors [1, 10, 11, 5], but little is
known about this question in general. In our main examples, we will give an
explicit differential finite generating set for $\mathcal{N}$ when
$\text{char}\ K=0$.

### Invariant theory for arc spaces

Let $G$ be an algebraic group over $K$ and let $V$ be a finite-dimensional
$G$-module. Then $J_{\infty}(G)$ is again an algebraic group, and there is an
induced action of $J_{\infty}(G)$ on $J_{\infty}(V)$. The quotient morphism
$V\to V/\\!\\!/G$ induces a morphism $J_{\infty}(V)\to
J_{\infty}(V/\\!\\!/G)$, so we have a morphism

(1.7) $J_{\infty}(V)/\\!\\!/J_{\infty}(G)\to J_{\infty}(V/\\!\\!/G).$

In particular, we have a ring homomorphism

(1.8) $K[J_{\infty}(V/\\!\\!/G)]\rightarrow K[J_{\infty}(V)]^{J_{\infty}(G)}.$

In the case $K=\mathbb{C}$, if $V/\\!\\!/G$ is smooth or a complete
intersection, it was shown in [9] that (1.8) is an isomorphism, although in
general it is neither injective nor surjective.

As above, let $W=K^{\oplus h}$ be the standard representation of $SL_{h}(K)$
and let $V=W^{\oplus p}\bigoplus{W^{*}}^{\oplus q}$, whose affine coordinate
ring is

$B^{K}=K[a^{(0)}_{il},b^{(0)}_{jl}|\ 0\leq i\leq p,\ 1\leq j\leq p,\ 1\leq
l\leq h].$

The action of $SL_{h}(K)$ on $V$ induces the action of $J_{\infty}(SL_{h}(K))$
on $J_{\infty}(V)$. Let

$\mathcal{S}_{h}=\\{a^{(k)}_{il},b^{(k)}_{jl}|\ 1\leq i\leq p,\ 1\leq j\leq
q,\ 1\leq l\leq h,\ k\in\mathbb{Z}_{\geq 0}\\}.$

The affine coordinate ring of $J_{\infty}(V)$ is
$\mathfrak{B}^{K}=K[\mathcal{S}_{h}]$, so $J_{\infty}(SL_{h}(K))$ acts on
$\mathfrak{B}^{K}$.

Let $\mathfrak{B}=\mathbb{Z}[\mathcal{S}_{h}]$, which has a derivation
$\partial$ defined on generators by $\partial
a^{(k)}_{ij}=(k+1)a^{(k+1)}_{ij}$ and $\partial
b^{(k)}_{ij}=(k+1)b^{(k+1)}_{ij}$. As above,
$\frac{1}{k!}\partial^{k}:\mathfrak{B}\to\mathfrak{B}$ is well defined and
induces a linear map
${\bar{\partial}}^{k}:\mathfrak{B}^{K}\to\mathfrak{B}^{K}$, where
$\mathfrak{B}^{K}=\mathfrak{B}\otimes K$.

Our main result is the following.

######  Theorem 1.2.

Let ${\mathfrak{B}^{K}}^{J_{\infty}({SL}_{h}(K))}$ be the ring of invariants
under the action of $J_{\infty}(SL_{h}(K))$ on $\mathfrak{B}^{K}$.

  1. (1)

${\mathfrak{B}^{K}}^{J_{\infty}({SL}_{h}(K))}$ is generated by

$\displaystyle{\bar{\partial}}^{k}X^{(0)}_{ij},\quad{\bar{\partial}}^{k}Y^{(0)}_{u_{1},\dots,u_{h}},\quad{\bar{\partial}}^{k}Z^{(0)}_{v_{1},\dots,v_{h}},\quad
k\in\mathbb{Z}_{\geq 0}$

with $1\leq i\leq p,\ 1\leq j\leq q,\ 1\leq u_{1}<u_{2}<\cdots<u_{h}\leq p,\
1\leq v_{1}<v_{2}<\cdots<v_{h}\leq q$.

  2. (2)

The ideal of relations among the generators in (1) is generated by

(1.9)
${\bar{\partial}}^{n}\left|\begin{array}[]{cccc}X^{(0)}_{u_{1}v_{1}}&X^{(0)}_{u_{1}v_{2}}&\cdots&X^{(0)}_{u_{1}v_{h}}\\\
X^{(0)}_{u_{2}v_{1}}&X^{(0)}_{u_{2}v_{2}}&\cdots&X^{(0)}_{u_{2}v_{h}}\\\
\vdots&\vdots&\vdots&\vdots\\\
X^{(0)}_{u_{h}v_{1}}&X^{(0)}_{u_{h}v_{2}}&\cdots&X^{(0)}_{u_{h}v_{h}}\\\
\end{array}\right|-\sum_{k=0}^{n}{\bar{\partial}}^{k}Y^{(0)}_{u_{1},\dots,u_{h}}{\bar{\partial}}^{n-k}Z^{(0)}_{v_{1},\dots,v_{h}},\qquad
n\geq 0,$

(1.10)
$\begin{split}&\sum_{k=0}^{n}\sum_{i=0}^{h}(-1)^{i}{\bar{\partial}}^{k}X^{(0)}_{u_{i}v}{\bar{\partial}}^{n-k}Y^{(0)}_{u_{0},\dots,u_{i-1},u_{i+1},\dots,u_{h}},\qquad
n\geq 0,\\\
&\sum_{k=0}^{n}\sum_{j=0}^{h}(-1)^{j}{\bar{\partial}}^{k}X^{(0)}_{uv_{j}}{\bar{\partial}}^{n-k}Z^{(0)}_{v_{0},\dots,v_{j-1},v_{j+1},\dots,v_{h}},\qquad
n\geq 0,\end{split}$

and for any $0\leq l<i\leq h$,

(1.11)
$\sum_{k=0}^{n}\sum_{\sigma}C_{k}^{l}\frac{\operatorname{sign}(\sigma)}{h!i!}{\bar{\partial}}^{n-k}Y^{(0)}_{u^{\prime}_{h},\dots,u^{\prime}_{i+1},u_{\sigma(h+i)},\dots,u_{\sigma(h+1)}}{\bar{\partial}}^{k}Y^{(0)}_{u_{\sigma(h)},\dots,u_{\sigma(1)}};$
(1.12)
$\sum_{k=0}^{n}\sum_{\sigma}C_{k}^{l}\frac{\operatorname{sign}(\sigma)}{h!i!}{\bar{\partial}}^{n-k}Z^{(0)}_{v^{\prime}_{h},\dots,v^{\prime}_{i+1},v_{\sigma(h+i)},\dots,v_{\sigma(h+1)}}{\bar{\partial}}^{k}Z^{(0)}_{v_{\sigma(h)},\dots,v_{\sigma(1)}}.$

Here the summations are over all permutations $\sigma$ of $1,2,\dots,h+i$.

  3. (3)

${\mathfrak{B}^{K}}^{J_{\infty}(SL_{h}(K))}$ has a $K$-basis given by standard
monomials of $\mathcal{J}_{h}^{+}$.

The first statement in Theorem 1.2 shows that the map (1.8) is surjective.
Relations (1.10) are obtained from relations (1.2) and (1.3), and relations
(1.11) and (1.12) for $l=0$ are obtained from relations (1.4), by taking
${\bar{\partial}}^{n}$. However, relations (1.11) and (1.12) for $0<l<h$ are
not consequences of (1.2) and (1.3) and their derivatives. Therefore (1.8) is
not injective if $\text{max}(p,q)-2>h$. We will see later that when
$\text{char}\ K=0$, the nilradical $\mathcal{N}$ in the affine coordinate ring
$K[J_{\infty}(V/\\!\\!/SL_{h}(K)]$ coincides with the kernel of (1.8).
Therefore $\mathcal{N}$ is generated by the relations (1.11) and (1.12) in the
case $n=0$ and $0<l<h$, together with their normalized derivatives; in
particular, $\mathcal{N}$ is differentially finitely generated.

The main idea in the proof is to construct a standard monomial basis for
${\mathfrak{B}^{K}}^{J_{\infty}({SL}_{h}(K))}$ which extends the above
standard monomial basis for the subring
${B^{K}}^{SL_{h}(K)}\subseteq{\mathfrak{B}^{K}}^{J_{\infty}({SL}_{h}(K))}$.
Note that Theorem 1.2 generalizes the results in Section 4 of [9], which deal
with the following special cases for $K=\mathbb{C}$.

  1. (1)

$p<n$ and $q\leq n$, or $p=n$ and $q<n$, so that $V/\\!\\!/SL_{h}(\mathbb{C})$
is an affine space,

  2. (2)

$p=n=q$, so that $V/\\!\\!/SL_{h}(\mathbb{C})$ is a hypersurface,

  3. (3)

$p=n+1$ and $q<n$, or $p<n$ and $q=n+1$, so that $V/\\!\\!/SL_{h}(\mathbb{C})$
is a complete intersection.

##  2\. Standard monomials

### A ring and its derivation

Let $\mathfrak{R}=\mathbb{Z}[x^{(k)}_{ij}|\ 1\leq i\leq p,\ 1\leq j\leq q,\
k\in\mathbb{Z}_{\geq 0}]$. For $h>0$, let $\mathcal{J}_{h}^{L}$ be the set of
sequences

(2.1) $J^{L}={\bar{\partial}}^{k}(u_{h},\dots,u_{2},u_{1}|,\quad\text{with
}1\leq u_{1}<u_{2}<\cdots<u_{h}\leq p,\,k\in\mathbb{Z}_{\geq 0},$

and $\mathcal{J}_{h}^{R}$ be the set of sequences

(2.2) $J^{R}={\bar{\partial}}^{k}|v_{1},v_{2},\dots,v_{h}),\quad\text{with
}1\leq v_{1}<v_{2}<\cdots<v_{h}\leq q,\,k\in\mathbb{Z}_{\geq 0}.$

We call $wt(J^{L})=wt(J^{R})=k$ the weights of $J^{L}$ and $J^{R}$.

For a permutation $\sigma$ of $1,2,\dots,h$, let

${\bar{\partial}}^{k}|u_{\sigma(h)},\dots,u_{\sigma(2)},u_{\sigma(1)})=\operatorname{sign}(\sigma)J^{L},\quad{\bar{\partial}}^{k}|v_{\sigma(1)},v_{\sigma(2)},\dots,v_{\sigma(h)})=\operatorname{sign}(\sigma)J^{R}.$

If there is $1\leq i<j\leq h$ with $u_{i}=u_{j}$, let
${\bar{\partial}}^{k}(u_{h},\dots,u_{2},u_{1}|=0$. If there is $1\leq i<j\leq
h$ with $v_{i}=v_{j}$, let ${\bar{\partial}}^{k}|v_{1},v_{2},\dots,v_{h})=0$.

There is a derivation $\partial$ on the polynomial ring
$\mathfrak{R}^{+}=\mathfrak{R}[\mathcal{J}_{h}^{L}\cup\mathcal{J}_{h}^{R}]$
given by

$\displaystyle\partial x^{(k)}_{ij}$ $\displaystyle=$
$\displaystyle(k+1)x^{(k+1)}_{ij},$
$\displaystyle\partial\,{\bar{\partial}}^{k}(u_{h},\dots,u_{2},u_{1}|$
$\displaystyle=$
$\displaystyle(k+1){\bar{\partial}}^{k+1}(u_{h},\dots,u_{2},u_{1}|,$
$\displaystyle\partial\,{\bar{\partial}}^{k}|v_{1},v_{2},\dots,v_{h})$
$\displaystyle=$
$\displaystyle(k+1){\bar{\partial}}^{k+1}|v_{1},v_{2},\dots,v_{h}).$

Let ${\bar{\partial}}^{k}=\frac{1}{k!}\partial^{k}$ be the normalized
$k$-derivation. The following propositions are easy to verify.

######  Proposition 2.1.

For any $a,b\in\mathfrak{R}^{+}$,

${\bar{\partial}}^{l}(ab)=\sum_{i=0}^{l}{\bar{\partial}}^{i}a\,{\bar{\partial}}^{l-i}b$

and ${\bar{\partial}}^{l}a\in\mathfrak{R}^{+}$.

Let

(2.3)
$B=\left|\begin{array}[]{cccc}x^{(0)}_{u_{1}v_{1}}&x^{(0)}_{u_{1}v_{2}}&\cdots&x^{(0)}_{u_{1}v_{h}}\\\
x^{(0)}_{u_{2}v_{1}}&x^{(0)}_{u_{2}v_{2}}&\cdots&x^{(0)}_{u_{2}v_{h}}\\\
\vdots&\vdots&\vdots&\vdots\\\
x^{(0)}_{u_{h}v_{1}}&x^{(0)}_{u_{h}v_{2}}&\cdots&x^{(0)}_{u_{h}v_{h}}\\\
\end{array}\right|.$

######  Proposition 2.2.

(2.4) ${\bar{\partial}}^{n}B=\sum_{\begin{subarray}{c}n_{1}+\cdots+n_{h}=n\\\
n_{i}\in\mathbb{Z}_{\geq
0}\end{subarray}}\sum_{\sigma}\operatorname{sign}(\sigma)x^{(n_{1})}_{u_{1}v_{\sigma(1)}}x^{(n_{2})}_{u_{2}v_{\sigma(2)}}\cdots
x^{(n_{h})}_{u_{h}v_{\sigma(h)}}.$

Let $\mathfrak{R}^{+}_{h}=\mathfrak{R}^{+}/\mathcal{I}_{h}^{+}$, where
$\mathcal{I}_{h}^{+}$ is the ideal generated by

(2.5)
${\bar{\partial}}^{n}B-{\bar{\partial}}^{n}({\bar{\partial}}^{0}(u_{h},\dots,u_{2},u_{1}|\,\,\,{\bar{\partial}}^{0}|v_{1},v_{2},\dots,v_{h}))=0.$
(2.6)
${\bar{\partial}}^{n}\sum_{i=0}^{h}(-1)^{i}x^{(0)}_{u_{i}v}\,{\bar{\partial}}^{0}(u_{h},\dots,u_{i+1},u_{i-1},\dots,u_{0}|=0;$
(2.7)
${\bar{\partial}}^{n}\sum_{j=0}^{h}(-1)^{j}x^{(0)}_{uv_{j}}\,{\bar{\partial}}^{0}|v_{0},\dots,v_{j-1},v_{j+1},\dots,v_{h})=0;$

For any $0\leq l<i$,

(2.8)
${\bar{\partial}}^{n}\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{h!i!}{\bar{\partial}}^{0}(u^{\prime}_{h},\dots,u^{\prime}_{i+1},u_{\sigma(h+i)},\dots,u_{\sigma(h+1)}|\,\,{\bar{\partial}}^{l}(u_{\sigma(h)},\dots,u_{\sigma(1)}|=0;$
(2.9)
${\bar{\partial}}^{n}\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{h!i!}{\bar{\partial}}^{0}|v^{\prime}_{h},\dots,v^{\prime}_{i+1},v_{\sigma(h+i)},\dots,v_{\sigma(h+1)})\,\,{\bar{\partial}}^{l}|v_{\sigma(h)},\dots,v_{\sigma(1)})=0.$

Here the summations are over all permutations $\sigma$ of $1,2,\dots,h+i$.

Then $\mathfrak{R}^{+}_{h}$ inherits the derivation $\partial$ and normalized
$k$-derivation ${\bar{\partial}}^{k}$ from $\mathfrak{R}^{+}$.

### Generators

Let the sequence

(2.10)
$J={\bar{\partial}}^{k}(u_{h},\dots,u_{2},u_{1}|v_{1},v_{2},\dots,v_{h})$

with $1\leq u_{i}<u_{i+1}\leq p$ and $1\leq v_{i}<v_{i+1}\leq q$ represent
${\bar{\partial}}^{k}B\in\mathfrak{R}$, the $k^{\text{th}}$ normalized
derivative of the minor $B$ of the form of 2.3. We call $wt(J)=k$ the weight
of $J$ and call $sz(J)=h$ the size of $J$. Let $\mathcal{J}$ be the set of
these sequences and

$\mathcal{J}_{h}=\\{J\in\mathcal{J}|\ sz(J)\leq h\\}$

be the set of elements in $\mathcal{J}$ with size less than or equal to $h$.
Let

$\mathcal{J}_{h}^{+}=\mathcal{J}_{h-1}\cup\mathcal{J}_{h}^{L}\cup\mathcal{J}_{h}^{R}.$

$\mathfrak{R}_{h}^{+}$ is generated by $\mathcal{J}_{h}^{+}$. Let

$J(L)={\bar{\partial}}^{0}(u_{h},\dots,u_{1}|\in\mathcal{J}^{L}_{h},\quad
J(R)={\bar{\partial}}^{0}|v_{1},\dots,v_{h})\in\mathcal{J}^{R}_{h^{\prime}}$

be the left hand part and the right hand part of $J$, respectively.

Let $\mathcal{E}$ be the set of sequences

(2.11)
$E=((u_{h},k_{h}),\dots,(u_{2},k_{2}),(u_{1},k_{1})|(v_{1},l_{1}),(v_{2},l_{2}),\dots,(v_{h},l_{h}))$

with $1\leq u_{i}\leq p$, $u_{i}\neq u_{j}$ if $i\neq j$; $1\leq v_{i}\leq q$,
$v_{i}\neq v_{j}$ if $i\neq j$ and $k_{i},l_{j}\in\mathbb{Z}_{\geq 0}$. Let

$||E||=\frac{1}{n!}\partial^{n}(u_{\sigma(h)},\dots,u_{\sigma(2)},u_{\sigma(1)}|v_{\sigma^{\prime}(1)},v_{\sigma^{\prime}(2)},\dots,v_{\sigma^{\prime}(h)})\in\mathcal{J}.$

Here $n=\sum k_{i}+\sum l_{i}$ and $\sigma$, $\sigma^{\prime}$ are
permutations of $1,2,\dots,h$ such that $u_{\sigma(i)}<u_{\sigma(i+1)}$ and
$v_{\sigma(i)}<v_{\sigma(i+1)}$. Let

$wt(E)=wt(||E||),\quad\quad sz(E)=sz(||E||).$

Let

$\mathcal{E}_{h}=\\{E\in\mathcal{E}|\,\,sz(E)\leq h\\}.$

For $h>0$, let $\mathcal{E}_{h}^{L}$ be the set of sequences

(2.12) $E^{L}=((u_{h},k_{h}),\dots,(u_{2},k_{2}),(u_{1},k_{1})|,\quad 1\leq
u_{i}\leq p,\,k_{i}\geq\mathbb{Z}_{\geq 0}$

with $1\leq u_{i}\leq p$ and $u_{i}\neq u_{j}$ if $i\neq j$. Similarly, let
$\mathcal{E}_{h}^{R}$ be the set of sequences

(2.13) $E^{R}=|(v_{1},l_{1}),(v_{2},l_{2}),\dots,(v_{h},l_{h})),\quad 1\leq
v_{j}\leq q,\,l_{j}\geq\mathbb{Z}_{\geq 0}$

with $1\leq v_{i}\leq q$ and $v_{i}\neq v_{j}$ if $i\neq j$. Let

$||E^{L}||={\bar{\partial}}^{k}(u_{\sigma(h)},\dots,u_{\sigma(2)},u_{\sigma(1)}|\in\mathcal{J}^{L}_{h},\quad||E^{L}||={\bar{\partial}}^{k}|v_{\sigma^{\prime}(1)},v_{\sigma^{\prime}(h)},\dots,v_{\sigma^{\prime}(h)})\in\mathcal{J}^{R}_{h}.$

Here $k=\sum k_{i}$, $l=\sum k_{j}$, $\sigma$ and $\sigma^{\prime}$ are
permutations of $1,2,\dots,h$ such that $u_{\sigma(i)}<u_{\sigma(i+1)}$ and
$v_{\sigma^{\prime}(i)}<v_{\sigma^{\prime}(i+1)}$. Let

$wt(E^{L})=wt(||E^{L}||),\quad wt(E^{R})=wt(||E^{R}||).$

Let
$\mathcal{E}_{h}^{+}=\mathcal{E}_{h}^{L}\cup\mathcal{E}_{h}^{R}\cup\mathcal{E}_{h-1}$.
For $J\in\tilde{\mathcal{J}}_{h}^{+}$, let

$\mathcal{E}(J)=\\{E\in\mathcal{E}_{h}^{+}|\,\,||E||=J\\}.$

We can use the elements in $\mathcal{E}(J)$ to represent $J$.

Let

$F:\mathcal{E}^{L}_{h}\times\mathcal{E}^{R}_{h}\to\mathcal{E},\quad
F(E^{L},E^{R})=E.$

Let $E(L)=E^{L}$ and $E(R)=E^{R}$ be the left hand part and the right hand
part of $E$, respectively. Then for any $E\in\mathcal{E}$, $F(E(L),E(R))=E$.

### Ordering

For any set $\mathcal{S}$, let $\mathcal{M}(\mathcal{S})$ be the set of
ordered products of elements of $\mathcal{S}$. If $\mathcal{S}$ is an ordered
set, we order the set $\mathcal{M}(\mathcal{S})$ lexicographically, that is

$S_{1}S_{2}\cdots S_{m}\prec S^{\prime}_{1}S^{\prime}_{2}\cdots
S^{\prime}_{n}\quad\text{ if }S_{i}=S^{\prime}_{i},\ i<i_{0},\text{ with
}S_{i_{0}}\prec S^{\prime}_{i_{0}}\text{ or }i_{0}=m+1,\ n>m.$

We order $\mathcal{M}(\mathbb{Z})$, the set of ordered products of integers,
lexicographically.

We define an ordering for the set $\mathcal{J}_{h}^{+}$:

  1. (1)

${\bar{\partial}}^{k}(u_{r},\dots,u_{2},u_{1}|v_{1},v_{2},\dots,v_{r})\prec{\bar{\partial}}^{k^{\prime}}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{2},u^{\prime}_{1}|v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h^{\prime}})$,
if $h^{\prime}<r$, or $h^{\prime}=r$ and $k<k^{\prime}$, or $h^{\prime}=r$,
$k=k^{\prime}$ and $u_{r}\cdots u_{1}v_{r}\cdots v_{1}\prec
u^{\prime}_{r}\cdots u^{\prime}_{1}v^{\prime}_{r}\cdots v^{\prime}_{1}$.

  2. (2)

${\bar{\partial}}^{k}(u_{h},\dots,u_{2},u_{1}|\prec{\bar{\partial}}^{k^{\prime}}|v_{1},v_{2},\dots,v_{h})\prec{\bar{\partial}}^{k^{\prime\prime}}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{2},u^{\prime}_{1}|v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h^{\prime}})$,

  3. (3)

${\bar{\partial}}^{k}(u_{h},\dots,u_{2},u_{1}|\prec{\bar{\partial}}^{k^{\prime}}(u^{\prime}_{h},\dots,u^{\prime}_{2},u^{\prime}_{1}|$,
if $k<k^{\prime}$, or $k=k^{\prime}$ and $u_{h}\cdots u_{1}\prec
u^{\prime}_{h}\cdots u^{\prime}_{1}$;
${\bar{\partial}}^{k}|v_{1},v_{2},\dots,v_{h})\prec{\bar{\partial}}^{k^{\prime}}|v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h})$,
if $k<k^{\prime}$, or $k=k^{\prime}$ and $v_{h}\cdots v_{1}\prec
v^{\prime}_{h}\cdots v^{\prime}_{1}$.

We order $\mathcal{M}(J_{h}^{+})$, the words of elements of
$\mathcal{J}_{h}^{+}$ lexicographically.

We order the pairs $(u,h)\in\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}$ by

$(u,h)\leq(u^{\prime},h^{\prime}),\text{ if }h<h^{\prime}\text{ or
}h=h^{\prime}\text{ and }u\leq u^{\prime}.$

We order $\mathcal{M}(\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0})$, the
words of $\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}$ lexicographically.

There is an ordering for the set $\mathcal{E}_{h}^{+}$:

  1. (1)

$((u_{r},k_{r}),\dots,(u_{1},k_{1})|(v_{1},l_{1}),\dots,(v_{r},l_{r}))\prec((u^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}}),\dots,(u^{\prime}_{1},k^{\prime}_{1})|(v^{\prime}_{1},l^{\prime}_{1}),\dots,(v^{\prime}_{h^{\prime}},l^{\prime}_{h^{\prime}}))$,
if $r>h^{\prime}$, or $r=h^{\prime}$ and
$\sum(k_{i}+l_{i})<\sum(k^{\prime}_{i}+l^{\prime}_{i})$, or $r=h^{\prime}$,
$\sum(k_{i}+l_{i})=\sum(k^{\prime}_{i}+l^{\prime}_{i})$ and
$(u_{r},k_{r})\cdots(u_{1},k_{1})(v_{r},l_{r})\cdots(v_{1},l_{1})\prec(u^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}})\cdots(u^{\prime}_{1},k^{\prime}_{1})(v^{\prime}_{h^{\prime}},l^{\prime}_{h^{\prime}})\cdots(v^{\prime}_{1},l^{\prime}_{1})$,

  2. (2)

$((u_{h},k_{h}),\dots,(u_{1},k_{1})|\prec|(v_{1},l_{1}),\dots,(v_{h},l_{h}))\prec((u^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}}),\dots,(u^{\prime}_{1},k^{\prime}_{1})|(v^{\prime}_{1},l^{\prime}_{1}),\dots,(v^{\prime}_{h^{\prime}},l^{\prime}_{h^{\prime}}))$,

  3. (3)

$((u_{h},k_{h}),\dots,(u_{1},k_{1})|\prec((u^{\prime}_{h},k^{\prime}_{h}),\dots,(u^{\prime}_{1},k^{\prime}_{1})|$
if $\sum k_{i}<\sum k^{\prime}_{i}$ or $\sum k_{i}=\sum k^{\prime}_{i}$ and
$(u_{h},k_{h})\cdots(u_{1},k_{1})\prec(u^{\prime}_{h},k^{\prime}_{h})\cdots(u^{\prime}_{1},k^{\prime}_{1})$,

  4. (4)

$|(v_{1},l_{1}),\dots,(v_{h},l_{h}))\prec|(v^{\prime}_{1},l^{\prime}_{1}),\dots,(v^{\prime}_{h},l^{\prime}_{h}))$
if $\sum l_{i}<\sum l^{\prime}_{i}$ or $\sum l_{i}=\sum l^{\prime}_{i}$ and
$(v_{h},l_{h})\cdots(v_{1},l_{1})\prec(v^{\prime}_{h},l^{\prime}_{h})\cdots(v^{\prime}_{1},l^{\prime}_{1})$.

There is a partial ordering for the set $\mathcal{E}_{h}^{+}$:

  1. (1)

$((u_{r},k_{r}),\dots,(u_{1},k_{1})|(v_{1},l_{1}),\dots,(v_{r},l_{r}))\leq((u^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}}),\dots,(u^{\prime}_{1},k^{\prime}_{1})|(v^{\prime}_{1},l^{\prime}_{1}),\dots,(v^{\prime}_{h^{\prime}},l^{\prime}_{h^{\prime}}))$,
if $h^{\prime}\leq r$ and $(u_{i},k_{i})\leq(u^{\prime}_{i},k^{\prime}_{i})$,
$(v_{i},l_{i})\leq(v^{\prime}_{i},l^{\prime}_{i})$, for $1\leq i\leq
h^{\prime}$,

  2. (2)

$((u_{h},k_{h}),\dots,(u_{1},k_{1})|\leq((u^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}}),\dots,(u^{\prime}_{1},k^{\prime}_{1})|(v^{\prime}_{1},l^{\prime}_{1}),\dots,(v^{\prime}_{h^{\prime}},l^{\prime}_{h^{\prime}}))$
if $(u_{i},k_{i})\leq(u^{\prime}_{i},k^{\prime}_{i})$, for $1\leq i\leq
h^{\prime}$;

  3. (3)

$|(v_{1},l_{1}),\dots,(v_{h},l_{h}))\leq((u^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}}),\dots,(u^{\prime}_{1},k^{\prime}_{1})|(v^{\prime}_{1},l^{\prime}_{1}),\dots,(v^{\prime}_{h^{\prime}},l^{\prime}_{h^{\prime}}))$
if $(v_{i},l_{i})\leq(v^{\prime}_{i},l^{\prime}_{i})$, for $1\leq i\leq
h^{\prime}$;

  4. (4)

$((u_{h},k_{h}),\dots,(u_{1},k_{1})|\leq((u^{\prime}_{h},k^{\prime}_{h}),\dots,(u^{\prime}_{1},k^{\prime}_{1})|$
if $(u_{i},k_{i})\leq(u^{\prime}_{i},k^{\prime}_{i})$, for $1\leq i\leq h$;

  5. (5)

$|(v_{1},l_{1}),\dots,(v_{h},l_{h}))\leq|(v^{\prime}_{1},l^{\prime}_{1}),\dots,(v^{\prime}_{h},l^{\prime}_{h}))$
if $(v_{i},l_{i})\leq(v^{\prime}_{i},l^{\prime}_{i})$, for $1\leq i\leq h$;

The following statements are immediate from the definition of the partial
order.

  1. (1)

For any $E,E^{\prime}\in\mathcal{E}$ with $h^{\prime}=sz(E^{\prime})$, if
$E\leq E^{\prime}$, then $E(h^{\prime})(L)\leq E^{\prime}(L)$ and
$E(h^{\prime})(R)\leq E^{\prime}(R)$;

  2. (2)

For $E_{1},E_{2}\in\mathcal{E}^{L}_{h}$, if $E_{1}\leq E_{2}$, then for any
$E_{3}\in\mathcal{E}^{R}_{h}$, $F(E_{1},E_{3})\leq F(E_{2},E_{3})$.

  3. (3)

For $E_{1},E_{2}\in\mathcal{E}^{R}_{h}$, if $E_{1}\leq E_{2}$, then for any
$E_{3}\in\mathcal{E}^{L}_{h}$, $F(E_{3},E_{1})\leq F(E_{3},E_{2})$.

######  Lemma 2.3.

If $E\leq E^{\prime}$, then $||E||\prec||E^{\prime}||$.

###### Proof.

If $E\in\mathcal{E}_{h}^{L}$ and
$E^{\prime}\in\mathcal{E}_{h}^{R}\cup\mathcal{E}_{h-1}$ or
$E\in\mathcal{E}_{h}^{R}$ and $E^{\prime}\in\mathcal{E}_{h=1}$, then
$||E||\prec||E^{\prime}||$. Otherwise, $E,E^{\prime}\in\mathcal{E}^{L}_{h}$,
$E,E^{\prime}\in\mathcal{E}^{R}_{h}$ or $E,E^{\prime}\in\mathcal{E}_{h-1}$, so
if $wt(E)<wt(E^{\prime})$, then $||E||\prec||E^{\prime}||$.

If $E,E^{\prime}\in\mathcal{E}^{L}_{h}$ and $wt(E)=wt(E^{\prime})$, we must
have $k_{i}=k_{i}^{\prime}$, so $u_{i}\leq u^{\prime}_{i}$. If
$E,E^{\prime}\in\mathcal{E}^{R}_{h}$ and $wt(E)=wt(E^{\prime})$, we must have
$l_{j}=l_{j}^{\prime}$, so $v_{j}\leq v^{\prime}_{j}$. If
$E,E^{\prime}\in\mathcal{E}_{h-1}$ and $sz(E)=sz(E^{\prime})$, we must have
$k_{i}=k_{i}^{\prime}$ and $l_{j}=l_{j}^{\prime}$. Then $u_{i}\leq
u^{\prime}_{i}$ and $v_{j}\leq v^{\prime}_{j}$, and we have
$||E||\prec||E^{\prime}||$. ∎

### Straightening relations

######  Lemma 2.4.

For any $0\leq l<i$,

(2.14)
$\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{h!i!}{\bar{\partial}}^{l}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i+1},u_{\sigma(h+i)},\dots,u_{\sigma(h+1)}|v_{1},\dots
v_{h^{\prime}})\,\,{\bar{\partial}}^{0}(u_{\sigma(h)},\dots,u_{\sigma(1)}|=0;$
(2.15)
$\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{h!i!}{\bar{\partial}}^{l}(u_{h^{\prime}},\dots,u_{1}|v^{\prime}_{h^{\prime}},\dots,v^{\prime}_{i+1},v_{\sigma(h+i)},\dots,v_{\sigma(h+1)})\,\,{\bar{\partial}}^{0}|v_{\sigma(h)},\dots,v_{\sigma(1)})=0.$

The summations are over all permutations $\sigma$ of $1,2,\dots,h+i$

###### Proof.

By Proposition 2.2,

${\bar{\partial}}^{l}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i+1},u_{\sigma(h+i)},\dots,u_{\sigma(h+1)}|v_{1},\dots
v_{h^{\prime}})=\sum_{l=1}^{i}\sum_{j=1}^{h}x^{(0)}_{u_{\sigma(h+l)v_{j}}}f_{lj}.$

By Equation (2.8),
$\sum_{\sigma}x^{(0)}_{u_{\sigma(h+l)v_{j}}}f_{lj}{\bar{\partial}}^{0}(u_{\sigma(h)},\dots,u_{\sigma(1)}|=0$.
So Equation (2.14) is satisfied. The proof of Equation (2.15) is similar. ∎

######  Lemma 2.5.

The elements of $\mathcal{J}_{h}^{+}$ satisfy the following relations:

  1. (1)

(2.16)
${\bar{\partial}}^{n}(u_{h},\dots,u_{2},u_{1}|v_{1},v_{2},\dots,v_{h})=\sum_{k=0}^{n}{\bar{\partial}}^{k}(u_{h},\dots,u_{2},u_{1}|\,\,\,{\bar{\partial}}^{n-k}|v_{1},v_{2},\dots,v_{h}).$

  2. (2)

For $i_{1},i_{2},k_{0},m,h,h^{\prime}\in\mathbb{Z}_{\geq 0}$ with $k_{0}\leq
m$ and $h\geq h^{\prime}$, let $l_{0}=i_{1}+i_{2}-h-1$. For any integers
$a_{k_{0}+l}$, $0\leq l\leq l_{0}$, there are integers $a_{k}$, $0\leq
k<k_{0}$ or $k_{0}+l_{0}<l\leq m$, such that

$\sum_{k=0}^{m}a_{k}\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{i_{1}!i_{2}!}\left(\begin{array}[]{cc}{\bar{\partial}}^{m-k}(u_{h},\dots,u_{i_{1}+1},\sigma(u_{i_{1}}),\dots,\sigma(u_{1})|&\\\
{\bar{\partial}}^{k}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1},\sigma(u^{\prime}_{i_{2}}),\dots,\sigma(u^{\prime}_{1})|&v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h^{\prime}})\end{array}\right)=0;$

Here the second summation is over all permutations $\sigma$ of
$u_{i_{1}},\dots,u_{1},u^{\prime}_{i_{2}},\dots,u^{\prime}_{1}$. For
simplicity, we write the above equation in the form:

(2.17) $\sum\epsilon
a_{k}\left(\begin{array}[]{cc}{\bar{\partial}}^{m-k}(u_{h},\dots,u_{i_{1}+1},\underline{u_{i_{1}},\dots,u_{1}}|&\\\
{\bar{\partial}}^{k}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1},\underline{u^{\prime}_{i_{2}},\dots,u^{\prime}_{1}}|&v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h^{\prime}})\end{array}\right)=0.$

Similarly, we have the following relations:

(2.20) $\displaystyle\sum\epsilon
a_{k}\left(\begin{array}[]{ccc}{\bar{\partial}}^{m-k}&&|\underline{v_{1},\dots,v_{i_{1}}},v_{i_{1}+1},\dots,v_{h})\\\
{\bar{\partial}}^{k}(&u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{2},u^{\prime}_{1}&|\underline{v^{\prime}_{1},\dots,v^{\prime}_{i_{2}}},v^{\prime}_{i_{2}+1},\dots,v^{\prime}_{h^{\prime}})\end{array}\right)=0;$
(2.23) $\displaystyle\sum\epsilon
a_{k}\left(\begin{array}[]{c}{\bar{\partial}}^{m-k}(u_{h},\dots,u_{i_{1}+1},\underline{u_{i_{1}},\dots,u_{1}}|\\\
{\bar{\partial}}^{k}(u^{\prime}_{h},\dots,u^{\prime}_{i_{2}+1},\underline{u^{\prime}_{i_{2}},\dots,u^{\prime}_{1}}|\end{array}\right)=0;$
(2.26) $\displaystyle\sum\epsilon
a_{k}\left(\begin{array}[]{c}{\bar{\partial}}^{m-k}|\underline{v_{1},\dots,v_{i_{1}}},v_{i_{1}+1},\dots,v_{h})\\\
{\bar{\partial}}^{k}|\underline{v^{\prime}_{1},\dots,v^{\prime}_{i_{2}}},v^{\prime}_{i_{2}+1},\dots,v^{\prime}_{h})\end{array}\right)=0.$

  3. (3)

For $i_{1},i_{2},j_{1},j_{2},r,h^{\prime},k_{0},m\in\mathbb{Z}_{\geq 0}$ with
$r\geq h^{\prime}$, $i_{1},j_{1}\leq r$, $i_{2},j_{2}\leq h^{\prime}$ and
$k_{0}\leq m$, let $l_{0}=i_{1}+i_{2}+j_{1}+j_{2}-2r-1$. Given any integers
$a_{k}$, $k_{0}\leq k\leq k_{0}+l_{0}$, there are integers $a_{k}$, $0\leq
k<k_{0}$ or $k_{0}+l_{0}<k\leq m$, such that

(2.27)
$\displaystyle\sum_{k=0}^{m}a_{k}\sum_{\sigma,\sigma^{\prime}}\frac{1}{i_{1}!i_{2}!j_{1}!j_{2}!}\operatorname{sign}(\sigma)\operatorname{sign}(\sigma^{\prime})\hskip
142.26378pt$ (2.30)
$\displaystyle\left(\begin{array}[]{ccc}{\bar{\partial}}^{m-k}(u_{r},\dots,u_{i_{1}+1},\sigma(u_{i_{1}}),\dots,\sigma(u_{1})&|&\sigma^{\prime}(v_{1}),\dots,\sigma^{\prime}(v_{j_{1}}),v_{j_{1}+1},\dots,v_{r})\\\
{\bar{\partial}}^{k}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1},\sigma(u^{\prime}_{i_{2}}),\dots,\sigma(u^{\prime}_{1})&|&\sigma^{\prime}(v^{\prime}_{1}),\dots,\sigma^{\prime}(v^{\prime}_{j_{2}}),v^{\prime}_{j_{2}+1},\dots,v^{\prime}_{h^{\prime}})\end{array}\right)$
$\displaystyle\in\mathfrak{R}[r+1].$

Here the second summation is over all pairs of permutations $\sigma$ of
$u_{i_{1}},\dots,u_{1},u^{\prime}_{i_{2}},\dots,u^{\prime}_{1}$ and
permutations $\sigma^{\prime}$ of
$v_{i_{1}},\dots,v_{1},v^{\prime}_{i_{2}},\dots,v^{\prime}_{1}$.
$\operatorname{sign}(\sigma)$ and $\operatorname{sign}(\sigma^{\prime})$ are
the signs of the permutations. $\mathfrak{R}[r+1]$ is the ideal of
$\mathfrak{R}$ generated by all $J\in\mathcal{J}$ with $sz(J)=r+1$.

######  Remark 2.6.

Since the summation is over all permutations of the underlined elements, each
monomial appear in Equations (2.17)–(2.26) will appear $i_{1}!i_{2}!$ times,
so the coefficient of each monomial will be $\pm a_{k}$. Since the second
summation in Equation (2.27) is over all pairs of permutations $\sigma$ and
$\sigma^{\prime}$, each monomial in the equation will appear
$i_{1}!i_{2}!j_{1}!j_{2}!$ times, so the coefficient of each monomial will be
$\pm a_{k}$.

###### Proof.

The statement (1) is directly induced from Equation (2.5) and Proposition 2.1.

Let
$\mathcal{F}_{l}(u_{h},\dots,u_{i_{1}+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1})$

$=\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{i_{1}!i_{2}!}\left(\begin{array}[]{cc}{\bar{\partial}}^{(0)}(u_{h},\dots,u_{i_{1}+1},\sigma(u_{i_{1}}),\dots,\sigma(u_{1})|&\\\
{\bar{\partial}}^{l}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1},\sigma(u^{\prime}_{i_{2}}),\dots,\sigma(u^{\prime}_{1})|&v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h^{\prime}})\end{array}\right).$

We have
$\mathcal{F}_{l}(u_{h},\dots,u_{i_{1}};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1})$

$=\mathcal{F}_{l}(u_{h},\dots,u_{i_{1}+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1})\pm\mathcal{F}_{l}(u_{h},\dots,u_{i_{1}+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1},u_{i_{1}}).$

If $i_{1}=h$ and $0\leq l\leq l_{0}$, by Equation (2.14),
$\mathcal{F}_{l}(\,\,;u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1})=0$.
By induction on $h-i_{1}$, we can see that
$\mathcal{F}_{l}(u_{h},\dots,u_{i_{1}+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1})=0.$
Thus

$0={\bar{\partial}}^{m-l}\mathcal{F}_{l}(u_{h},\dots,u_{i_{1}+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1})$
$=\sum_{k=0}^{m}C_{k}^{l}\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{i_{1}!i_{2}!}\left(\begin{array}[]{cc}{\bar{\partial}}^{m-k}(u_{h},\dots,u_{i_{1}+1},\sigma(u_{i_{1}}),\dots,\sigma(u_{1})|&\\\
{\bar{\partial}}^{k}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1},\sigma(u^{\prime}_{i_{2}}),\dots,\sigma(u^{\prime}_{1})|&v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h^{\prime}})\end{array}\right)$

Now the $(l_{0}+1)\times(l_{0}+1)$ integer matrix with entries
$c_{ji}=C_{k_{0}+j}^{i}$, $0\leq i,j\leq l_{0}$ is invertible since the
determinant of this matrix is $\pm 1$. Let $b_{ij}\in\mathbb{Z}$ be the
entries of the inverse matrix. Let
$a_{k}=\sum_{l=0}^{l_{0}}\sum_{j=0}^{l_{0}}C_{k}^{l}b_{l,j}a_{k_{0}+j}$. So

$0=\sum_{l=0}^{l_{0}}\sum_{j=0}^{l_{0}}b_{l,j}a_{k_{0}+j}{\bar{\partial}}^{m-l}\mathcal{F}_{l}(u_{h},\dots,u_{i_{1}+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{i_{2}+1})=\text{right
hand side of Equation (\ref{eqn:relation1+})}$

The proofs for Equations (2.20, 2.23, 2.26) are similar. Equation (2.27) is
the Lemma 2.4 in [8]. ∎

### Standard monomials

Now we give a definition of the standard monomials of $\mathcal{J}_{h}^{+}$.

######  Definition 2.7.

An ordered product $E^{L}_{1}E^{L}_{2}\cdots E^{L}_{m}E^{R}_{1}\cdots
E^{R}_{n}E_{1}\cdots E_{t}$ of elements of $\mathcal{E}_{h}^{+}$ is said to be
standard if

  1. (1)

$E^{L}_{i}\in\mathcal{E}_{h}^{L}$, $E^{R}_{i}\in\mathcal{E}_{h}^{R}$,
$E_{i}\in\mathcal{E}_{h-1}$;

  2. (2)

for $1\leq a<m$, $E^{L}_{a}\leq E^{L}_{a+1}$ and $E^{L}_{a+1}$ is the largest
element in $\mathcal{E}(||E^{L}_{a+1}||)$ such that $E^{L}_{a}\leq
E^{L}_{a+1}$;

  3. (3)

for $1\leq a<n$, $E^{R}_{a}\leq E^{R}_{a+1}$ and $E^{R}_{a+1}$ is the largest
element in $\mathcal{E}(||E^{R}_{a+1}||)$ such that $E^{R}_{a}\leq
E^{R}_{a+1}$;

  4. (4)

for $1\leq a<t$, $E_{a}\leq E_{a+1}$ and $E_{a+1}$ is the largest element in
$\mathcal{E}(||E_{a+1}||)$ such that $E_{a}\leq E_{a+1}$.

  5. (5)

$E^{L}_{1}$ and $E^{R}_{1}$ are the largest elements in
$\mathcal{E}(||E^{L}_{1}||)$ and $\mathcal{E}(||E^{R}_{1}||)$ respectively.
$E_{1}$ is the largest element in $\mathcal{E}(||E_{1}||)$ such that
$E^{L}_{m}\leq E_{1}$ if $m\geq 1$ and $E^{n}_{l}\leq E_{1}$ if $n\geq 1$,

An ordered product $J_{1}J_{2}\cdots J_{k}$ of elements of
$\mathcal{J}_{h}^{+}$ is said to be standard if there is a standard ordered
product $E_{1}E_{2}\cdots E_{k}$ such that
$E_{i}\in\mathcal{E}_{h}^{+}(J_{i})$.

######  Remark 2.8.

Let $E_{0}^{L}=((h,0),\dots,(2,0),(1,0)|$ and
$E^{R}_{0}=|(1,0),(2,0),\dots,(h,0)$. Let $E=F(E_{m}^{L},E_{n}^{R})$, then the
condition ” $E^{L}_{m}\leq E_{1}$ if $m\geq 1$ and $E^{n}_{l}\leq E_{1}$ if
$n\geq 1$”, is equivalent to ”$E\leq E_{1}$”.

Let
$\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})\subset\mathcal{M}(\mathcal{J}_{h}^{+})$
be the set of standard monomials of $\mathcal{J}_{h}^{+}$. Let
$\mathcal{S}\mathcal{M}(\mathcal{E}_{h}^{+})\subset\mathcal{M}(\mathcal{E}_{h}^{+})$
be the set of standard monomials of $\mathcal{E}_{h}^{+}$.

By definition, the standard monomial $E^{L}_{1}E^{L}_{2}\cdots
E^{L}_{m}E^{R}_{1}\cdots E^{R}_{n}E_{1}\cdots
E_{k-m-n}\in\mathcal{S}\mathcal{M}(\mathcal{E}_{h}^{+})$ corresponding to the
standard monomial $J_{1}J_{2}\cdots
J_{k}\in\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ is unique and $E^{L}_{1}$
(if $m=0$, left part of $E_{1}$) has the form
$((u_{h^{\prime}},k),(u_{h^{\prime}-1},0),\dots,(u_{1},0)|$ and $E^{R}_{1}$
(if $n=0$, the right part of $E_{1}$) has the form
$|(v_{1},0),\dots,(v_{h^{\prime}-1},0),(v_{h^{\prime}},l))$ (if $n=0$, $l=0$)
with $u_{i}<u_{i+1}$ and $v_{i}<v_{i+1}$. So the map

$\pi_{h}^{+}:\mathcal{S}\mathcal{M}(\mathcal{E}_{h}^{+})\to\mathcal{S}\mathcal{M}(\mathcal{J}^{+}_{h}),\quad
E_{1}E_{2}\cdots E_{m}\mapsto||E_{1}||||E_{2}||\cdots||E_{m}||$

is a bijection.

The following lemma will be proved later in Section 6.

######  Lemma 2.9.

If $J=J_{1}\cdots J_{b}\in\mathcal{M}(\mathcal{J}_{h}^{+})$, is not standard,
$J$ can be written as a linear combination of
$\mathcal{M}(\mathcal{J}_{h}^{+})$ preceding $J_{1}\cdots J_{b-1}$ with
integer coefficients.

Since $\mathfrak{R}_{h}^{+}$ is generated by $\mathcal{J}_{h}^{+}$, by the
above lemma, we immediately have

######  Lemma 2.10.

Any element of $\mathfrak{R}^{+}_{h}$ can be written as a linear combination
of standard ordered products of elements of $\mathcal{J}^{+}_{h}$ with integer
coefficients.

###### Proof.

If the lemma is not true, there must be a smallest element
$J\in\mathcal{M}(\mathcal{J}_{h}^{+})$, which cannot be written as a linear
combination of elements of $\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ with
integer coefficients. So $J$ is not standard. By Lemma 2.9,
$J=\sum_{\alpha}\pm J_{\alpha}$ with
$J_{\alpha}\in\mathcal{M}(\mathcal{J}_{h}^{+})$ and $J_{\alpha}\prec J$. By
assumption, $J_{\alpha}$ can be written as a linear combination of elements of
$\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ with integer coefficients. Thus
$J$ can be written as a linear combination of elements of
$\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ with integer coefficients, which
is a contradiction. ∎

##  3\. A canonical basis

### A homomorphism

Let
$\tilde{Q}_{h}^{+}:\mathfrak{R}[\mathcal{J}_{h}^{L}\cup\mathcal{J}_{h}^{R}]\to\mathfrak{B}$
be a ring homomorphism given by

(3.1) $\tilde{Q}_{h}^{+}(x^{(k)}_{ij})={\bar{\partial}}^{k}X^{(0)}_{ij};$
(3.2)
$\tilde{Q}_{h}^{+}({\bar{\partial}}^{k}(u_{h},\dots,u_{1}|)={\bar{\partial}}^{k}Y^{(0)}_{u_{1},\dots,u_{h}}$
(3.3)
$\tilde{Q}_{h}^{+}({\bar{\partial}}^{k}|v_{1},\dots,v_{h}))={\bar{\partial}}^{k}Z^{(0)}_{v_{1},\dots,v_{h}}$

######  Lemma 3.1.

$\partial\tilde{Q}_{h}^{+}=\tilde{Q}_{h}^{+}\partial$ and
$\mathcal{I}^{+}_{h}\subset\ker(\tilde{Q}_{h}^{+})$.

###### Proof.

The first statement is obvious from the definition of $\tilde{Q}_{h}^{+}$. For
the second statement, we only need to show $\tilde{Q}_{h}^{+}$ maps the right
hand sides of Equations (2.5-2.9) to zero when $n=0$. This is obvious for
Equations (2.5-2.7). The proof for Equations (2.8) and (2.9) is similar to the
proof of Equation (2.14), since

$\sum_{i=0}^{h}(-1)^{i}\tilde{Q}_{h}^{+}({\bar{\partial}}^{0}(u_{h},\dots,u_{i+1},u_{i-1},\dots,u_{0}))a_{u_{i},j}=0.$

∎

By Lemma 3.1, $\tilde{Q}_{h}^{+}$ induces a ring homomorphism
$Q_{h}^{+}:\mathfrak{R}_{h}^{+}\to\mathfrak{B}$ and $\partial
Q_{h}^{+}=Q_{h}^{+}\partial$.

### Double tableaux

We use the double tableaux to represent monomials of $\mathfrak{B}$. Let
$\mathcal{T}_{h}^{+}$ be the set of the following tableaux:

(3.4) $\left|\begin{array}[]{ccc}y_{1,h},\cdots,y_{2,2},y_{1,1}&|&\\\
\vdots&|&\\\ y_{m,h},\cdots,y_{m,2},y_{m,1}&|&\\\
&|&z_{1,1},z_{1,2},\cdots,z_{1,h}\\\ &|&\vdots\\\
&|&z_{n,1},z_{n,2},\cdots,z_{n,h}\\\
y^{\prime}_{1,h},\cdots,y^{\prime}_{2,2},y^{\prime}_{1,1}&|&z^{\prime}_{1,1},z^{\prime}_{1,2},\cdots,z^{\prime}_{1,h}\\\
\vdots&\vdots&\vdots\\\
y^{\prime}_{t,h},\cdots,y^{\prime}_{t,2}y^{\prime}_{t,1}&|&z^{\prime}_{t,1},z^{\prime}_{t,2}\cdots,z^{\prime}_{t,h}\\\
\end{array}\right|.$

Here $y_{s,h}$ are some $a^{(k)}_{ih}$ and $z_{s,h}$ are some $b^{(k)}_{jh}$;
$y^{\prime}_{s,h}$ and $z^{\prime}_{s,h}$ are $*$, $b^{(k)}_{jh}$; for $1\leq
l\leq h-1$, $y_{s,l}$, $y^{\prime}_{s,l}$ are some $a^{(k)}_{il}$ or $*$ and
$z_{s,l}$, $z^{\prime}_{s,l}$ are some $b^{(k)}_{jl}$ or $*$; every row of the
tableau has elements in $\mathcal{S}_{h}$; and

$y_{s,j}\leq y_{s+1,j}\leq y^{\prime}_{s^{\prime},j}\leq
y^{\prime}_{s^{\prime}+1,j},\quad z_{s,j}\leq z_{s+1,j}\leq
z^{\prime}_{s^{\prime},j}\leq z^{\prime}_{s^{\prime}+1,j}.$

We use the tableau (3.4) to represent a monomial in $\mathfrak{B}$, which is
the product of ${a^{(k)}_{ij}}^{\prime}s$ and ${b^{(k)}_{ij}}^{\prime}s$ in
the tableau. It is easy to see that the representation is a one-to-one
correspondence between $\mathcal{T}_{h}^{+}$ and the set of monomials of
$\mathfrak{B}$. We associate to the tableau (3.4) the word:

$y_{1,h}y_{1,h-1}\cdots y_{1,1}\cdots y_{m,2}y_{m,1}z_{1,h}\cdot z_{1,1}\cdots
z_{n,1}y^{\prime}_{1,h-1}\cdots y^{\prime}_{1,1}z^{\prime}_{1,h-1}\cdot
z^{\prime}_{1,1}\cdots z^{\prime}_{t,h-1}.$

and order these words lexicographically. For a polynomial $f\in\mathfrak{B}$,
let $Ld^{+}(f)$ be its leading monomial in $f$ under the order we defined on
$\mathcal{T}_{h}^{+}$.

For $E_{1}\cdots E_{k}\in\mathcal{S}\mathcal{M}(\mathcal{E})$ with

$\displaystyle E_{i}$
$\displaystyle=((u^{i}_{h_{1}},k^{i}_{h_{1}}),\dots,(u^{i}_{2},k^{i}_{2}),(u^{i}_{1},k^{i}_{1})|\in\mathcal{E}^{L}_{h},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle 1\leq i\leq m;$ $\displaystyle E_{i+m}$
$\displaystyle=|(v^{i}_{1},l^{i}_{1}),(v^{i}_{2},l^{i}_{2}),\dots,(v^{i}_{h_{1}},l^{i}_{h_{1}}))\in\mathcal{E}^{R}_{h},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle 1\leq i\leq n;$ $\displaystyle E_{i+m+n}$
$\displaystyle=((u^{i}_{h_{1}},k^{i}_{h_{1}}),\dots,(u^{i}_{2},k^{i}_{2}),(u^{i}_{1},k^{i}_{1})|(v^{i}_{1},l^{i}_{1}),(v^{i}_{2},l^{i}_{2}),\dots,(v^{i}_{h_{1}},l^{i}_{h_{1}}))\in\mathcal{E}^{L}_{h},\quad$
$\displaystyle 1\leq i\leq t,$

and $k=m+n+t$, we use the following double tableau to represent it.

(3.5)
$\left(\begin{array}[]{ccc}(u^{1}_{h},k^{1}_{h}),\cdots,(u^{1}_{2},k^{1}_{2}),(u^{1}_{1},k^{1}_{1})&|&\\\
\vdots&|&\\\
(u^{m}_{h},k^{m}_{h}),\cdots,(u^{m}_{2},k^{m}_{2}),(u^{m}_{1},k^{m}_{1})&|&\\\
\----------------&|&\--------------\\\
&|&(v^{1}_{1},l^{1}_{1}),(v^{1}_{2},l^{1}_{2}),\cdots,(v^{1}_{h},l^{1}_{h})\\\
&|&\vdots\\\
&|&(v^{n}_{1},l^{n}_{1}),(v^{n}_{2},l^{n}_{2}),\cdots,(v^{n}_{h},l^{n}_{h})\\\
\----------------&|&\--------------\\\ (u^{\prime 1}_{h_{1}},k^{\prime
1}_{h_{1}}),\cdots,(u^{\prime 1}_{2},k^{\prime 1}_{2}),(u^{\prime
1}_{1},k^{\prime 1}_{1})&|&(v^{\prime 1}_{1},l^{\prime 1}_{1}),(v^{\prime
1}_{2},l^{\prime 1}_{2}),\cdots,(v^{\prime 1}_{h^{\prime}_{1}},l^{\prime
1}_{h_{1}})\\\ (u^{\prime 2}_{h_{2}},k^{\prime 2}_{h_{2}}),\cdots,(u^{\prime
2}_{2},k^{\prime 2}_{2}),(u^{\prime 2}_{1},k^{\prime 2}_{1})&|&(v^{\prime
2}_{1},l^{\prime 2}_{1}),(v^{\prime 2}_{2},l^{\prime 2}_{2}),\cdots,(v^{\prime
2}_{h_{2}},l^{\prime 2}_{h_{2}})\\\ &\vdots&\\\ (u^{\prime
t}_{h_{t}},k^{\prime t}_{h_{t}}),\cdots,(u^{\prime t}_{2},k^{\prime
t}_{2}),(u^{\prime t}_{1},k^{\prime t}_{1})&|&(v^{\prime t}_{1},l^{\prime
t}_{1}),(v^{\prime t}_{2},l^{\prime t}_{2}),\cdots,(v^{\prime
t}_{h_{t}},l^{\prime t}_{h_{t}})\\\ \end{array}\right).$

Let $T^{+}:\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+}))\to\mathcal{T}^{+}$
with

$T^{+}(E_{1}\cdots
E_{k})=\left(\begin{array}[]{ccc}a_{u^{1}_{h}h}^{(k^{1}_{h})},\,\,\,\cdots,\,\,\,a_{u^{1}_{2}2}^{(k^{1}_{2})},\,\,a_{u^{1}_{1}1}^{(k^{1}_{1})}&|&\\\
\vdots&|&\\\
a_{u^{m}_{h}h}^{(k^{m}_{h})},\,\,\,\cdots,\,\,\,a_{u^{m}_{2}2}^{(k^{m}_{2})},\,\,a_{u^{m}_{1}1}^{(k^{m}_{1})}&|&\\\
&|&b_{v^{1}_{1}1}^{(l^{1}_{1})},\,\,b_{v^{1}_{2}2}^{(l^{1}_{2})},\,\,\,\cdots,\,\,\,b_{v^{1}_{h}h}^{(l^{1}_{h})}\\\
&|&\vdots\\\
&|&b_{v^{n}_{1}1}^{(l^{n}_{1})},\,\,b_{v^{n}_{2}2}^{(l^{n}_{2})},\,\,\,\cdots,\,\,\,b_{v^{n}_{h}h}^{(l^{n}_{h})}\\\
,\cdots,*,a_{u^{\prime 1}_{h^{\prime}_{1}}h^{\prime}_{1}}^{(k^{\prime
1}_{h^{\prime}_{1}})},\cdots,a_{u^{\prime 1}_{1}1}^{(k^{\prime
1}_{1})}&|&b_{v^{\prime 1}_{1}1}^{(l^{\prime 1}_{1})},\cdots,b_{v^{\prime
1}_{h^{\prime}_{1}}h^{\prime}_{1}}^{(l^{\prime
1}_{h^{\prime}_{1}})},*,\cdots,*\\\ \vdots&|&\vdots\\\ ,\cdots,*,a_{u^{\prime
t}_{h^{\prime}_{t}}h^{\prime}_{t}}^{(k^{\prime
t}_{h^{\prime}_{t}})},\cdots,a_{u^{\prime
t}_{1}1}^{(^{\prime}k^{t}_{1})}&|&b_{v^{\prime t}_{1}1}^{(l^{\prime
t}_{1})},\cdots,b_{v^{\prime t}_{h^{\prime}_{t}}h^{\prime}_{t}}^{(l^{\prime
t}_{h^{\prime}_{t}})},*,\cdots,*\end{array}\right).$

Then $T^{+}$ is injective and $T(E_{1})\prec T(E_{2})$ if $E_{1}\prec E_{2}$.

######  Lemma 3.2.

Let $J_{1}\cdots J_{k}\in\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ and
$E_{1}\cdots E_{k}\in\mathcal{S}\mathcal{M}(\mathcal{E}_{h}^{+})$ be its
associated standard order product of elements of $\mathcal{E}_{h}^{+}$. Assume
the double tableau representing $E_{1}\cdots E_{k}$ is (3.5). Then the leading
monomial of $Q_{h}^{+}(J_{1}\cdots J_{k})$ is represented by the double
tableau $T^{+}(E_{1}\cdots E_{k})$. So

$Ld^{+}\circ
Q_{h}^{+}=T^{+}\circ(\pi_{h}^{+})^{-1}:\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})\to\mathcal{T}_{h}^{+}$

is injective. The coefficient of the leading monomial of
$Q_{h}^{+}(J_{1}\cdots J_{m})$ is $\pm 1$.

###### Proof.

Let $W_{r}$ be the monomial corresponding to the double tableau $T(E_{1}\cdots
E_{r})$. Let $M_{r}$ be the monomial corresponding to the double tableau
$T(E_{r})$. Then $W_{r}=W_{r-1}M_{r}$. We prove the lemma by induction on $r$.
If $r=1$, the lemma is obviously true. Assume the lemma is true for
$J_{1}\cdots J_{r-1}$. Then $Ld(Q_{h}^{+}(J_{1}\cdots J_{r-1}))=W_{n-1}$, the
monomial corresponds to $T(E_{1}\cdots E_{r-1})$, and the coefficient of
$W_{n-1}$ in $Q_{h}^{+}(J_{1}\cdots J_{r-1})$ is $\pm 1$. Let $h_{r}=h$ if
$r\leq m+n$.

$\displaystyle Q_{h}^{+}(J_{r})$ $\displaystyle=$ $\displaystyle\sum\pm
a_{u^{r}_{1}s_{1}}^{(k_{1})}a_{u^{r}_{2}s_{2}}^{(k_{2})}\cdots
a_{u^{r}_{h_{r}}s_{h_{r}}}^{(k_{h_{r}})},\quad\quad\quad\quad\quad\quad\quad\quad
r\leq m;$ $\displaystyle Q_{h}^{+}(J_{r})$ $\displaystyle=$
$\displaystyle\sum\pm
b_{v^{r}_{1}t_{1}}^{(l_{1})}b_{v^{r}_{2}t_{2}}^{(l_{2})}\cdots
b_{v^{r}_{h_{r}}t_{h_{r}}}^{(l_{h_{r}})},\quad\quad\quad\quad\quad\quad\quad\quad\quad
m<r\leq m+n;$ $\displaystyle Q_{h}^{+}(J_{r})$ $\displaystyle=$
$\displaystyle\sum\pm
a_{u^{r}_{1}s_{1}}^{(k_{1})}a_{u^{r}_{2}s_{2}}^{(k_{2})}\cdots
a_{u^{r}_{h_{r}}s_{h_{r}}}^{(k_{h_{r}})}b_{v^{r}_{1}t_{1}}^{(l_{1})}b_{v^{r}_{2}t_{2}}^{(l_{2})}\cdots
b_{v^{r}_{h_{r}}t_{h_{r}}}^{(l_{h_{r}})},\quad r>m+n.$

If $r<m$, the summation is over all $k_{i}\geq 0$ with $\sum k_{i}=wt(E_{r})$,
and all $s_{i}$ with $1\leq s_{1},s_{2},\dots,s_{h_{r}}\leq h$ and they are
different from each other. If $m<r\leq m+n$, the summation is over all
$l_{i}\geq 0$ with $\sum l_{i}=wt(E_{r})$, and all $t_{i}$ with $1\leq
t_{1},t_{2},\dots,t_{h_{r}}\leq h$ and they are different from each other. If
$r>m+n$, the summation is over all $l_{i},k_{i}\geq 0$ with
$\sum(l_{i}+k_{i})=wt(E_{r})$, all $s_{i}$ with $1\leq
s_{1},s_{2},\dots,s_{h_{r}}\leq h$ and they are different from each other, and
all $t_{1},\dots,t_{h_{r}}$, which are permutations of
$s_{1},\dots,s_{h_{r}}$. $M_{r}$ is one of the monomials in $Q_{h}^{+}(J_{r})$
with coefficient $\pm 1$. All of the monomials in $Q_{h}^{+}(J_{1}\cdots
J_{r-1})$ except $W_{n-1}$ are less than $W_{n-1}$, so any monomial in
$Q_{h}^{+}(J_{1}\cdots J_{r-1})$ except $W_{n-1}$ times any monomial in
$Q_{h}^{+}(J_{r})$ is less than $W_{r-1}$. The coefficient of $W_{r}$ in
$Q^{+}_{h}(J_{1}\cdots J_{r})$ is $\pm 1$ since $W_{r-1}\prec W_{r}$, and the
coefficients of $W_{r-1}$ in $Q^{+}_{h}(J_{1}\cdots J_{r-1})$ and $M_{r}$ in
$Q^{+}_{h}(J_{r})$ are $\pm 1$. Now

$W_{r-1}\prec W_{r}\prec Ld(Q_{h}^{+}(J_{1}\cdots J_{r})).$

The leading monomial $Ld(Q_{h}^{+}(J_{1}\cdots J_{r}))$ must have the form

$W=W_{r-1}a_{u^{r}_{1}s_{1}}^{(k_{1})}a_{u^{r}_{2}s_{2}}^{(k_{2})}\cdots
a_{u^{r}_{h_{r}}s_{h_{r}}}^{(k_{h_{r}})},r\leq m,\quad
W=W_{r-1}b_{v^{r}_{1}t_{1}}^{(l_{1})}b_{v^{r}_{2}t_{2}}^{(l_{2})}\cdots
b_{v^{r}_{h_{r}}t_{h_{r}}}^{(l_{h_{r}})},m<r\leq m+n$
$W=W_{r-1}a_{u^{r}_{1}s_{1}}^{(k_{1})}a_{u^{r}_{2}s_{2}}^{(k_{2})}\cdots
a_{u^{r}_{h_{r}}s_{h_{r}}}^{(k_{h_{r}})}b_{v^{r}_{1}t_{1}}^{(l_{1})}b_{v^{r}_{2}t_{2}}^{(l_{2})}\cdots
b_{v^{r}_{h_{r}}t_{h_{r}}}^{(l_{h_{r}})},\quad r>m+n.$

If some $s_{i}$ or $t_{i}$ is greater than $h_{r-1}$, then $W\prec W_{n-1}$.
If there is some $h_{r-1}\geq s_{i}>h_{r}$, there is $1\leq j\leq h_{r}$, with
$j\notin\\{s_{1},\dots s_{h_{r}}\\}$; if we replace $s_{i}$ by $j$ in $W$, we
get a larger monomial in $Q_{h}^{+}(J_{1}\cdots J_{r})$. So we can assume
$s_{1},\dots,s_{h_{r}}$ is a permutation of $1,2,\dots,h_{r}$. Similarly, we
can assume $t_{1},\dots,t_{h_{r}}$ is a permutation of $1,2,\dots,h_{r}$. We
must have $a_{u^{r}_{i}s_{i}}^{(k_{i})}\geq
a_{u^{r-1}_{s_{i}}s_{i}}^{(k^{r-1}_{s_{i}})}$ if $1<r\leq m$ or $r>m+n+1$,
$b_{v^{r}_{i}t_{i}}^{(k_{i})}\geq
b_{v^{r-1}_{t_{i}}t_{i}}^{(k^{r-1}_{t_{i}})}$ if $m+1<r$, and
$a_{u^{r}_{i}s_{i}}^{(k_{i})}\geq a_{u^{m}_{i}s_{i}}^{(k^{m}_{s_{i}})}$ if
$r=m+n+1$ and $m>0$, otherwise $W\prec W_{r-1}$. Such monomials in
$Q_{h}^{+}(J_{r})$ are in one-to-one correspondence with
$E^{\prime}_{r}\in\mathcal{E}(J_{r})$ which satisfy the property
(P):  $E_{r-1}\leq E^{\prime}_{r}$ if $r\neq 1,m+1,m+n+1$ and $E_{m}\leq
E^{\prime}_{r}$ and $E_{r-1}\leq E^{\prime}_{r}$ if $r=m+n+1$ and $m>0$.
$E_{r}$ is the largest element in $\mathcal{E}(J_{r})$ satisfying the property
(P), since $E$ is standard, so $W_{r}$ is the leading term of
$Q_{h}^{+}(J_{1}\cdots J_{r})$. ∎

Let $\mathfrak{C}^{+}=Q_{h}^{+}(\mathfrak{R}_{h}^{+})$. It is a subring of
$\mathfrak{B}$ generated by (3.2),(3.3) and
${\bar{\partial}}^{k}\sum_{i=1}^{n}a^{(0)}_{il}b^{(0)}_{jl}$. We have

######  Theorem 3.3.

$Q_{h}^{+}:\mathfrak{R}_{h}^{+}\to\mathfrak{B}$ is injective. The set of
standard monomials $\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ is a
$\mathbb{Z}$-basis of $\mathfrak{R}_{h}^{+}$. So
$Q_{h}^{+}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+}))$ is a
$\mathbb{Z}$-basis of the subring $\mathfrak{C}^{+}$.

###### Proof.

By Lemma 3.2, $Ld^{+}(Q_{h}^{+}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})))$
are linearly independent, so $\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ are
linearly independent. By Lemma 2.10,
$\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ generates
$\mathfrak{R}_{h}^{+}$. So $\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$ is a
$\mathbb{Z}$-basis of $\mathfrak{R}_{h}^{+}$ and $Q_{h}^{+}$ is injective. ∎

Since $Q_{h}^{+}$ is injective and $\mathfrak{B}$ is an integral domain, we
obtain

######  Corollary 3.4.

$\mathfrak{R}^{+}_{h}$ is an integral domain.

Let $K$ be an algebraically closed field. Let
$\mathfrak{R}_{h}^{+K}=\mathfrak{R}^{+}_{h}\otimes K$,
$Q^{+K}_{h}=Q^{+}_{h}\otimes Id:R_{h}^{+}\otimes K\to\mathfrak{B}\otimes
K=\mathfrak{B}^{K}$. Let $\mathfrak{C}^{+K}$ be the image of $Q^{+K}_{h}$. By
Theorem 3.3, we immediately have

######  Corollary 3.5.

$\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ is a $K$-basis of
$\mathfrak{R}_{h}^{+K}$.

######  Corollary 3.6.

${Q^{+K}_{h}}$ is injective. So $\mathfrak{R}_{h}^{+K}$ is integral and
$Q_{h}^{+}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+}))$ is a $K$-basis of the
subring $\mathfrak{C}^{+K}$.

###### Proof.

By Lemma 3.2, $Ld^{+}(Q_{h}^{+}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})))$
are linearly independent, since $\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$
is a $K$-basis of ${\mathfrak{R}_{h}^{+}}^{K}$. So $Q_{h}^{+}$ is injective
and $Q_{h}^{+}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+}))$ is a $K$-basis of
the subring ${\mathfrak{C}^{+}}^{K}$. since $\mathfrak{B}^{K}$ is an integral
domain, so
$\mathfrak{R}_{h}^{+K}\cong\mathfrak{C}^{+K}\subset\mathfrak{B}^{K}$ is an
integral domain. ∎

##  4\. Application

In this section, we use the standard monomial basis we have constructed to
prove Theorem 1.2. Let $G=SL_{h}(K)$, and let $W=K^{\oplus h}$ be its standard
representation. As before, for $p,q\geq 0$, let $V=W^{\oplus
p}\bigoplus{W^{*}}^{\oplus q}$. The group structure $G\times G\to G$ induces
the group structure on its arc space

$J_{\infty}(G)\times J_{\infty}(G)\to J_{\infty}(G).$

Therefore $J_{\infty}(G)$ is an algebraic group. The action of $G$ on $V$,
$G\times V\to V$, induces the action of $J_{\infty}(G)$ on $J_{\infty}(V)$,

$J_{\infty}(G)\times J_{\infty}(V)\to J_{\infty}(V).$

The affine quotient map $q:V\to V/\\!\\!/G$ induces
$q_{\infty}:J_{\infty}(V)\to J_{\infty}(V/\\!\\!/G)$ and

$\bar{q}_{\infty}:J_{\infty}(V)/\\!\\!/J_{\infty}(G)\to
J_{\infty}(V/\\!\\!/G).$

The affine coordinate ring of $J_{\infty}(V/\\!\\!/G)$ is

$\tilde{\mathfrak{R}}_{h}^{+}=K[X^{(k)}_{ij},Y^{(k)}_{u_{1},\dots,u_{h}},Z^{(k)}_{v_{1},\dots,v_{h}}|\
k\in\mathbb{Z}_{\geq 0}]/\tilde{\mathcal{I}}_{h}^{+},$

where $\tilde{\mathcal{I}}_{h}^{+}$ is the ideal generated by relations (1.2),
(1.3), and (1.4), and their $n^{\text{th}}$ normalized derivatives. Then
$q_{\infty}$ induces the ring homomorphism
$q_{\infty}^{*}:\tilde{\mathfrak{R}}_{h}^{+}\to\mathfrak{B}$ given by

(4.1) $X^{(k)}_{ij}\mapsto{\bar{\partial}}^{k}X^{(0)}_{ij},\quad
Y^{(k)}_{u_{1},\dots,u_{h}}\mapsto{\bar{\partial}}^{k}Y^{(0)}_{u_{1},\dots,u_{h}},\quad
Z^{(k)}_{v_{1},\dots,v_{h}}\mapsto{\bar{\partial}}^{k}Z^{(0)}_{v_{1},\dots,v_{h}}.$

So $\mathfrak{C}^{+K}$ is the image of $q_{\infty}^{*}$, and the elements of
$\mathfrak{C}^{+K}$ are $J_{\infty}(G)$-invariant. Let
$\Gamma=Q_{h}^{+}({\bar{\partial}}^{0}(h,\dots,2,1|)$.
$\mathfrak{B}^{K}_{\Gamma}$ and $\mathfrak{C}^{+K}_{\Gamma}$ are localizations
of $\mathfrak{B}^{K}$ and $\mathfrak{C}^{+K}$ at $\Gamma$.

######  Lemma 4.1.

If $p\geq h$,

${\mathfrak{B}^{K}_{\Gamma}}^{J_{\infty}({SL}_{h}(K))}=\mathfrak{C}^{+K}_{\Gamma}.$

###### Proof.

Let $G_{\infty}=J_{\infty}(SL_{h}(K))$. Let $B^{K}_{\Gamma}$ be the
localization of $B^{K}$ at $\Gamma$ and
$V_{\Gamma}=\operatorname{Spec}B^{K}_{\Gamma}$. Let $H$ be a subvariety of
$V_{\Gamma}$ given by the ideal generated by $a_{il}-\delta_{i}^{l}$ with
$1\leq i,l\leq h$ and $i+l>2$. The composition of the embedding
$\iota:H\hookrightarrow V_{\Gamma}$ and the affine quotient
$q^{\Gamma}:V_{\Gamma}\to V_{\Gamma}/\\!\\!/G=(V/\\!\\!/G)_{\Gamma}$ gives the
isomorphism $q^{\Gamma}\circ\iota:H\to(V/\\!\\!/G)_{\Gamma}$. So we have an
isomorphism of arc spaces

$q^{\Gamma}_{\infty}\circ\iota_{\infty}:J_{\infty}(H)\to
J_{\infty}((V/\\!\\!/G)_{\Gamma})=J_{\infty}(V/\\!\\!/G)_{\Gamma}.$

$q^{\Gamma}_{\infty}$ induces a morphism
$\bar{q}^{\Gamma}_{\infty}:J_{\infty}(V_{\Gamma})/\\!\\!/G_{\infty}\to
J_{\infty}(V/\\!\\!/G)_{\Gamma}$. The action of $G$ on $V$ gives a
$G$-equivariant isomorphism

$G\times H\to V_{\Gamma}.$

So we have a $G_{\infty}$-equivariant isomorphism

$G_{\infty}\times J_{\infty}(H)\to
J_{\infty}(V_{\Gamma})=J_{\infty}(V)_{\Gamma}.$

and an isomorphism of their affine quotients

$i:J_{\infty}(H)=G_{\infty}\times J_{\infty}(H)/\\!\\!/G\cong
J_{\infty}(V)/\\!\\!/G_{\infty}.$

$\bar{q}^{\Gamma}_{\infty}\circ i=q^{\Gamma}_{\infty}\circ\iota_{\infty}$ is
an isomorphism, so $\bar{q}^{\Gamma}_{\infty}$ is an isomorphism since $i$ is
an isomorphism. So
${\mathfrak{B}^{K}_{\Gamma}}^{J_{\infty}({SL}_{h}(K))}=\bar{q}^{\Gamma*}_{\infty}((\tilde{\mathfrak{R}}_{h}^{+})_{\Gamma})=\mathfrak{C}^{+K}_{\Gamma}$.
∎

######  Theorem 4.2.

$\mathfrak{B}^{J_{\infty}({SL}_{h}(K))}=\mathfrak{C}^{+K}$.

###### Proof.

Let $G=SL_{h}(K)$ and $G_{\infty}=J_{\infty}(SL_{h}(K))$. If $p\geq h$, regard
$\mathfrak{B}^{K}$ and $\mathfrak{C}^{+K}_{\Gamma}$ as subrings of
$\mathfrak{B}^{K}_{\Gamma}$. By Lemma 4.1, we have

${\mathfrak{B}^{K}}^{G_{\infty}}=\mathfrak{B}^{K}\cap\mathfrak{C}^{+K}_{\Gamma}.$

Now for any $f\in\mathfrak{B}^{K}\cap\mathfrak{C}^{+K}_{\Gamma}$. Therefore
$f=\frac{g}{\Gamma^{n}}$ with $\Gamma^{n}f=g\in\mathfrak{C}^{+K}$. The leading
monomial of $g$ is

$L(g)=(a^{(0)}_{11}\cdots a^{(0)}_{hh})^{n}L(f)$

with coefficient $C_{0}\neq 0$. Since $g\in\mathfrak{C}^{+K}$, there is a
standard monomial $J\in\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$, with
$L(I^{+}_{h}(J))=L(g)$. $J$ has the factor $(n,\dots,1|^{n}$, so
$I^{+}_{h}(J)$ has the factor $\Gamma^{n}$. Thus
$f-C_{0}\frac{I^{+}_{h}(J)}{{\Gamma}^{n}}\in\mathfrak{B}^{K}\cap\mathfrak{C}^{+K}_{\Gamma}$
with a lower leading monomial and
$\frac{I(J)}{{\Gamma}^{n}}\in\mathfrak{C}^{+K}$, by induction on the leading
monomial of $f$, we can show that
$\mathfrak{B}^{K}\cap\mathfrak{C}^{+K}_{\Gamma}=\mathfrak{C}^{+K}$. That is
${\mathfrak{B}^{K}}^{G_{\infty}}=\mathfrak{C}^{+K}$.

In general, let $\mathfrak{B}^{\prime}=K[a^{(k)}_{il},b^{(k)}_{jl}|\ 1\leq
i\leq p+h,\ 1\leq j\leq q,\ k\in\mathbb{Z}_{\geq 0}]$. Let
$\mathfrak{C}^{\prime}$ be the subspace of $G_{\infty}$-invariant elements. It
is generated by
${\bar{\partial}}^{k}X^{(0)}_{ij}={\bar{\partial}}^{k}\sum_{l}a^{(0)}_{il}b^{(0)}_{jl}$
and $I_{h}^{+}(J)$ with $J={\bar{\partial}}^{k}(i_{h},\dots,i_{1}|$ or
$J={\bar{\partial}}^{k}|j_{1},\dots,j_{h})$ by the previous argument. Let
${\mathcal{I}^{\prime}}$ be the ideal of $\mathfrak{B}^{\prime}$ generated by
$a^{(k)}_{il}$ with $i>p$. ${\mathcal{I}^{\prime}}$ is a
$G_{\infty}$-invariant subspace of $\mathfrak{B}^{\prime}$. Then
$\mathfrak{B}^{\prime}=\mathfrak{B}^{K}\oplus{\mathcal{I}^{\prime}}$.

$\mathfrak{C}^{\prime}={\mathfrak{B}^{\prime}}^{G_{\infty}}={\mathfrak{B}}^{G_{\infty}}\oplus{\mathcal{I}^{\prime}}^{G_{\infty}}.$

If $i>p$,
${\bar{\partial}}^{k}X^{(0)}_{ij}\in{\mathcal{I}^{\prime}}^{G_{\infty}}$. If
$J={\bar{\partial}}^{k}(i_{h},\dots,i_{1}|$ with $i_{h}>p$, So
${\mathfrak{B}^{\prime}}^{G_{\infty}}\cong\mathfrak{C}^{\prime}/{\mathcal{I}^{\prime}}^{G_{\infty}}$
is generated by ${\bar{\partial}}^{k}X^{(0)}_{ij}$, $1\leq i\leq p$, $1\leq
j\leq q$ and $I^{+}_{h}(J)$, $J\in\mathcal{J}_{h}^{L}\cup\mathcal{J}_{h}^{R}$.
And ${\mathfrak{B}^{K}}^{G_{\infty}}=\mathfrak{C}^{+K}$. ∎

###### Proof of Theorem 1.2.

By Corollary 3.6 and Theorem 4.2, $\mathfrak{R}_{h}^{+}$ is isomorphic to
$\mathfrak{B}^{J_{\infty}({SL}_{h}(K))}$ through $Q_{h}^{+}$. Then according
to the definition of $\mathfrak{B}_{h}^{+}$ and Corollary 3.6, the theorem is
true. ∎

Now by Theorem 4.2, $\mathfrak{C}^{+K}=K[J_{\infty}(V)/\\!\\!/J_{\infty}(G)]$.
The map $\bar{q}_{\infty}:J_{\infty}(V)/\\!\\!/J_{\infty}(G)\to
J_{\infty}(V/\\!\\!/G)$ induces a ring homomorphism
$\bar{q}^{*}_{\infty}:\tilde{\mathfrak{R}}_{h}^{+}\to\mathfrak{C}^{+K}$ of
affine coordinate rings given by (4.1). By Theorem 1.2, $\bar{q}^{*}$ is
surjective and the kernel of $\bar{q}^{*}$ is generated by Equation (1.11) and
(1.12) for $l\neq 0$.

######  Corollary 4.3.

If $2<h<\max(p,q)-2$, $\bar{q}_{\infty}:J_{\infty}(V)/\\!\\!/J_{\infty}(G)\to
J_{\infty}(V/\\!\\!/G)$ is not an isomorphism and $J_{\infty}(V/\\!\\!/G)$ is
not integral. Otherwise, $\bar{q}_{\infty}$ is an isomorphism and
$J_{\infty}(V/\\!\\!/G)$ is integral.

###### Proof.

$\mathfrak{A}=K[X^{(k)}_{ij},Y^{(k)}_{u_{1},\dots,u_{h}},Z^{(k)}_{v_{1},\dots,v_{h}}]$
has a $\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}$
grading given by

$\deg X^{(k)}_{ij}=(1,1,k),\quad\deg
Y^{(k)}_{u_{1},\dots,u_{h}}=(h,0,k),\quad\deg
Z^{(k)}_{v_{1},\dots,v_{h}}=(0,h,k).$

If $p>h-2$, we consider $\mathfrak{A}[2h,0,1]$, the subspace of elements with
degree $(2h,0,1)$. It is a linear combination of
$Y^{(0)}_{u_{1},\dots,u_{h}}Y^{(1)}_{u^{\prime}_{1},\dots,u^{\prime}_{h}}$.
Its intersection with the ideal
$\mathfrak{A}[2h,0,1]\cap\tilde{\mathcal{I}}_{h}^{+}$ is a linear combination
of

(4.2)
$\sum_{i}(-1)^{i}(Y^{(1)}_{u^{\prime}_{h},\dots,u^{\prime}_{2},u_{i}}Y^{(0)}_{u_{1},\dots,u_{i-1},u_{i+1},\dots,u_{h}}+Y^{(0)}_{u^{\prime}_{h},\dots,u^{\prime}_{2},u_{i}}Y^{(1)}_{u_{1},\dots,u_{i-1},u_{i+1},\dots,u_{h}}).$

Let

$f=\sum_{1\leq i<j\leq
h+2}(-1)^{i+j}Y^{(0)}_{h+3,\dots,6,i,j}Y^{(1)}_{1,\dots,i-1,i+1,\dots,j-1,j+1,\dots,h+2},$

then $\bar{q}^{*}_{\infty}(f)=0$ by Equation (2.8). But $f$ is not in
$\tilde{\mathcal{I}}_{h}^{+}$ since in (4.2),
$Y^{(1)}_{h+3,\dots}Y^{(0)}_{\dots}$ and $Y^{(0)}_{h+3,\dots}Y^{(1)}_{\dots}$
appear in pairs with the same sign, while in $f$ only
$Y^{(0)}_{h+3,\dots}Y^{(1)}_{\dots}$ appears. So $\bar{q}^{*}_{\infty}$ is not
injective and $\bar{q}_{\infty}$ is not an isomorphism.

By Lemma 4.1,
$\bar{q}^{\Gamma*}_{\infty}:(\tilde{\mathfrak{R}}_{h}^{+})_{\Gamma}\to\mathfrak{C}^{+K}_{\Gamma}$
is an isomorphism, so $f$ is zero in
$(\tilde{\mathfrak{R}}_{h}^{+})_{\Gamma}$. Then there is $n\in\mathbb{Z}_{\geq
0}$, $\Gamma^{n}f\in\tilde{\mathcal{I}}^{+}_{h}$. So
$\tilde{\mathfrak{R}}_{h}^{+}$ is not integral.

Similarly, if $q>h-2$, we can show that $\bar{q}_{\infty}$ is not an
isomorphism and $\tilde{\mathfrak{R}}_{h}^{+}$ is not integral.

$\bar{q}_{h}^{*}$ maps the generators of $\tilde{\mathfrak{R}}_{h}^{+}$ to the
generators $\mathfrak{C}_{h}^{+}$, and maps the generators of the ideal
$\tilde{\mathcal{I}}_{h}^{+}$ to the relations (1.10), and relations (1.11)
and (1.12) for $l=0$. If $h\leq 2$ or $h\geq\max(p,q)-2$, it is easy to check
that relations (1.11) and (1.12) are trivial for $2\leq l<i$. Also, these
relations are the same for $l=0$ and $l=1$. So $\tilde{\mathfrak{R}}_{h}^{+}$
and $\mathfrak{C}_{h}^{+K}$ have the same generators and relations, so
$\bar{q}^{*}_{h}$ is an isomorphism. Finally, $\tilde{\mathfrak{R}}_{h}^{+}$
is integral since $\mathfrak{C}_{h}^{+}$ is integral. ∎

######  Corollary 4.4.

If $\text{char}\ K=0$, the nilradical $\mathcal{N}$ in
$\tilde{\mathfrak{R}}_{h}^{+}$ is the kernel of $\bar{q}^{*}$, which is
generated by Equation (1.11) and (1.12) for $l\neq 0$.

###### Proof.

Since $\mathfrak{C}^{+K}$ is integral, $\mathcal{N}$ is in the kernel. By
Lemma 4.1, $J_{\infty}(V_{\Gamma})/\\!\\!/J_{\infty}(G)\to
J_{\infty}(V/\\!\\!/G)_{\Gamma}$ is an isomorphism. If there is
$f\in\tilde{\mathfrak{R}}_{h}^{+}$, $\bar{q}^{*}_{\infty}(f)=0$, we must have
$(Y^{(0)}_{h,\dots,1})^{n}f=0$. If $\text{ char }K=0$,
$\tilde{\mathfrak{R}}_{h}^{+}$ is irreducible, so $f\in\mathcal{N}$ since
$Y^{(0)}_{h,\cdots,1}$ is not in the nilradical. ∎

##  5\. Some properties of standard monomials

In this section, we study the properties of $||\mathcal{E}(E_{i+1})||$ and
$E_{i+1}$ that need to be satisfied to make $E_{1}E_{2}\cdots E_{n}$ a
standard monomial. These properties are similar to the properties of standard
monomials in [8], and so are their proofs.

Let

$E^{L}=((u_{h},k_{h}),\dots,(u_{1},k_{1})|\in\mathcal{E}^{L}_{h},\quad
J^{\prime
L}={\bar{\partial}}^{k^{\prime}}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{2},u^{\prime}_{1}|\in\mathcal{J}^{L}_{h^{\prime}},$
$E^{R}=|(v_{1},l_{1}),\dots,(v_{h},l_{h}))\in\mathcal{E}^{R}_{h},\quad
J^{\prime
R}={\bar{\partial}}^{l^{\prime}}|v^{\prime}_{1},v^{\prime}_{2},\dots,v^{\prime}_{h^{\prime}})\in\mathcal{J}^{R}_{h^{\prime}},$
$E=((u_{h},k_{h}),\dots,(u_{1},k_{1})|(v_{1},l_{1}),\dots,(v_{h},l_{h}))\in\mathcal{E},\quad
J^{\prime}={\bar{\partial}}^{n^{\prime}}(u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{1}|v^{\prime}_{1},\dots,v^{\prime}_{h^{\prime}})\in\mathcal{J}.$

For $h^{\prime}\leq h$, let

$E^{L}(h^{\prime})=((u_{h^{\prime}},k_{h^{\prime}}),\dots,(u_{1},k_{1})|,\quad
E^{R}(h^{\prime})=|(v_{1},l_{1}),\dots,(v_{h^{\prime}},l_{h^{\prime}})),$
$E(h^{\prime})=((u_{h^{\prime}},k_{h^{\prime}}),\dots,(u_{1},k_{1})|(v_{1},l_{1}),\dots,(v_{h^{\prime}},l_{h^{\prime}})).$

Then if $E,E^{\prime}\in\mathcal{E}$, $E\leq E^{\prime}$ if and only if
$sz(E)\geq sz(E^{\prime})=h^{\prime}$ and $E(h^{\prime})\leq E^{\prime}$.

###  $L(E^{L},J^{\prime L})$ and $R(E^{R},J^{\prime R})$

For $h^{\prime}\leq h$, let $\sigma_{L}$ and $\sigma_{R}$ be the permutations
of $\\{1,2,\dots,h^{\prime}\\}$ such that
$u_{\sigma_{L}(i)}<u_{\sigma_{L}(i+1)}$ and
$v_{\sigma_{R}(i)}<v_{\sigma_{R}(i+1)}$. Let $L(E^{L},J^{\prime
L})=L(E,J^{\prime})$ and $R(E^{R},J^{\prime R})=R(E,J^{\prime})$ be the
smallest non-negative integers $i_{0}$ and $j_{0}$ such that
$u^{\prime}_{i}\geq u_{\sigma_{L}(i-i_{0})}$, $i_{0}<i\leq h^{\prime}$ and
$v^{\prime}_{j}\geq v_{\sigma_{L}(j-j_{0})}$, $j_{0}<j\leq h^{\prime}$,
respectively.

Then

$L(E,J^{\prime})=L(E(L),J^{\prime}(L)),\quad
R(E,J^{\prime})=R(E(R),J^{\prime}(R)),$ $L(E^{L},J^{\prime
L})=L(E^{L}(h^{\prime}),J^{\prime L}),\quad R(E^{R},J^{\prime
R})=R(E^{R}(h^{\prime}),J^{\prime L}),$
$L(E,J^{\prime})=L(E(h^{\prime}),J^{\prime}),\quad
R(E,J^{\prime})=R(E(h^{\prime}),J^{\prime}).$

The following lemma is obvious.

######  Lemma 5.1.

For $J^{\prime\prime
L}={\bar{\partial}}^{k}(u^{\prime\prime}_{h^{\prime}},\dots,u^{\prime\prime}_{1})\in\mathcal{J}_{h^{\prime}}^{L}$,
if there are at least $s$ elements in
$\\{u^{\prime\prime}_{h^{\prime}},\dots,u^{\prime\prime}_{1}\\}$ from the set
$\\{u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{1}\\}$, then
$L(E^{L},J^{\prime\prime L})\geq L(E^{L},J^{\prime L})-h^{\prime}+s$;
For $J^{\prime\prime
R}={\bar{\partial}}^{k}(v^{\prime\prime}_{1},\dots,v^{\prime\prime}_{h^{\prime}})\in\mathcal{J}_{h^{\prime}}^{R}$,
if there are at least $s$ elements in
$\\{v^{\prime\prime}_{h^{\prime}},\dots,v^{\prime\prime}_{1}\\}$ from the set
$\\{v^{\prime}_{h^{\prime}},\dots,v^{\prime}_{1}\\}$, then
$R(E^{R},J^{\prime\prime R})\geq R(E^{R},J^{\prime R})-h^{\prime}+s$.

### A criterion for $J^{\prime}$ greater than $E$

We say that $J^{\prime L}$ is greater than $E^{L}$ if there is an element
$E^{\prime L}\in\mathcal{E}(J^{\prime L})$ with $E^{L}(h^{\prime})\leq
E^{\prime L}$. Similarly, we say that $J^{\prime R}$ is greater than $E^{R}$
if there is an element $E^{\prime R}\in\mathcal{E}(J^{\prime R})$ with
$E^{R}(h^{\prime})\leq E^{\prime R}$. We say that $J^{\prime}$ is greater than
$E$ if there is an element $E^{\prime}\in\mathcal{E}(J^{\prime})$ with $E\leq
E^{\prime}$. Then $J^{\prime}$ is greater than $E$ if and only if $J^{\prime}$
is greater than $E(h^{\prime})$.

The following lemma is from [8],

######  Lemma 5.2.

$J^{\prime}$ is greater than $E$ if and only if
$wt(E(h^{\prime}))-wt(J^{\prime})\geq L(E,J^{\prime})+R(E,J^{\prime})$.

The next result is analogous.

######  Lemma 5.3.

$J^{\prime L}$ is greater than $E^{L}$ if and only if
$wt(E^{L}(h^{\prime}))-wt(J^{\prime L})\geq L(E^{L},J^{\prime L})$. Similarly,
$J^{\prime R}$ is greater than $E^{R}$ if and only if
$wt(E^{R}(h^{\prime}))-wt(J^{\prime R})\geq R(E^{R},J^{\prime R})$.

###### Proof.

$J^{\prime L}$ is greater then $E^{L}$ if and only if there is an
$E^{\prime}\in\mathcal{E}(J^{\prime L})$ with $E^{L}\leq E^{\prime}$. Let

$E_{0}=|(h,0),\dots,(2,0),(1,0)),\qquad E_{1}=F(E^{L},E_{0}),\qquad
E_{2}=F(E^{\prime},E_{0}).$

Then $E^{L}\leq E^{\prime}$ if and only if $E_{1}\leq E_{2}$. We have
$wt(E_{1}(h^{\prime}))=wt(E^{L}(h^{\prime}))$, $wt(E_{2})=wt(J^{\prime L})$,
$L(E_{1},||E_{2}||)=L(E^{L},J^{\prime L})$ and $R(E_{1},E_{2})=0$. By Lemma
5.2, $E_{1}\leq E_{2}$ if and only if $wt(E^{L}(h^{\prime}))-wt(J^{\prime
L})\geq L(E^{L},J^{\prime L})$. The proof of the second statement is similar.
∎

######  Corollary 5.4.

$J^{\prime L}$ is greater than $E^{L}$ if and only if
$||E^{L}(h^{\prime})||J^{\prime L}$ is standard; $J^{\prime R}$ is greater
than $E^{R}$ if and only if $||E^{R}(h^{\prime})||J^{\prime R}$ is standard;
$J^{\prime}$ is greater than $E$ if and only if $||E(h^{\prime})||J^{\prime}$
is standard.

###### Proof.

We only prove the third statement, since the proofs of the others are similar.
By Lemma 5.3, $J^{\prime}$ is greater than $E$ if and only if
$wt(E(h^{\prime}))-wt(J^{\prime})\geq L(E,J^{\prime})+R(E,J^{\prime})$ and
$||E(h^{\prime})||J^{\prime}$ is standard if and only if
$wt(E(h^{\prime}))-wt(J^{\prime})\geq
L(E(h^{\prime}),J^{\prime})+R(E(h^{\prime}),J^{\prime})=L(E,J^{\prime})+R(E,J^{\prime})$.
∎

### The “largest” property

Let

$\displaystyle\mathcal{W}^{L}_{s}(E^{L},J^{\prime L})$ $\displaystyle=$
$\displaystyle\\{J^{L}={\bar{\partial}}^{k}(u^{\prime}_{i_{s}},\dots,u^{\prime}_{i_{1}}||\
1\leq i_{l}\leq h^{\prime},\ J^{L}\text{ is greater than }E^{L}\\};$
$\displaystyle\mathcal{W}^{R}_{s}(E^{R},J^{\prime R})$ $\displaystyle=$
$\displaystyle\\{J^{R}={\bar{\partial}}^{k}|v^{\prime}_{j_{1}},\dots,v^{\prime}_{j_{s}})|\
1\leq j_{l}\leq h^{\prime},\ J^{R}\text{ is greater than }E^{R}\\};$
$\displaystyle\mathcal{W}_{s}(E,J^{\prime})$ $\displaystyle=$
$\displaystyle\\{J={\bar{\partial}}^{k}(u^{\prime}_{i_{s}},\dots,u^{\prime}_{i_{1}}|v^{\prime}_{j_{1}},\dots,v^{\prime}_{j_{s}})|\
1\leq i_{l},j_{l}\leq h^{\prime},\ J\text{ is greater than }E\\}.$

Let $W_{s}^{L}(E,J^{\prime})=W_{s}^{L}(E(L),J^{\prime}(L))$ and
$W_{s}^{R}(E,J^{\prime})=W_{s}^{R}(E(R),J^{\prime}(R)).$

######  Lemma 5.5.

If $E^{\prime L}$ is the largest element in $\mathcal{E}(J^{\prime L})$ such
that $E^{L}\leq E^{\prime L}$, then for $s<h^{\prime}$, $||E^{\prime L}(s)||$
is the smallest element in $\mathcal{W}_{s}^{L}(E^{L},J^{\prime L})$. If
$E^{\prime R}$ is the largest element in $\mathcal{E}(J^{\prime R})$ such that
$E^{R}\leq E^{\prime R}$, then for $s<h^{\prime}$, $||E^{\prime R}(s)||$ is
the smallest element in $\mathcal{W}_{s}^{R}(E^{R},J^{\prime R})$. If
$E^{\prime}$ is the largest element in $\mathcal{E}(J^{\prime})$ such that
$E\leq E^{\prime}$, then for $s<h^{\prime}$, $||E^{\prime}(s)(L)||$ is the
smallest element in $\mathcal{W}_{s}^{L}(E,J^{\prime})$ and
$||E^{\prime}(s)(R)||$ is the smallest element in
$\mathcal{W}_{s}^{R}(E,J^{\prime})$, so $||E^{\prime}(s)||$ is the smallest
element in $\mathcal{W}_{s}(E,E^{\prime})$.

###### Proof.

The third statement is Lemma 5.4 of [8]. We use a similar method to prove the
first statement. The proof of the second statement is similar and is omitted.
Assume

$E^{\prime
L}=((u^{\prime}_{h^{\prime}},k_{h^{\prime}}),\dots,(u^{\prime}_{2},k_{2}),(u^{\prime}_{1},k_{1})|.$

For $s<h^{\prime}$, let $J^{L}_{s}$ be the smallest element in
$\mathcal{W}_{s}^{L}(E^{L},J^{\prime L})$. Let

$E_{s}=((u^{\prime}_{i_{s}},\tilde{k}_{s}),\dots,(u^{\prime}_{i_{2}},\tilde{k}_{2}),(u^{\prime}_{i_{1}},\tilde{k}_{1})|$

be the largest element in $\mathcal{E}(J^{L}_{s})$ such that $E^{L}(s)\leq
E_{s}$.

Assume $l$ is the largest number such that
$(u^{\prime}_{j},k^{\prime}_{j})=(u^{\prime}_{i_{j}},\tilde{k}_{j})$ for
$j<l$. If $l=s+1$, then $E_{s}=E^{\prime L}(s)$, so $||E^{\prime L}(s)||$ is
the smallest element in $\mathcal{W}_{s}^{L}(E^{L},E^{\prime L})$. Otherwise
$l\leq s$, then $i_{l}\geq l$ and
$(u^{\prime}_{i_{l}},\tilde{k}_{l})\neq(u^{\prime}_{l},k^{\prime}_{l})$. If
$i_{l}=l$, by the maximality of $E^{\prime L}$ and the minimality of
$J_{s}^{L}$, we must have
$(u^{\prime}_{i_{l}},\tilde{k}_{l})=(u^{\prime}_{l},k^{\prime}_{l})$. This is
a contradiction, so $i_{l}>l$.

If $(u^{\prime}_{i_{l}},\tilde{k}_{l})<(u^{\prime}_{l},k^{\prime}_{l}))$, then
$(u^{\prime}_{l},k^{\prime}_{l}+k^{\prime}_{i_{l}}-\tilde{k}_{l})>(u^{\prime}_{i_{l}},k^{\prime}_{i_{l}})$.
Let $E^{\prime\prime}$ be the element in
$\mathcal{E}_{h^{\prime}}^{L}(J^{\prime L})$ by replacing
$(u^{\prime}_{l},k^{\prime}_{l})$ and
$(u^{\prime}_{i_{l}},k^{\prime}_{i_{l}})$ in $E^{\prime L}$ by
$(u^{\prime}_{i_{l}},\tilde{k}_{l})$ and
$(u^{\prime}_{l},k^{\prime}_{l}+k^{\prime}_{i_{l}}-\tilde{k}_{l})$,
respectively. We have $E^{\prime L}\prec E^{\prime\prime}$ and
$E^{L}(h^{\prime})\leq E^{\prime\prime}$. But $E^{\prime L}\neq
E^{\prime\prime}$ is the largest element in $\mathcal{E}(||E^{\prime L}||)$
such that $E^{L}\leq E^{\prime L}$, a contradiction.

Assume $(u^{\prime}_{i_{l}},\tilde{k}_{l})>(u^{\prime}_{l},k^{\prime}_{l})$.
If $l\notin\\{i_{1},\dots,i_{s}\\}$, replacing
$(u^{\prime}_{i_{l}},\tilde{k}_{l})$ in $E_{s}$ by
$(u^{\prime}_{l},k^{\prime}_{l})$, we get $E^{\prime}_{s}$ with $E^{L}(s)\leq
E^{\prime}_{s}$ and $||E^{\prime}_{s}||\prec J_{s}$. This is impossible since
$J_{s}\neq||E^{\prime}_{s}||$ is the smallest element in
$\mathcal{W}_{s}(E^{L},E^{\prime L})$. If $l=i_{j}\in\\{i_{1},\dots,i_{s}\\}$,
$(u^{\prime}_{i_{l}},\tilde{k}_{l}+\tilde{k}_{j}-k^{\prime}_{l})>(u^{\prime}_{i_{j}},\tilde{k}_{j})$.
Let $E^{\prime}_{s}$ be the sequence in $\mathcal{E}(J_{s})$ obtained by
replacing $(u^{\prime}_{i_{l}},\tilde{k}_{l})$ and
$(u^{\prime}_{i_{j}},\tilde{k}_{j})$ in $E^{\prime}_{s}$ by
$(u^{\prime}_{l},k^{\prime}_{l})$ and
$(u^{\prime}_{i_{l}},\tilde{k}_{l}+\tilde{k}_{j}-k^{\prime}_{l})$,
respectively. We have $E_{s}\prec E^{\prime}_{s}$ and $E^{L}\leq
E^{\prime}_{s}$. But $E^{\prime}_{s}\neq E_{s}$ is the largest element in
$\mathcal{E}(J_{s})$ such that $E^{L}\leq E^{\prime}_{s}$, a contradiction. ∎

######  Corollary 5.6.

If $E^{\prime L}$ is the largest element in $\mathcal{E}(||E^{\prime L}||)$
such that $E^{L}\leq E^{\prime L}$, then for $s<h^{\prime}$,

$L(E,||E^{\prime}(s)||)=wt(E^{\prime L}(s))-wt(E^{L}(s));$

if $E^{\prime R}$ is the largest element in $\mathcal{E}(||E^{\prime R}||)$
such that $E^{R}\leq E^{\prime R}$, then for $s<h^{\prime}$,

$R(E^{R},||E^{\prime R}(s)||)=wt(E^{\prime R}(s))-wt(E^{R}(s));$

If $E^{\prime}$ is the largest element in $\mathcal{E}(||E^{\prime}||)$ such
that $E\leq E^{\prime}$, then for $s<h^{\prime}$,

$L(E,||E^{\prime}(s)||)+R(E,||E^{\prime}(s)||)=wt(E^{\prime}(s))-wt(E(s)).$

###### Proof.

We only prove the first statement since the proofs of the other statements are
similar. Since $E^{L}\leq E^{\prime L}$, $E^{L}\leq E^{\prime L}(s)$. By Lemma
5.5, $||E^{\prime}(s)||$ is the smallest element in
$\mathcal{W}^{L}_{s}(E^{L},J^{\prime L})$. By Lemma 5.3,
$L(E,||E^{\prime}(s)||)=wt(E^{\prime L}(s))-wt(E^{L}(s))$. ∎

######  Corollary 5.7.

If $E^{\prime L}$ is the largest element in $\mathcal{E}(||E^{\prime L}||)$
such that $E^{L}\leq E^{\prime L}$, then for $s<h^{\prime}$, and any
$J^{L}\in\mathcal{W}_{s}^{L}(E^{L},E^{\prime L})$,

$L(E^{L},||E^{\prime L}(s)||)\leq L(E^{L},J^{L});$

If $E^{\prime R}$ is the largest element in $\mathcal{E}(||E^{\prime R}||)$
such that $E^{R}\leq E^{\prime R}$, then for $s<h^{\prime}$, and
$J^{R}\in\mathcal{W}_{s}^{R}(E^{R},E^{\prime R})$,

$R(E^{R},||E^{\prime R}(s)||)\leq R(E^{R},J^{R});$

If $E^{\prime}$ is the largest element in $\mathcal{E}(||E^{\prime}||)$ such
that $E\leq E^{\prime}$, then for $s<h^{\prime}$ and any
$J\in\mathcal{W}_{s}(E,E^{\prime})$,

$L(E,||E^{\prime}(s)||)\leq L(E,J),\quad R(E,||E^{\prime}(s)||)\leq R(E,J).$

###### Proof.

We only prove the first statement since the proofs of the others are similar.
By Lemma 5.5, $||E^{\prime L}(s)||$ is the smallest element in
$\mathcal{W}_{s}^{L}(E^{L},E^{\prime L})$, so $wt(E^{\prime L}(s))\leq
wt(J^{L})$. By Lemma 5.6 and Lemma 5.3,

$L(E^{L},||E^{\prime L}(s)||)=wt(E^{\prime L}(s))-wt(E^{L}(s))\leq
wt(J^{L})-wt(E^{L}(s))\leq L(E^{L},J^{L}).$

∎

We have the following lemma from [8].

######  Lemma 5.8.

Let
$E_{i}=((u^{i}_{h_{i}},k^{i}_{h_{i}}),\dots,(u^{i}_{1},k^{i}_{1})|(v^{i}_{1},l^{i}_{1}),\dots,(v^{i}_{h_{i}},l^{i}_{h_{i}}))$,
$i=a,b$. Suppose that $E_{b}\leq E_{a}$, and that $E_{a}$ is the largest
element in $\mathcal{E}(||E_{a}||)$ such that $E_{b}\leq E_{a}$. Let $1\leq
h<h_{a}$ and let $\sigma_{i}$, $\sigma^{\prime}_{i}$ be permutations of
$\\{1,\dots,h\\}$, such that
$u^{i}_{\sigma_{i}(1)}<u^{i}_{\sigma_{i}(2)}<\cdots<u^{i}_{\sigma_{i}(h)}$ and
$v^{i}_{\sigma^{\prime}_{i}(1)}<v^{i}_{\sigma^{\prime}_{i}(2)}<\cdots<v^{i}_{\sigma^{\prime}_{i}(h)}$.
Let $v_{1}^{\prime},\dots,v^{\prime}_{h_{a}}$ be a permutation of
$v^{a}_{1},\dots,v^{a}_{h_{a}}$ such that
$v_{1}^{\prime}<v_{2}^{\prime}<\cdots<v^{\prime}_{h_{a}}$. Let
$u_{1}^{\prime},\dots,u^{\prime}_{h_{a}}$ be a permutation of
$u^{a}_{1},\dots,u^{a}_{h_{a}}$ such that
$u_{1}^{\prime}<u_{2}^{\prime}<\cdots<u^{\prime}_{h_{2}}$.

  1. (1)

Assume $u^{\prime}_{i_{2}}=v^{a}_{\sigma(i_{1})}$ with $i_{2}>i_{1}$, then for
any

$K={\bar{\partial}}^{k}(u_{h},\dots,u_{s+1},u^{\prime}_{t_{s}},\dots,u^{\prime}_{t_{1}}|v_{1},\dots,v_{h}),$

with $t_{1}<t_{2}<\cdots<t_{s}<i_{2}$,
$L(E_{b},K)>L(E_{b},||E_{a}(h)||)+s-i_{1}$.

  2. (2)

Assume $v^{\prime}_{j_{2}}=v^{a}_{\sigma(j_{1})}$ with $j_{2}>j_{1}$, then for
any

$K={\bar{\partial}}^{k}(u_{h},\dots,u_{1}|v^{\prime}_{t_{1}},\dots,v^{\prime}_{t_{s}},v_{s+1},\dots,v_{h}),$

with $t_{1}<t_{2}<\cdots<t_{s}<j_{2}$,
$R(E_{b},K)>R(E_{b},||E_{a}(h)||)+s-j_{1}$.

The following lemmas are obvious:

######  Lemma 5.9.

If $J_{1}\prec J_{2}\prec\cdots\prec J_{n}$, $\sigma$ is a permutation of
$\\{1,\dots,n\\}$, then

$J_{1}J_{2}\cdots J_{n}\prec J_{\sigma(1)}\cdots J_{\sigma(n)}.$

######  Lemma 5.10.

If $K_{1}K_{2}\cdots K_{k}\prec J_{1}\cdots J_{l}$, then

$K_{1}K_{2}\cdots K_{s-1}JK_{s}\cdots K_{k}\prec J_{1}\cdots
J_{s-1}JJ_{s}\cdots J_{l}.$

##  6\. Proof of Lemma 2.9

In this section we prove Lemma 2.9. By Lemma 5.9, we can assume each monomial
is expressed as an ordered product $J_{1}J_{2}\cdots J_{b}$ with $J_{a}\prec
J_{a+1}$. For $\alpha\in\mathcal{M}(\mathcal{J}_{h}^{+})$, let

$\mathfrak{R}_{h}^{+}(\alpha)=\\{\sum
c_{i}\beta_{i}\in\mathfrak{R}_{h}^{+}|c_{i}\in\mathbb{Z},\beta_{i}\in\mathcal{M}(\mathcal{J}_{h}^{+}),\beta_{i}\prec\alpha,\beta_{i}\neq\alpha\\},$

the space of linear combinations of elements preceding $\alpha$ in
$\mathcal{M}(\mathcal{J}^{+}_{h})$ with integer coefficients. The following
lemma is from [8].

######  Lemma 6.1.

Let $E\in\mathcal{E}$, $J_{a}$ and $J_{b}$ in $\mathcal{J}$ with $J_{a}\prec
J_{b}$, and suppose that $E_{a}$ is the largest element in
$\mathcal{E}(J_{a})$ such that $E\leq E_{a}$. If $J_{b}$ is not greater than
$E_{a}$, then $J_{a}J_{b}=\sum K_{i}f_{i}$ with $K_{i}\in\mathcal{J}$,
$f_{i}\in\mathfrak{R}$ such that $K_{i}$ is either smaller than $J_{a}$, or
$K_{i}$ is not greater than $E$.

######  Lemma 6.2.

Assume $J_{1}J_{2}$ is not standard. If $J_{1},J_{2}\in\mathcal{J}_{h}^{L}$,
$J_{1}J_{2}$ can be written as a linear combination of ordered products of
elements of $\mathcal{J}_{h}^{L}$ preceding $J_{1}$ with integer coefficients.
If $J_{1},J_{2}\in\mathcal{J}_{h}^{R}$ is not standard, $J_{1}J_{2}$ can be
written as a linear combination of ordered products of elements of
$\mathcal{J}_{h}^{R}$ preceding $J_{1}$ with integer coefficients.

###### Proof.

We only prove the first statement, since the proof of the second statement is
similar. Assume
$J_{i}={\bar{\partial}}^{n_{i}}(u^{i}_{h},\dots,u^{i}_{2},u^{i}_{1}|$ ,
$i=1,2$. Let $n=n_{2}-n_{1}\geq 0$. Since $J_{1}J_{2}$ is not standard, by
Lemma 5.3, there is an integer $i>n$, $u^{2}_{i}<u^{1}_{i-n}$. By Equation
(2.23),

$\sum\epsilon
a_{k}\left(\begin{array}[]{c}{\bar{\partial}}^{m-k}(\underline{u^{1}_{h},\dots,u^{1}_{i-n}},u^{1}_{i-n-1},\dots,u^{1}_{1}|\\\
{\bar{\partial}}^{k}(u^{2}_{h},\dots,u^{2}_{i+1},\underline{u^{2}_{i},\dots,u^{2}_{1}}|\end{array}\right)=0.$

Here $m=n_{1}+n_{2}$, $a_{n_{2}-l}=\delta_{0,l}$ for $0\leq l\leq n$. In the
above equation:

  * •

All the terms with $k=n_{2}$ are preceding $J_{1}$ in the lexicographic order
except $J_{1}J_{2}$ itself;

  * •

All the terms with $k=n_{2}-1,\dots,n_{1}$ vanish since $a_{k}=0$;

  * •

All the terms with $k=n_{2}+1,\dots,m$ are preceding $J_{1}$ since the weight
of the upper sequence is $m-k<n_{1}$.

  * •

All the terms with $k=0,\dots,n_{1}-1$ are preceding $J_{1}$ after exchanging
the upper sequence and the lower sequence since the weight of the lower
sequence is $k<n_{1}$.

Thus, $J_{1}J_{2}$ can be written as a linear combination of ordered products
of elements of $\mathcal{J}_{h}^{L}$ preceding $J_{1}$ with integer
coefficients. ∎

###### Proof of Lemma 2.9..

Since $J_{1}$ is always standard, we can asssume $b\geq 2$. Assume
$J_{1}\cdots J_{b}$ is not standard.

If $J_{b}\in J_{h}^{R}\cup J_{h}^{L}$, then $J_{i}\in J_{h}^{R}\cup J_{h}^{L}$
for $1\leq i\leq b$. By Lemma 6.2 and Corollary 5.4, it is easy to see that
$J_{1}\cdots J_{b}\in\mathfrak{R}_{h}^{+}(J_{1}\cdots J_{b-1})$.

If
$J_{b}={\bar{\partial}}^{n_{b}}(u^{b}_{h_{b}},\dots,u^{b}_{1}|v^{b}_{1},v^{b}_{2},\dots,v^{b}_{h_{b}})\notin
J_{h}^{R}\cup J_{h}^{L}$, then $h_{b}<h$. We show the lemma by induction on
$b$.

For $b=2$, $J_{1}J_{2}$ is not standard by assumption.

  1. (1)

If $J_{1}={\bar{\partial}}^{n_{1}}(u^{1}_{h},\dots,u^{1}_{1}|$, there is an
integer $i>n_{2}$ such that $u^{2}_{i}<u^{1}_{i-n_{2}}$. Let $m=n_{1}+n_{2}$,
by Equation (2.17),

$\sum\epsilon
a_{k}\left(\begin{array}[]{ccc}{\bar{\partial}}^{m-k}(\underline{u^{1}_{h},\dots,u^{1}_{i-n_{2}}},u^{1}_{i-n_{2}-1},\dots,u^{1}_{1}&|&\\\
{\bar{\partial}}^{k}(u^{2}_{h_{2}},\dots,u^{2}_{i+1},\underline{u^{2}_{i},\dots,u^{2}_{1}}&|&v^{2}_{1},v^{2}_{2},\dots,v^{2}_{h_{2}})\end{array}\right)=0$

with $a_{n_{2}-l}=\delta_{0,l}$ for $0\leq l\leq n_{2}$. In the above
equation,

    1. (a)

All the terms with $k=n_{2}$ are preceding $J_{1}$ in the lexicographic order
except $J_{1}J_{2}$ itself;

    2. (b)

All the terms with $k=0,\dots,n_{2}-1$ vanish since $a_{k}=0$;

    3. (c)

All the terms with $k=n_{2}+1,\dots,m$ are preceding $J_{1}$, since the weight
of the upper sequence is $m-k<n_{1}$.

Thus, $J_{1}J_{2}\in\mathfrak{R}_{h}^{+}(J_{1})$.

  2. (2)

If $J_{1}={\bar{\partial}}^{n_{1}}|v^{1}_{1},\dots,v^{1}_{h})$, similar to
case (1), we can show $J_{1}J_{2}\in\mathfrak{R}_{h}^{+}(J_{1})$.

  3. (3)

If $J_{1}\notin\mathcal{J}_{h}^{R}\cup\mathcal{J}_{h}^{L}$, then
$J_{1}\in\mathcal{J}$. By Lemma 7.1 of [8],
$J_{1}J_{2}\in\mathfrak{R}(J_{1})$, so
$J_{1}J_{2}\in\mathfrak{R}_{h}^{+}(J_{1})$.

For $b>2$, let $a=b-1$. By induction, the lemma is true for $J_{1}\cdots
J_{a}$: if $J_{1}\cdots J_{a}$ is not standard, then $J_{1}\cdots
J_{a}\in\mathfrak{R}_{h}^{+}(J_{1}\cdots J_{a-1})$. So we can assume
$J_{1}\cdots J_{a}$ and $J_{1}\cdots J_{b-2}J_{b}$ are standard by Lemma 5.10.
Let $E_{1}\cdots E_{a}\in\mathcal{S}\mathcal{M}(\mathcal{J}_{h}^{+})$ be the
standard ordered product of elements of $\mathcal{E}_{h}^{+}$ corresponding to
$J_{1}\cdots J_{a}$. To show $J_{1}\cdots
J_{b}\in\mathfrak{R}_{h}^{+}(J_{1}\cdots J_{a})$, we only need to show
$(*)$ $J_{a}J_{b}=\sum K_{i}f_{i}$ with $K_{i}\in\mathcal{J}_{h}^{+}$ and
$f_{i}\in\mathfrak{R}_{h}^{+}$ such that either $K_{i}$ is smaller than
$J_{a}$ or $J_{1}\cdots J_{b-2}K_{i}$ is not standard.
In the first case $J_{1}\cdots
J_{b-2}K_{i}f_{i}\in\mathfrak{R}_{h}^{+}(J_{1}\cdots J_{a})$ and in the second
case $J_{1}\cdots J_{b-2}K_{i}\in\mathfrak{R}_{h}^{+}(J_{1}\cdots J_{b-2})$ by
induction, then $J_{1}\cdots
J_{b-2}K_{i}f_{i}\in\mathfrak{R}_{h}^{+}(J_{1}\cdots J_{a})$.

Now let us show the statement $(*)$. The proof is divided into four cases.
Case 1: $J_{a}\in\mathcal{J}_{h}^{L}$, then $J_{i}\in\mathcal{J}_{h}^{L}$ for
$1\leq i\leq a$.

Assume $E_{i}=((u^{i}_{h},k^{i}_{h}),\dots,(u^{i}_{1},k^{i}_{1})|$. Let
$m_{i}=wt(E_{i}(h_{b}))$ and $n_{i}=wt(E_{i})$. Let $\sigma_{i}$ be the
permutations of ${1,2,\dots,h_{b}}$ such that
$u^{i}_{\sigma_{i}(j)}<u^{i}_{\sigma_{i}(j+1)}$ for $1\leq j\leq h_{b-1}$. Let
$u^{\prime}_{h},\dots,u^{\prime}_{1}$ be a permutation of
$u^{a}_{h},\dots,u^{a}_{1}$ such that $u^{\prime}_{i}\leq u^{\prime}_{i+1}$.
Let $m=n_{a}+n_{b}$.

  1. (1)

If $n_{b}<m_{a}$. By Equation (2.17),

(6.1) $\sum_{0\leq
i<h_{b}}\sum_{\sigma}\frac{(-1)^{i}\operatorname{sign}(\sigma)}{i!(h_{b}-i)!}\sum_{k=0}^{m}\epsilon
a^{i}_{k}\left(\begin{array}[]{ccc}{\bar{\partial}}^{m-k}(\underline{u_{h}^{a},\dots,u_{i+1}^{a}},u^{b}_{\sigma{(i)}},\dots,u_{\sigma(2)}^{b},u^{b}_{\sigma(1)}&|&\\\
{\bar{\partial}}^{k}(\underline{u_{\sigma(h_{b})}^{b},\dots,u^{b}_{\sigma{(i+1)}},u^{a}_{i},\dots,u^{a}_{1}}&|&v^{b}_{1},\dots,v^{b}_{h_{b}})\end{array}\right)=0.$

Here $a_{k}^{i}$ are integers and $a_{n_{b}-l}^{i}=\delta_{0,l}$ for $0\leq
l<h_{b}-i$, and $\sigma$ are permutations of $\\{1,\dots,h_{b}\\}$. In the
above equation:

    1. (a)

All the terms with $k=n_{b}+1,\dots,n$ are preceding $J_{a}$ since the weight
of the upper sequence is $m-k<n_{a}$.

    2. (b)

The terms with $k=n_{b}$ are $J_{a}J_{b}$ and the terms with the lower
sequences

$K_{0}={\bar{\partial}}^{n_{b}}(u_{i_{h_{b}}}^{\prime},\dots,u_{i_{2}}^{\prime},u^{\prime}_{i_{1}}|v^{b}_{1},v^{b}_{2},\dots,v^{b}_{h_{b}}).$

And all of the other terms cancel. Let

$K^{\prime}_{0}={\bar{\partial}}^{n_{b}}(u_{i_{h_{b}}}^{\prime},\dots,u_{i_{2}}^{\prime},u^{\prime}_{i_{1}}|.$

By Corollary 5.6 and 5.7

$L(E_{b-2},K^{\prime}_{0}))\geq
L(E_{b-2},||E_{a}(h_{b})||))=wt(E_{a}(h_{b}))-wt(E_{b-2}(h_{b}))>n_{b}-m_{b-2}.$

By Lemma 5.3, $K^{\prime}_{0}$ is not greater than $E_{b-2}$. So $J_{1}\cdots
J_{b-2}K_{0}$ is not standard.

    3. (c)

The terms with $k<n_{b}$ in Equation (6.1) vanish unless $h_{b}-i\leq
n_{b}-k$. In this case the lower sequences of the terms are

$K_{1}={\bar{\partial}}^{k}(\underline{u_{\sigma(h_{b})}^{b},\dots,u^{b}_{\sigma{(i+1)}}},u^{\prime}_{s_{i}},\dots,u^{\prime}_{s_{1}}|v^{b}_{1},v^{b}_{2},\dots,v^{b}_{h_{b}}).$

Let

$K^{\prime}_{1}={\bar{\partial}}^{k}(\underline{u_{\sigma(h_{b})}^{b},\dots,u^{b}_{\sigma{(i+1)}}},u^{\prime}_{s_{i}},\dots,u^{\prime}_{s_{1}}|.$

By Lemma 5.1,

$L(E_{b-2},||K^{\prime}_{1}||)\geq
L(E_{b-2},||E_{a}(h_{b})||)-(h_{b}-i)=m_{a}-m_{b-2}-(h_{b}-i)>k-m_{b-2}.$

By Lemma 5.3, $K^{\prime}_{1}$ is not greater than $E_{b-2}$. So $J_{1}\cdots
J_{b-2}K_{1}$ is not standard.

By the above analysis, $J_{1}\cdots J_{b}\in\mathfrak{R}_{h}^{+}(J_{1}\cdots
J_{a})$.

  2. (2)

If $n_{b}\geq m_{a}$.

Let $J^{\prime}_{b}={\bar{\partial}}^{n_{b}}(u^{b}_{h_{b}},\dots,u^{b}_{1}|$
and $i_{0}=L(E_{a},J^{\prime}_{b}).$ Since $J_{1}\cdots J_{b}$ is not
standard, $J^{\prime}_{b}$ is not greater than $E_{a}$. By Lemma 5.3,

(6.2) $i_{0}>n_{b}-m_{a}\geq 0.$

There is $i_{0}\leq i_{1}\leq h_{b}$, such that
$u^{b}_{i_{1}}<u^{a}_{\sigma_{a}(i_{1}-i_{0}+1)}$.

Assume $u^{\prime}_{i_{2}}=u^{a}_{\sigma_{a}(i_{1}-i_{0}+1)}$, then $i_{2}\geq
i_{1}-i_{0}+1$. By Equation (2.17),

$0=\sum_{\begin{subarray}{c}i_{1}>s\\\ i_{2}-1\geq
s\end{subarray}}\sum_{\sigma}\frac{(-1)^{s}\operatorname{sign}(\sigma)}{s!(i_{2}-1-s)!}\sum_{k=0}^{n}\epsilon
a^{s}_{k}$ (6.3)
$\left(\begin{array}[]{ccc}{\bar{\partial}}^{m-k}(\underline{u^{\prime}_{h},\dots,u^{\prime}_{i_{2}},u^{\prime}_{\sigma(i_{2}-1)},\dots,u^{\prime}_{\sigma(s+1)},u_{s}^{b},\dots,u^{b}_{1}}&|&\\\
{\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{i_{1}+1},\underline{u^{b}_{i_{1}},\dots,u^{b}_{s+1}},u^{\prime}_{\sigma(s)},\dots,u^{\prime}_{\sigma(1)}&|&v^{b}_{1},\dots,v^{b}_{h_{b}})\end{array}\right).$

Here $a_{k}^{s}$ are integers and $a_{n_{b}-l}^{s}=\delta_{0,l}$ for $0\leq
l<i_{1}-s$, and $\sigma$ are permutations of $\\{1,\dots,i_{2}-1\\}$.

$0=\sum_{\begin{subarray}{c}i\geq i_{1}>s\\\ i_{2}-1\geq
s\end{subarray}}\sum_{\sigma,\sigma^{\prime}}\frac{(-1)^{i+s}\operatorname{sign}(\sigma^{\prime})\operatorname{sign}(\sigma)}{(i-i_{1})!(h_{b}-i)!s!(i_{2}-1-s)!}\sum_{k=0}^{n}\epsilon
a^{i,s}_{k}$

(6.4)
$\left(\begin{array}[]{ccc}{\bar{\partial}}^{n-k}(u^{b}_{\sigma^{\prime}(h_{b})},\dots,\underline{u^{b}_{\sigma^{\prime}(i)},\dots
u^{b}_{\sigma^{\prime}(i_{1}+1)},u^{b}_{i_{1}},\dots
u^{b}_{1},u^{\prime}_{h_{a}},\dots u^{\prime}_{h_{b}+1}}&|&\\\
{\bar{\partial}}^{k}(\underline{u^{\prime}_{h_{b}},\dots,u^{\prime}_{i_{2}},u^{\prime}_{\sigma(i_{2}-1)},\dots,u^{\prime}_{\sigma(s+1)}},u^{\prime}_{\sigma(s)},\dots,u^{\prime}_{\sigma(1)}&|&v^{b}_{1},v^{b}_{2},\dots
v^{b}_{h_{b}})\end{array}\right).$

Here $a_{k}^{i,s}$ are integers, $a_{n_{b}-l}^{i,s}=\delta_{0,l}$ for $0\leq
l<i-s$, $\sigma$ are permutations of $\\{1,\dots,i_{2}-1\\}$, and
$\sigma^{\prime}$ are permutations of $\\{i_{1}+1,i_{1}+2,\dots,h_{b}\\}$.

If $i_{2}>i_{1}-i_{0}+1$, we use Equation (6.3) and if $i_{2}=i_{1}-i_{0}+1$
we use Equation (6.4).

In the above Equations:

    1. (a)

All the terms with $k=n_{b}+1,\dots,n$ are preceding $J_{a}$ since the weight
of the upper sequence is $m-k<n_{a}$.

    2. (b)

The terms with $k=n_{b}$ are $J_{a}J_{b}$, the terms with the upper sequences
preceding $J_{a}$ (the upper sequences are the sequences given by replacing
some $u^{\prime}_{i}$, $i\geq i_{2}$ in $J_{a}$ by some $u^{b}_{k}$, $k\leq
i_{1}$), and the terms with the lower sequences

$K_{0}={\bar{\partial}}^{n_{b}}(u_{h_{b}}^{b},\dots,u_{i_{1}+1}^{b},u^{\prime}_{\sigma_{1}(i_{1})},\dots
u^{\prime}_{\sigma_{1}(1)}|v^{b}_{1},v^{b}_{2},\dots,u^{b}_{h_{b}}).$

All of the other terms cancel. Let

$K^{\prime}_{0}={\bar{\partial}}^{n_{b}}(u_{h_{b}}^{b},\dots,u_{i_{1}+1}^{b},u^{\prime}_{\sigma_{1}(i_{1})},\dots
u^{\prime}_{\sigma_{1}(1)}|.$

If $i_{2}-1<i_{1}$, the terms of the form $K_{0}$ do not appear in Equations
(6.3) and (6.4). Otherwise $i_{2}-1\geq i_{1}$, by Lemma 5.8,

$L(E_{b-2},K^{\prime}_{0})>L(E_{b-2},||E_{a}(h_{b})||)+i_{1}-(i_{1}-i_{0}+1).$

By Corollary 5.6,

(6.5) $L(E_{b-2},||E_{a}(h_{b})||)=m_{a}-m_{b-2}.$

Then by Equation (6.2),

$L(E_{b-2},K^{\prime}_{0})\geq m_{a}-m_{b-2}+i_{0}>n_{b}-m_{b-2}.$

By Lemma 5.3, $K^{\prime}_{0}$ is not greater than $E_{b-2}$. So there is no
$E_{b}^{\prime}\in\mathcal{E}(K_{0})$, such that $E_{b-2}\leq E_{b}^{\prime}$.
$J_{1}\cdots J_{b-2}K_{0}$ is not standard.

    3. (c)

If $i_{2}>i_{1}-i_{0}+1$, the terms with $k<n_{b}$ in Equation (6.3) vanish
unless $i_{1}-s\leq n_{b}-k$. In this case the lower sequences of the terms
are

$K_{1}={\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{i_{1}+1},\underline{u^{b}_{i_{1}},\dots,u^{b}_{s+1}},u^{\prime}_{\sigma(s)},\dots,u^{\prime}_{\sigma(1)}|v^{b}_{1},\dots,v^{b}_{h_{b}}).$

The underlined elements can be any underlined elements in Equation (6.3). Let

$K^{\prime}_{1}={\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{i_{1}+1},\underline{u^{b}_{i_{1}},\dots,u^{b}_{s+1}},u^{\prime}_{\sigma(s)},\dots,u^{\prime}_{\sigma(1)}|.$

By Lemma 5.8,

$L(E_{b-2},K^{\prime}_{1})>L(E_{b-2},||E_{a}(h_{b})||)+s-(i_{1}-i_{0}+1).$

Then by Equations (6.5) and (6.2),

$L(E_{b-2},K^{\prime}_{0})\geq m_{a}-m_{b-2}+i_{0}-(i_{1}-s)>k-m_{b-2}.$

By Lemma 5.3, $K^{\prime}_{1}$ is not greater than $E_{b-2}$. So there is no
$E_{b}^{\prime}\in\mathcal{E}_{h}^{b}(K_{1})$, such that $E_{b-2}\leq
E_{b}^{\prime}$. $J_{1}\cdots J_{b-2}K_{1}$ is not standard.

    4. (d)

If $i_{2}=i_{1}-i_{0}+1$, the terms with $k<n_{b}$ in Equation (6.4) vanish
unless $i-s\leq n_{b}-k$. In this case the lower sequences of the terms are

$K_{1}={\bar{\partial}}^{k}(\underline{u^{\prime}_{h_{b}},\dots,u^{\prime}_{i_{2}},u^{\prime}_{\sigma(i_{2}-1)},\dots,u^{\prime}_{\sigma(s+1)}},u^{\prime}_{\sigma(s)},\dots,u^{\prime}_{\sigma(1)}|v^{b}_{1},v^{b}_{2},\dots,v^{b}_{h_{b}})$

Underlined elements can be any underlined elements in Equation (6.4). Let

$K^{\prime}_{1}={\bar{\partial}}^{k}(\underline{u^{\prime}_{h_{b}},\dots,u^{\prime}_{i_{2}},u^{\prime}_{\sigma(i_{2}-1)},\dots,u^{\prime}_{\sigma(s+1)}},u^{\prime}_{\sigma(s)},\dots,u^{\prime}_{\sigma(1)}|$

Let

$K^{\prime}_{0}={\bar{\partial}}^{m_{a}-1}(u^{\prime}_{l_{h_{b}}},\dots,u^{\prime}_{l_{i_{2}}},u^{\prime}_{i_{2}-1},\dots,u^{\prime}_{1}|.$

Here $i_{2}\leq l_{i_{2}}<l_{j_{2}+1}<\cdots<l_{h_{b}}\leq h_{a}$. By Lemma
5.1, there is $K^{\prime}_{0}$ with

(6.6) $L(E_{b-2},K^{\prime}_{1})\geq
R(E_{b-2},K^{\prime}_{0})-(i_{2}-1-s)-(i-i_{1})$

since in $K^{\prime}_{1}$ the number of $u^{b}_{i}$ with
$u^{b}_{i}>v^{\prime}_{i_{2}}$ is at most $i-i_{1}$. By Corollary 5.6 and 5.7,

(6.7) $R(E_{b-2},K_{0}^{\prime})\geq
R(E_{b-2},||E_{a}(h_{b})||)=m_{a}-m_{b-2}.$

So by Equations (6.6),(6.7), and (6.2),

$L(E_{b-2},K^{\prime}_{1})\geq m_{a}-m_{b-2}+i_{0}-(i-s)>k-m_{b-2}.$

By Lemma 5.3, $K^{\prime}_{1}$ is not greater than $E_{b-2}$. So there is no
$E_{b}^{\prime}\in\mathcal{E}_{h}^{b}(K_{1})$, such that $E_{b-2}\leq
E_{b}^{\prime}$. $J_{1}\cdots J_{b-2}K_{1}$ is not standard.

Case 2: $J_{i}\in\mathcal{J}_{h}^{R}$ for $1\leq i\leq a$, the proof is
similar to the case of $J_{a}\in\mathcal{J}_{h}^{L}$.

Case 3: $J_{1},\dots,J_{c}\in\mathcal{J}_{h}^{L}$ and
$J_{c+1},\dots,J_{a}\in\mathcal{J}_{h}^{R}$, $1\leq c<a$.

Let $E_{1}\cdots E_{a}$ be the standard ordered product of elements of
$\mathcal{E}_{h}^{+}$ corresponding to $J_{1}\cdots J_{a}$. Assume
$E_{i}=((u^{i}_{h},k^{i}_{h}),\dots,(u^{i}_{1},k^{i}_{1})|$, for $1\leq i\leq
c$ and $E_{i}=|(v^{i}_{1},k^{i}_{1}),\dots,(v^{i}_{h},k^{i}_{h}))$, for
$c<i\leq a$. Let $m_{i}=wt(E_{i}(h_{b}))$ and $n_{i}=wt(E_{i})$. Let
$\sigma_{i}$ be a permutation of ${1,2,\dots,h_{b}}$ such that
$u_{\sigma_{i}(j)}<u_{\sigma_{i}(j+1)}$ or
$v_{\sigma_{i}(j)}<v_{\sigma_{i}(j+1)}$ for $1\leq j\leq h_{b-1}$. Let
$v^{\prime}_{h_{a}},\dots,v^{\prime}_{1}$ be a permutation of
$v^{a}_{h},\dots,v^{a}_{1}$ such that $v^{\prime}_{i}\leq v^{\prime}_{i+1}$.
Let $m=n_{a}+n_{b}$. Let $k_{0}=m_{c}+L(E_{c},J_{b}(L))$ and
$i_{0}=R(E_{a},J_{b}(R))$. Then by Lemma 5.3,

(6.8) $i_{0}>n_{b}-k_{0}-m_{a}.$

By assumption, $J_{1}\cdots J_{b-2}J_{b}$ is standard. By Lemma 5.3, $k_{0}$
is the smallest number such that there is $E\in\mathcal{E}(J_{b})$ with
$E_{c}\leq E$. Then $k_{0}\leq n_{b}$. Now the proof is similar to Case 1.

  1. (1)

If $n_{b}-k_{0}<m_{a}$, then $m_{a}>0$ and $c<b-2$. By Equation (2.20),

(6.9) $0=\sum_{0\leq
i<h_{b}}\sum_{\sigma}\frac{(-1)^{i}\operatorname{sign}(\sigma)}{i!(h_{b}-i)!}\sum_{k=0}^{m}\epsilon
a^{i}_{k}\left(\begin{array}[]{ccc}{\bar{\partial}}^{m-k}&|&v^{b}_{\sigma(1)},v_{\sigma(2)}^{b},\dots,v_{\sigma(i)}^{\prime},\underline{v_{i+1}^{\prime},\dots,v^{\prime}_{h}})\\\
{\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{1}&|&\underline{v^{\prime}_{1},\dots,v^{\prime}_{i},v^{b}_{\sigma(i+1)},\dots,v^{b}_{\sigma(h_{b})}})\end{array}\right)$

Here $a_{k}^{i}$ are integers and $a_{n_{b}-l}^{i}=\delta_{0,l}$ for $0\leq
l<h_{b}-i$, and $\sigma$ are permutations of $\\{1,\dots,h_{b}\\}$. In the
above equations:

    1. (a)

All the terms with $k=n_{b}+1,\dots,n$ are preceding $J_{a}$, since the weight
of the upper sequence is $m-k<n_{a}$.

    2. (b)

The terms with $k=n_{b}$ are $J_{a}J_{b}$ and the terms in Equation (6.9) with
the lower sequences

$K_{0}={\bar{\partial}}^{n_{b}}(u_{h_{b}}^{b},\dots,u_{2}^{b},u^{b}_{1}|v^{\prime}_{i_{1}},v^{\prime}_{i_{2}},\dots,v^{\prime}_{i_{h_{b}}}).$

And all of the other terms cancel.

By Lemma 5.3, if there is $E\in\mathcal{E}(K_{0})$ such that $E_{c}\leq E$ and
$E_{b-2}\leq E$, we must have $wt(E(L))\geq k_{0}$ and

$wt(E(R))-wt(E_{b-2}(h_{b}))\geq L(E_{b-2},||E(R)||).$

By Corollary 5.7 and Corollary 5.6,

(6.10) $L(E_{b-2},||E(R))||\geq L(E_{b-2},||E_{a}(h_{b})||)=m_{a}-m_{b-2}.$

Then

$n_{b}=wt(E(L))+wt(E(R))\geq k_{0}+m_{a}>n_{b}.$

This is a contradiction, so $J_{1}\cdots J_{a}K_{0}$ is not standard.

    3. (c)

The terms with $k<n_{b}$ in Equation (6.9) vanish unless $h_{b}-i\leq
n_{b}-k$. In this case the lower sequences of the terms are

$K_{1}={\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{1}|v^{\prime}_{s_{1}},\dots,v^{\prime}_{s_{i}},\underline{v^{b}_{\sigma(i+1)},\dots,v^{b}_{\sigma(h_{b})}}).$

By Lemma 5.3, if there is $E\in\mathcal{E}(K_{1})$, such that $E_{a}\leq E$
and $E_{c}\leq E$, we have $wt(E(L))\geq k_{0}$ and

$wt(E(R))-wt(E_{b-2})\geq L(E_{b-2},||E(R)||)).$

By Lemma 5.1 and Equation (6.10),

$L(E_{b-2},||E(R)||))\geq
L(E_{b-2},{\bar{\partial}}^{k}|v^{\prime}_{s_{1}},\dots,v^{\prime}_{s_{h}}))-(h_{b}-i)\geq
m_{a}-m_{b-2}-(h_{b}-i).$

Then $k=wt(E^{\prime}(L)+wt(E^{\prime}(R))\geq k_{0}+m_{a}-(h_{b}-i)>k$. This
is a contradiction, so $J_{1}\cdots J_{b-2}K_{1}$ is not standard.

  2. (2)

If $n_{b}-k_{0}\geq m_{a}$, then $i_{0}>n_{b}-k_{0}-m_{a}\geq 0$. There is
$i_{0}\leq i_{1}\leq h_{b}$, such that
$v^{b}_{i_{1}}<v^{a}_{\sigma_{a}(i_{1}-i_{0}+1)}$.

Assume $v^{\prime}_{i_{2}}=v^{a}_{\sigma_{a}(i_{1}-i_{0}+1)}$, then $i_{2}\geq
i_{1}-i_{0}+1$. By Equation (2.20,

$0=\sum_{\begin{subarray}{c}i_{1}>s\\\ i_{2}-1\geq
s\end{subarray}}\sum_{\sigma}\frac{(-1)^{s}\operatorname{sign}(\sigma)}{s!(i_{2}-1-s)!}\sum_{k=0}^{n}\epsilon
a^{s}_{k}$ (6.11)
$\left(\begin{array}[]{ccc}{\bar{\partial}}^{m-k}&|&\underline{v^{b}_{1},\dots,v^{b}_{s},v^{\prime}_{\sigma(s+1)},\dots,v^{\prime}_{\sigma(i_{2}-1)},v^{\prime}_{i_{2}},\dots,v^{\prime}_{h}}\\\
{\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{1}&|&v^{\prime}_{\sigma(1)},\dots,v^{\prime}_{\sigma(s)},\underline{v^{b}_{s+1},\dots,v^{b}_{i_{1}}},v^{b}_{i_{1}+1},\dots,v^{b}_{h_{b}})\end{array}\right).$

Here $a_{k}^{s}$ are integers and $a_{n_{b}-l}^{s}=\delta_{0,l}$ for $0\leq
l<i_{1}-s$, and $\sigma$ are permutations of $\\{1,\dots,i_{2}-1\\}$.

$0=\sum_{\begin{subarray}{c}i\geq i_{1}>s\\\ i_{2}-1\geq
s\end{subarray}}\sum_{\sigma,\sigma^{\prime}}\frac{(-1)^{i+s}\operatorname{sign}(\sigma^{\prime})\operatorname{sign}(\sigma)}{(i-i_{1})!(h_{b}-i)!s!(i_{2}-1-s)!}\sum_{k=0}^{n}\epsilon
a^{i,s}_{k}$

(6.12)
$\left(\begin{array}[]{ccc}{\bar{\partial}}^{n-k}&|&\underline{v^{\prime}_{h_{b}+1},\dots,v^{\prime}_{h_{a}},v^{b}_{1},\dots,v^{b}_{i_{1}},v^{b}_{\sigma^{\prime}(i_{1}+1)},\dots,v^{b}_{\sigma^{\prime}(i)}},\dots,v^{b}_{\sigma^{\prime}(h_{b})})\\\
{\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{1}&|&v^{\prime}_{\sigma(1)},\dots,v^{\prime}_{\sigma(s)},\underline{v^{\prime}_{\sigma(s+1)},\dots,v^{\prime}_{\sigma(i_{2}-1)},v^{\prime}_{i_{2}},\dots,v^{\prime}_{h_{b}}})\end{array}\right).$

Here $a_{k}^{i,s}$ are integers, $a_{n_{b}-l}^{i,s}=\delta_{0,l}$ for $0\leq
l<i-s$, $\sigma$ are permutations of $\\{1,\dots,i_{2}-1\\},$ and
$\sigma^{\prime}$ are permutations of $\\{i_{1}+1,i_{1}+2,\dots,h_{b}\\}$.

If $i_{2}>i_{1}-i_{0}+1$, we use Equation (6.11) and if $i_{2}=i_{1}-i_{0}+1$
we use Equation (6.12).

In the above equations:

    1. (a)

All the terms with $k=n_{b}+1,\dots,n$ are preceding $J_{a}$ since the weight
of upper sequence is $n-k<n_{a}$.

    2. (b)

The terms with $k=n_{b}$ are $J_{a}J_{b}$, the terms with upper sequence
preceding $J_{a}$ (the upper sequences are the sequences given by replacing
some $v^{\prime}_{i}$, $i\geq i_{2}$ in $J_{a}$ by some $v^{b}_{k}$, $k\leq
i_{1}$), and the terms with the lower sequences

$K_{0}={\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{1}|v^{\prime}_{\sigma(1)},\dots,v^{\prime}_{\sigma(i_{1})},v^{b}_{i_{1}+1},\dots,v^{b}_{h_{b}}).$

All of the other terms cancel. If $i_{2}-1<i_{1}$, the terms of the form
$K_{0}$ do not appear in Equations (6.11) and (6.12). Otherwise $i_{2}-1\geq
i_{1}$. In this case $i_{2}\geq i_{1}+1>i_{1}-i_{0}+1$, $E_{b-2}$ must be in
$\mathcal{E}_{h}^{R}$.

If $E\in\mathcal{E}_{h}^{b}(K_{0})$, such that $E_{b-2}\leq E$ and $E_{c}\leq
E$. Then $wt(E(L))\geq k_{0}$ and $wt(E(R))-wt(E_{b-2}(h_{b}))\geq
R(E_{b-2},||E(R)||)$ by Lemma 5.3. By Lemma 5.8

$R(E_{b-2},E(R))>R(E_{b-2},||E_{a}(h_{b}||))-i_{1}-(i_{1}-i_{0}+1).$

By Corollary 5.6,

(6.13) $R(E_{b-2},||E_{a}(h_{b}||))=m_{a}-m_{b-2}.$

Then $n_{b}=wt(E(L))+wt(E(R))\geq k_{0}+m_{a}+i_{0}>n_{b}$. This is a
contradiction, so $J_{1}\cdots J_{b-2}K_{0}$ is not standard.

    3. (c)

If $i_{2}>i_{1}-i_{0}+1$, the terms with $k<n_{b}$ in Equation (6.11) vanish
unless $i_{1}-s\leq n_{b}-k$. In this case the lower sequences of the terms
are

$K_{1}={\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots,u^{b}_{1}|v^{\prime}_{\sigma(1)},\dots,v^{\prime}_{\sigma(s)},\underline{v^{b}_{s+1},\dots,v^{b}_{i_{1}}},v^{b}_{i_{1}+1},\dots,v^{b}_{h_{b}})$

Underlined elements can be any underlined elements in Equation (6.11).

If $E\in\mathcal{E}_{h}^{b}(K_{1})$, such that $E_{b-2}\leq E$ and $E_{c}\leq
E$. Then $wt(E_{b}(L))\geq k_{0}$ and $wt(E(R))-wt(E_{b-2})\geq
R(E_{b-2},E(R))$ by Lemma 5.3. By Lemma 5.8

$R(E_{b-2},E(R))>R(E_{b-2},||E_{a}(h_{b}||))+s-(i_{1}-i_{0}+1).$

By Equation (6.13) and (6.8), then

$k=wt(E(L))+wt(E(R))\geq k_{0}+m_{a}+i_{0}+s-i_{1}>k.$

This is a contradiction, so $J_{1}\cdots J_{b-2}K_{1}$ is not standard.

    4. (d)

If $i_{2}=i_{1}-i_{0}+1$, the terms with $k<n_{b}$ in Equation (6.12) vanish
unless $i-s\leq n_{b}-k$. In this case the lower sequences of the terms are

$K_{1}={\bar{\partial}}^{k}(u^{b}_{h_{b}},\dots
u^{b}_{1}|v^{\prime}_{\sigma(1)},\dots,v^{\prime}_{\sigma(s)},\underline{v^{\prime}_{\sigma(s+1)},\dots,v^{\prime}_{\sigma(i_{2}-1)},v^{\prime}_{i_{2}},\dots,v^{\prime}_{h_{b}}})$

Underlined elements can be any underlined elements in Equation (6.12). If
there is $E\in\mathcal{E}(K_{1})$ such that $E_{c}\leq E$ and $E_{b-2}\leq E$.
Let

$K^{\prime}_{0}={\bar{\partial}}^{k}|v^{\prime}_{1},\dots,v^{\prime}_{i_{2}-1},v^{\prime}_{l_{i_{2}}},\dots,v^{\prime}_{l_{h_{b}}}).$

Here $i_{2}\leq l_{i_{2}}<l_{j_{2}+1}<\cdots<l_{h_{b}}\leq h_{a}$. We have
$wt(E(L))\geq k_{0}$ and

$R(E_{b-2},E(R))\geq R(E_{b-2},K^{\prime}_{0})-(i_{2}-1-s)-(i-i_{1})$

for some $K_{0}^{\prime}$ by Lemma 5.1 since in $K^{\prime}_{1}$ the number of
$v^{b}_{i}$ with $v^{b}_{i}>v^{\prime}_{i_{2}}$ is at most $i-i_{1}$. By
Corollary 5.6 and 5.7,

$R(E_{b-2},K_{0}^{\prime})\geq R(E_{b-2},||E_{a}(h_{b})||)=m_{a}-m_{b-2}.$

So by Lemma 5.3,

$wt(E(R))-wt(E_{b-2})\geq R(E_{b-2},E(R))\geq m_{a}-m_{b-2}+i_{0}-(i-s).$
$k=wt(E(L))+wt(E(R))\geq k_{0}+m_{a}+i_{0}-(i-s)>k.$

This is a contradiction, so $J_{1}\cdots J_{b-2}K_{1}$ is not standard.

Case 4:$J_{a}\notin J_{h}^{L}\cup J_{h}^{R}$. Let $E_{1}\cdots E_{a}$ be the
standard ordered product of elements of $\mathcal{E}_{h}^{+}$ corresponding to
$J_{1}\cdots J_{a}$. Assume $J_{i}\in\mathcal{J}^{L}_{h}$ for $1\leq i\leq c$,
$J_{i}\in\mathcal{J}^{R}_{h}$ for $c<i\leq d$ and $J_{i}\in\mathcal{J}_{h-1}$
for $d<i\leq a$.

  1. (1)

if $d<a-1$, let $E=E_{a-1}$;

  2. (2)

if $d=a-1$ and $d>c>0$, let $E=F(E_{c},E_{d})$;

  3. (3)

if $d=a-1=c$, let $E=F(E_{c},E_{0})$ with $E_{0}=|(1,0),(2,0),\dots,(h,0))$;

  4. (4)

if $d=a-1$and $d>c=0$, Let $E=F(E_{0},E_{d})$ such that
$E_{0}=((h,0),\dots,(2,0),(1,0)|$.

Then by Remark 2.8, $E_{a}$ is the largest element in $\mathcal{E}(J_{a})$
such that $E\leq E_{a}$. Since $J_{1}\cdots J_{b}$ is not standard, than
$J_{b}$ is not greater than $E_{a}$, by Lemma 6.1,
$J_{a}J_{b}=\sum\tilde{K}_{i}\tilde{f}_{i}$ with
$\tilde{K}_{i}\in\mathcal{J}$, $\tilde{f}_{i}\in\mathfrak{R}$ such that
$\tilde{K}_{i}$ is either smaller than $J_{a}$ or $\tilde{K}_{i}$ is not
greater than $E$. If $sz(\tilde{K}_{i})<h$,
$\tilde{K}_{i}\in\mathcal{J}_{h-1}^{+}$. If $sz(\tilde{K}_{i})\geq h$,
$\tilde{K}_{i}=\sum\tilde{K}_{ij}\tilde{g}_{j}$ with
$\tilde{K}_{ij}\in\mathcal{J}$ and $sz(\tilde{K}_{ij})=h$. By Equation (2.16),
$\tilde{K}_{ij}=\sum J_{k}^{L}J_{k}^{R}$ with
$J_{k}^{L}\in\mathcal{J}_{h}^{L}$ and $J_{k}^{R}\in\mathcal{J}_{h}^{R}$. So
$J_{a}J_{b}=\sum K_{i}f_{i}$ with $K_{ij}\in J_{h}^{+}$, such that $\ K_{i}$
is either smaller than $J_{a}$ or $K_{i}$ is not greater than $E$. ∎

## References

  * [1] D. Bourqui and J. Sebag, The radical of the differential ideal generated by $XY$ in the ring of two variable differential polynomials is not differentially finitely generated, J. Commut. Algebra 11 (2019), no. 2, 155-162.
  * [2] C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), no. 3, 330-354.
  * [3] L.Ein, M. Mustata, Jet schemes and Singularities, in: algebraic Geometry-Seattle 2005, Part 2, in: Proc. Sympos. Pure Math., vol. 80, Part2, Amer. Math. Soc., Providence, RI, 2009, pp.505-546.
  * [4] E. Kolchin, Differential algebra and algebraic groups, Academic Press, New York 1973.
  * [5] K. Kpognon, J. Sebag, Nilpotency in arc schemes of plane curves, Comm. in Algebra, Vol. 45 no 5 (2017), 2195-2221
  * [6] V. Lakshmibai, K. N. Raghavan, Standard Monomial Theory: Invariant Theoretic Approach. Encyclopaedia of Mathematical Sciences Volume 137. Springer-Verlag Berlin Heidelberg, 2008.
  * [7] V. Lakshmibai and P. Shukla, Standard monomial bases and geometric consequences for certain rings of invariants. Proc. Indian Acad. Sci. Math. Sci. 116 (2006), no. 1, 9-36.
  * [8] A. Linshaw and B. Song, Standard monomials and invariant theory for arc space I: general linear group, arXiv:2108.06864.
  * [9] A. Linshaw, G. Schwarz, B. Song, Jet schemes and invariant theory, Ann. I. Fourier 65 (2015) no. 6, 2571-2599.
  * [10] J. Sebag, Arcs schemes, derivations and Lipman’s theorem, J. Algebra 347 (2011) 173-183.
  * [11] J. Sebag, A remark on Berger’s conjecture, Kolchin’s theorem and arc schemes, Archiv der Math., Vol. 108 no (2017), 145-150.
  * [12] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939.