# Standard monomials and invariant theory of arc spaces II: Symplectic 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 second in a series of papers on standard monomial theory and
invariant theory of arc spaces. For any algebraically closed field $K$, we
construct a standard monomial basis for the arc space of the Pfaffian variety
over $K$. As an application, we prove the arc space analogue of the first and
second fundamental theorems of invariant theory for the symplectic group.

###### 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. 11101393

##  1\. Introduction

### Standard monomials and Pfaffian varieties

For a positive integer $p$, let

$R=\mathbb{Z}[x^{(0)}_{uv}|\ 1\leq u,v\leq
p]/(x^{(0)}_{uv}+x^{(0)}_{vu},x^{(0)}_{uu})$

be the ring of polynomial functions with integer coefficients on the space of
$p\times p$ skew-symmetric matrices. Let the sequence
$|u_{h},\dots,u_{2},u_{1}|$ with $1\leq u_{i}<u_{i+1}\leq p$ and $h\in
2\mathbb{Z}_{\geq 0}$ represent the Pfaffian $P(B)$ of the skew-symmetric
matrix

(1.1)
$B=\left(\begin{array}[]{cccc}x^{(0)}_{u_{1}u_{1}}&x^{(0)}_{u_{1}u_{2}}&\cdots&x^{(0)}_{u_{1}u_{h}}\\\
x^{(0)}_{u_{2}u_{1}}&x^{(0)}_{u_{2}u_{2}}&\cdots&x^{(0)}_{u_{2}u_{h}}\\\
\vdots&\vdots&\vdots&\vdots\\\
x^{(0)}_{u_{h}u_{1}}&x^{(0)}_{u_{h}u_{2}}&\cdots&x^{(0)}_{u_{h}u_{h}}\\\
\end{array}\right).$

There is a partial ordering on the set of these sequences given by

$|u_{h},\dots,u_{2},u_{1}|\leq|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{2},u^{\prime}_{1}|,\quad\text{if
}h^{\prime}\leq h,u_{i}\leq u^{\prime}_{i}.$

$R$ has a standard monomial basis (cf. [5]) with respect to this partially
ordered set; the ordered products $A_{1}A_{2}\cdots A_{k}$ of Pfaffians
$A_{i}$ with $A_{i}\leq A_{i+1}$, form a basis of $R$.

Similarly, let $R[h]$ be the ideal of $R$ generated by the Pfaffians of the
diagonal $h\times h$-minors, and let $R_{h}=R/R[h]$. Then $R_{h}$ has a basis
consisting of ordered products $A_{1}A_{2}\cdots A_{k}$ of the Pfaffians
$A_{i}$ with $(h-2,\dots,2,1)\leq A_{i}\leq A_{i+1}$.

For an arbitrary algebraically closed field $K$, let $SM_{p}(K)$ be the affine
space of $p\times p$ skew-symmetric matrices with entries in $K$. The
polynomial ring on $SM_{p}(K)$ is $R^{K}=R\otimes K$. Let $R^{K}[h]$ be the
ideal generated by the Pfaffians of the diagonal $h\times h$ minors. The
Pfaffian variety $Pf_{h}(K)$ is a closed subvariety of $SM_{p}(K)$ with
$R^{K}[h]$ as the defining ideal. Then for an even integer $h$,
$R^{K}/R^{K}[h]$, the affine coordinate ring of $Pf_{h}(K)$, has a standard
monomial basis: the ordered products of $A_{1}A_{2}\cdots A_{k}$ with
$(h-2,\dots,2,1)\leq A_{i}\leq A_{i+1}$ form a basis of $R^{K}/R^{K}[h]$.

### Classical invariant theory for the symplectic group

For an even integer $h$, let $W=K^{\oplus h}$ together with a non-degenerate,
skew-symmetric bilinear form given by $w=\sum_{i=1}^{h/2}dz_{2i-1}\wedge
dz_{2i}$. Let

$Sp_{h}(K)=\\{A\in SL_{h}(K)|\ A\ \text{ preserves}\ w\\}$

be the symplectic group over $K$. For $p\geq 1$, let $V=W^{\oplus p}$ be the
direct sum of $p$ copies of $W$. The affine coordinate ring of $V$ is

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

######  Theorem 1.1.

  1. (1)

The ring of invariants $(B^{K})^{Sp_{h}(K)}$ is generated by

$X^{(0)}_{uv}=\sum_{i=1}^{h/2}(a^{(0)}_{u{2i-1}}a^{(0)}_{v2i}-a^{(0)}_{v{2i-1}}a^{(0)}_{u2i}).$

  2. (2)

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

$P\left(\begin{array}[]{cccc}X^{(0)}_{u_{1}v_{1}}&X^{(0)}_{u_{1}v_{2}}&\cdots&X^{(0)}_{u_{1}v_{h+2}}\\\
X^{(0)}_{u_{2}v_{1}}&X^{(0)}_{u_{2}v_{2}}&\cdots&X^{(0)}_{u_{2}v_{h+2}}\\\
\vdots&\vdots&\vdots&\vdots\\\
X^{(0)}_{u_{h+2}v_{1}}&X^{(0)}_{u_{h+2}v_{2}}&\cdots&X^{(0)}_{u_{h+2}v_{h+2}}\\\
\end{array}\right).$

In particular, $V/\\!\\!/Sp_{k}(K)=\text{Spec}\ {B^{K}}^{Sp_{h}(K)}$ is
isomorphic to $Pf_{h+2}(K)$. Statements (1) and (2) are called the first and
second fundamental theorems of invariant theory for $Sp_{h}(K)$. In the case
$K=\mathbb{C}$, this theorem is classical and was proven by Weyl [8]. The
general case is due to de Concini and Procesi [2], and makes use of the
standard monomial basis. A uniform treatment of the first and second
fundamental theorems for all the classical groups using standard monomial
theory can be found in the book of Lakshmibai and Raghavan [5].

### Arc spaces

Given an irreducible scheme $X$ of finite type over $K$, the arc space
$J_{\infty}(X)$ is determined by its functor of points [3]. 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 $G$ is an algebraic group over $K$, $J_{\infty}(G)$ is again an algebraic
group over $K$. If $V$ is a finite-dimensional $G$-module, 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

(1.2) $J_{\infty}(V)/\\!\\!/J_{\infty}(G)\to J_{\infty}(V/\\!\\!/G),$

and the corresponding ring homomorphism

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

In the case $K=\mathbb{C}$, if $V/\\!\\!/G$ is smooth or a complete
intersection, it was shown in [6] that (1.3) is an isomorphism, although in
general it is neither injective nor surjective.

### Standard monomials for arc spaces

Let

$\mathfrak{R}=\mathfrak{R}^{p}=\mathbb{Z}[x^{(k)}_{uv}|\ 1\leq u,v\leq p,\
k\geq 0]/(x^{(k)}_{uv}+x^{(k)}_{vu},x^{(k)}_{uu}),$

with a derivation $\partial$ characterized by $\partial
x_{uv}^{(k)}=(k+1)x_{uv}^{(k+1)}$. It can be regarded as the ring of
polynomial functions with integer coefficients on the arc space of $p\times p$
skew-symmetric matrices.

Let $\mathfrak{R}[h]$ be the ideal of $\mathfrak{R}$ generated by the
Pfaffians of the diagonal $h\times h$ minors in the form of (1.1) and their
normalized derivatives $\frac{1}{n!}\partial^{n}P(B)$. Let
$\mathfrak{R}_{h}=\mathfrak{R}/\mathfrak{R}[h+2]$. Let $\mathcal{J}_{r}$ be
the set of Pfaffians of the matrices of the form of (1.1) with $h\leq r$ and
their normalized derivatives $\frac{1}{n!}\partial^{n}P(B)$. Note that $R$ and
$R_{h}$ are naturally subrings of $\mathfrak{R}$ and $\mathfrak{R}_{h}$,
respectively. In Section 2, we will define a notion of standard monomial on
$\mathcal{J}_{h}$ that extends the above notion on $R_{h}$, and we will prove
the following result.

######  Theorem 1.2.

For even integer $h$, $\mathfrak{R}_{h}$ has a $\mathbb{Z}$-basis given by the
standard monomials of $\mathcal{J}_{h}$.

Let $J_{\infty}(Pf_{h+2}(K))$ be the arc space of the Pfaffian variety
$Pf_{h+2}(K)$. The affine coordinate ring of $J_{\infty}(Pf_{h+2}(K))$ is
$\mathfrak{R}_{h}\otimes K$, so we immediately have

######  Corollary 1.3.

The affine coordinate ring of $J_{\infty}(Pf_{h+2}(K))$ has a $K$-basis given
by the standard monomials of $\mathcal{J}_{h}$.

### Application in invariant theory

Our main application of Theorem 1.2 is to prove the arc space analogue of
Theorem 1.1. As above, for an even integer $h\geq 2$, let $Sp_{h}(K)$ be the
symplectic group over $K$, $W=K^{\oplus h}$ its standard representation, and
$V=W^{\oplus p}$ the sum of $p$ copies of $W$. The affine coordinate ring of
$J_{\infty}(V)$ is

$\mathfrak{B}^{K}=K[a^{(k)}_{il}|\ 1\leq i\leq p,\ 1\leq l\leq h,\
k\in\mathbb{Z}_{\geq 0}],$

and the action of $Sp_{h}(K)$ on $V$ induces the action of
$J_{\infty}(Sp_{h}(K))$ on $\mathfrak{B}^{K}$.

######  Theorem 1.4.

Let
$X^{(k)}_{uv}={\bar{\partial}}^{k}\sum_{i=1}^{h/2}(a^{(0)}_{u{2i-1}}a^{(0)}_{v2i}-a^{(0)}_{v{2i-1}}a^{(0)}_{u2i})$.

  1. (1)

The ring of invariants ${\mathfrak{B}^{K}}^{J_{\infty}(GL_{h}(K))}$ is
generated by $X^{(k)}_{uv}$.

  2. (2)

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

(1.4)
${\bar{\partial}}^{k}P\left(\begin{array}[]{cccc}X^{(0)}_{u_{1}v_{1}}&X^{(0)}_{u_{1}v_{2}}&\cdots&X^{(0)}_{u_{1}v_{h+2}}\\\
X^{(0)}_{u_{2}v_{1}}&X^{(0)}_{u_{2}v_{2}}&\cdots&X^{(0)}_{u_{2}v_{h+2}}\\\
\vdots&\vdots&\vdots&\vdots\\\
X^{(0)}_{u_{h+2}v_{1}}&X^{(0)}_{u_{h+2}v_{2}}&\cdots&X^{(0)}_{u_{h+2}v_{h+2}}\\\
\end{array}\right).$

  3. (3)

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

######  Corollary 1.5.

For all $h\geq 1$ and $p\geq 1$, the map
$K[J_{\infty}(V/\\!\\!/Sp_{h}(K))]\rightarrow
K[J_{\infty}(V)]^{J_{\infty}(Sp_{h}(K))}$ given by (1.3), is an isomorphism.
In particular,

$J_{\infty}(V)/\\!\\!/J_{\infty}(Sp_{h}(K))\cong
J_{\infty}(V/\\!\\!/Sp_{h}(K)).$

Corollary 1.5 generalizes Theorem 4.5 of [6], which is the case $K=\mathbb{C}$
and $p\leq h+2$. A similar result was proven in [7] for the general linear
group $GL_{k}(K)$. The approach in this paper is similar to [7] but more
involved since we need a result of Bardsley and Richardson [1] which provides
a version of the Luna slice theorem in arbitrary characteristic.

##  2\. Standard monomials

### Derivation

Let $\mathbb{Z}_{\geq 0}$ be the set of non-negative integers. For any
$k,n\in\mathbb{Z}_{\geq 0}$, let

$C_{n}^{k}=\left\\{\begin{array}[]{cc}\frac{n!}{k!(n-k)!},&0\leq k\leq n;\\\
0,&\text{otherwise.}\end{array}\right.$

As above, let $\mathfrak{R}=\mathbb{Z}[x^{(k)}_{uv}|\ 1\leq u,v\leq p,\ k\geq
0]/(x^{(k)}_{uv}+x^{(k)}_{vu},x^{(k)}_{uu})$ be the ring with derivation
$\partial$ given on generators by $\partial x_{uv}^{(k)}=(k+1)x_{uv}^{(k+1)}$.
Let ${\bar{\partial}}^{l}=\frac{1}{l!}\partial^{l}$ be the $l^{\text{th}}$
normalized derivation. We have

${\bar{\partial}}^{l}x_{ij}^{(k)}=C_{k+l}^{l}x_{ij}^{(k+l)}\in\mathfrak{R}.$

The following statements 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 ${\bar{\partial}}^{l}a\in\mathfrak{R}$.

######  Proposition 2.2.

For a skew-symmetric matrix $B$ of (1.1) with $h=2l$,

(2.1)
${\bar{\partial}}^{n}P(B)=\sum_{\begin{subarray}{c}n_{1}+\cdots+n_{l}=n\\\
n_{i}\in\mathbb{Z}_{\geq
0}\end{subarray}}\frac{1}{l!2^{l}}\sum_{\sigma}\operatorname{sign}(\sigma)x^{(n_{1})}_{u_{\sigma(1)}u_{\sigma(2)}}x^{(n_{2})}_{u_{\sigma(3)}u_{\sigma(4)}}\cdots
x^{(n_{l})}_{u_{\sigma(h-1)}u_{\sigma(h)}}.$

### Generators

Let the sequence

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

with $1\leq u_{i}<u_{i+1}\leq p$ and $h\in 2\mathbb{Z}_{\geq 0}$ represent
${\bar{\partial}}^{n}P(B)\in\mathfrak{R}$, the $n^{\text{th}}$ normalized
derivative of the Pfaffian of $B$ in (1.1). We call $wt(J)=n$ 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 $\mathcal{E}$ be the set of sequences

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

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

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

Here $n=\sum k_{i}$, and $\sigma$ is the permutation of $1,2,\dots,h$ such
that $u_{\sigma(i)}<u_{\sigma(i+1)}$. Let

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

Let

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

For $J\in\mathcal{J}$, let

$\mathcal{E}(J)=\\{E\in\mathcal{E}|\ ||E||=J\\}.$

$\mathcal{J}$ is a set of generators of $\mathfrak{R}$ and we can use the
elements in $\mathcal{E}(J)$ to represent $J$.

### 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.

There is an ordering on the set $\mathcal{J}$:

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

if

  * •

$h^{\prime}<h$ ;

  * •

or $h^{\prime}=h$ and $k<k^{\prime}$;

  * •

or $h^{\prime}=h$, $k=k^{\prime}$ and $u_{h}\cdots u_{1}\prec
u^{\prime}_{h}\cdots u^{\prime}_{1}$. Here we order the words of natural
numbers lexicographically.

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

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

There is a partial ordering on the set $\mathcal{E}$:

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

if $h^{\prime}\leq h$ and $(u_{i},k_{i})\leq(u^{\prime}_{i},k^{\prime}_{i})$,
for $1\leq i\leq h^{\prime}$.
Finally, there is an ordering on $\mathcal{E}$:

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

if

  * •

$h>h^{\prime}$;

  * •

or $h=h^{\prime}$ and $\sum k_{i}<\sum k^{\prime}_{i}$;

  * •

or $h=h^{\prime}$, $\sum k_{i}=\sum k^{\prime}_{i}$ and

$(u_{h},k_{h})\cdots(u_{1},k_{1})\prec(u^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}})\cdots(u^{\prime}_{1},k^{\prime}_{1})$

Here we order the words of $\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}$
lexicographically.

######  Lemma 2.3.

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

###### Proof.

If $sz(E^{\prime})<sz(E)$ or $sz(E)=sz(E^{\prime})$ and
$wt(E)<wt(E^{\prime})$, then $||E||\prec||E^{\prime}||$.

If $sz(E)=sz(E^{\prime})$ and $wt(E)=wt(E^{\prime})$, we must have
$k_{i}=k_{i}^{\prime}$ . So $u_{i}\leq u^{\prime}_{i}$, we have
$||E||\prec||E^{\prime}||$. ∎

### Relations

Let

${\bar{\partial}}^{k}|u_{h},\dots,u_{1}|=0$

if there is $1\leq i<j\leq h$ such that $u_{i}=u_{j}$. For a sequence
$J\in\mathcal{J}$ of (2.2), let

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

Here $\sigma$ is a permutation of $1,2,\dots,h$, and
$\operatorname{sign}(\sigma)$ is the sign of the permutation.

######  Lemma 2.4.

For $0\leq k<s$,

$\displaystyle\sum_{\sigma}\frac{1}{h!s!}\operatorname{sign}(\sigma){\bar{\partial}}^{0}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)}|\,\,{\bar{\partial}}^{k}|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{s+1},u_{\sigma(s)},\dots,u_{\sigma(1)}|\in\mathfrak{R}[h+2].$

Here the summation is over all permutations $\sigma$ of $1,2,\dots,h+s$.

###### Proof.

It is easy to see that

(2.4) $\displaystyle\quad{\bar{\partial}}^{0}|u_{h},\dots,u_{1}|$
$\displaystyle=$
$\displaystyle\sum_{i=2}^{h}(-1)^{i}x^{(0)}_{1,i}{\bar{\partial}}^{0}|u_{h},\dots,u_{i+1},u_{i-1},\dots,u_{2}|;$
(2.5) $\displaystyle{\bar{\partial}}^{1}|u_{h},\dots,u_{1}|$ $\displaystyle=$
$\displaystyle\sum_{1\leq i<j\leq
h}(-1)^{i+j+1}x^{(0)}_{i,j}{\bar{\partial}}^{1}|u_{h},\dots,u_{j+1},u_{j-1},\dots,u_{i+1},u_{i-1},\dots,u_{1}|.$

Let $l^{\prime}=h^{\prime}/2$. For $0\leq k<i$,

${\bar{\partial}}^{k}|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{s+1},u_{\sigma(s)},\dots,u_{\sigma(1)}|=\sum_{k_{1}+\cdots+k_{l^{\prime}}=k}\sum_{\sigma^{\prime}}\frac{\operatorname{sign}(\sigma^{\prime})}{l^{\prime}!2^{l^{\prime}}}x^{(k_{1})}_{\sigma^{\prime}(u_{\sigma(1)})\sigma^{\prime}(u_{\sigma(2)})}\cdots
x^{(k_{l^{\prime}})}_{\sigma^{\prime}(u^{\prime}_{h^{\prime}-1})\sigma^{\prime}(u^{\prime}_{h^{\prime}})}$
(2.6)
$=\sum_{j=s+1}^{h^{\prime}}\sum_{i=1}^{s}x^{(0)}_{u_{\sigma(i)}u^{\prime}_{j}}f_{ij,\sigma}+\sum_{1\leq
i<j\leq s}x^{(0)}_{u_{\sigma(i)}u_{\sigma(j)}}g_{ij,\sigma}+\sum_{1\leq
i<j\leq s}x^{(1)}_{u_{\sigma(i)}u_{\sigma(j)}}h_{ij,\sigma}.$

Here $f_{ij,\sigma},g_{ij,\sigma},h_{ij,\sigma}\in\mathfrak{R}$, and if
$\sigma_{1}$ is a permutation of $1,2,\dots,s$ and $\sigma_{2}$ is a
permutation of $s+1,\dots,h+s$, then

$\displaystyle x^{(0)}_{u_{\sigma(i)}u}f_{iu,\sigma}$ $\displaystyle=$
$\displaystyle\operatorname{sign}(\sigma_{1})x^{(0)}_{u_{\sigma(i)}u}f_{\sigma_{1}^{-1}(i)u,\sigma\sigma_{1}\sigma_{2}},$
$\displaystyle x^{(0)}_{u_{\sigma(i)}u_{\sigma(j)}}g_{ij,\sigma}$
$\displaystyle=$
$\displaystyle\operatorname{sign}(\sigma_{1})x^{(0)}_{u_{\sigma(i)}u_{\sigma(j)}}g_{\sigma^{-1}_{1}(i)\sigma^{-1}_{1}(j),\sigma\sigma_{1}\sigma_{2}},$
$\displaystyle x^{(1)}_{u_{\sigma(i)}u_{\sigma(j)}}h_{ij,\sigma}$
$\displaystyle=$
$\displaystyle\operatorname{sign}(\sigma_{1})x^{(1)}_{u_{\sigma(i)}u_{\sigma(j)}}h_{\sigma^{-1}_{1}(i)\sigma^{-1}_{1}(j),\sigma\sigma_{1}\sigma_{2}}.$

We can also require that if $\sigma_{3}$ is a permutation of
$i,s+1,\dots,h+s$, $f_{iu,\sigma}=f_{iu,\sigma\sigma_{3}}$ and if $\sigma_{4}$
is a permutation of $i,j,s+1,\dots,h+s$,
$g_{ij,\sigma}=g_{ij,\sigma\sigma_{4}}$,
$h_{ij,\sigma}=h_{ij,\sigma\sigma_{4}}$. So by Equation (2.4), (2.5) and
(2.6),

$\displaystyle\sum_{\sigma}$
$\displaystyle\frac{1}{h!s!}\operatorname{sign}(\sigma){\bar{\partial}}^{0}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)}|\,\,{\bar{\partial}}^{k}|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{s+1},u_{\sigma(s)},\dots,u_{\sigma(1)}|$
$\displaystyle=$
$\displaystyle\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{h!s!}\sum_{j=s+1}^{h^{\prime}}\sum_{i=1}^{s}\frac{-1}{h+1}{\bar{\partial}}^{0}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)},u_{\sigma(i)},u^{\prime}_{j}|\,f_{ij,\sigma}$
$\displaystyle+\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{h!s!}\sum_{1\leq
i<j\leq
s}\frac{1}{(h+1)}{\bar{\partial}}^{0}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)},u_{\sigma(j)},u_{\sigma(i)}|\,g_{ij,\sigma}$
$\displaystyle+\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{h!s!}\sum_{1\leq
i<j\leq
s}\frac{1}{(h+1)(h+2)}{\bar{\partial}}^{1}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)},u_{\sigma(j)},u_{\sigma(i)}|\,h_{ij,\sigma}$
$\displaystyle=$
$\displaystyle\sum_{j=s+1}^{h^{\prime}}\sum_{\begin{subarray}{c}\sigma(h+s)>\cdots>\sigma(s)\\\
\sigma(s-1)>\cdots>\sigma(1)\end{subarray}}-\operatorname{sign}(\sigma){\bar{\partial}}^{0}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)},u_{\sigma(s)},u^{\prime}_{j}|\,f_{sj,\sigma}$
$\displaystyle+\sum_{\begin{subarray}{c}\sigma(h+s)>\cdots>\sigma(s)\\\
\sigma(s-2)>\cdots>\sigma(1)\end{subarray}}{\bar{\partial}}^{0}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)},u_{\sigma(s)},u_{\sigma(s-1)}|\,g_{ij,\sigma}$
$\displaystyle+\sum_{\begin{subarray}{c}\sigma(h+s)>\cdots>\sigma(s-1)\\\
\sigma(s-2)>\cdots>\sigma(1)\end{subarray}}-\operatorname{sign}(\sigma){\bar{\partial}}^{0}|u_{\sigma(h+s)},\dots,u_{\sigma(s+1)},u_{\sigma(s)},u_{\sigma(s-1)}|\,h_{ij,\sigma}$
$\displaystyle\in\mathfrak{R}[h+2]$

∎

######  Lemma 2.5.

For $i,j,h,h^{\prime},k_{0},m\in\mathbb{Z}_{\geq 0}$ with $h\geq h^{\prime}$,
$i\leq h$, $j\leq h^{\prime}$ and $k_{0}\leq m$, let $l_{0}=i+j-h-1$. 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

$\displaystyle\sum_{k=0}^{m}a_{k}\sum_{\sigma}\frac{1}{i!j!}\operatorname{sign}(\sigma)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle{\bar{\partial}}^{m-k}|u_{h},\dots,u_{i+1},\sigma(u_{i}),\dots,\sigma(u_{1})|\,\,{\bar{\partial}}^{k}|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{j+1},\sigma(u^{\prime}_{j}),\dots,\sigma(u^{\prime}_{1})|$
$\displaystyle\in\mathfrak{R}[h+2].$

Here the second summation is over all permutations $\sigma$ of
$u_{i},\dots,u_{1},u^{\prime}_{j},\dots,u^{\prime}_{1}$ and
$\operatorname{sign}(\sigma)$ is the sign of the permutation.

For simplicity, we write Equation (2.5) in the following way,

(2.8) $\sum\epsilon
a_{k}{\bar{\partial}}^{m-k}|u_{h},\dots,u_{i+1},\underline{u_{i},\dots,u_{1}}|{\bar{\partial}}^{k}|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{j+1},\underline{u^{\prime}_{j},\dots,u^{\prime}_{1}}|\in\mathfrak{R}[h+2].$

######  Remark 2.6.

Since the second summation in Equation (2.5) is over all permutations, each
monomial in the equation will appear $i!j!$ times, and the coefficient of each
monomial will be $\pm a_{k}$.

###### Proof of Lemma 2.5.

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

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

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

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

If $i=h$ and $0\leq l\leq l_{0}$, by Lemma (2.4),
$\mathcal{F}_{l}(\,\,;u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{j+1})\in\mathfrak{R}[h+2]$.
By induction on $h-i$, we can see that
$\mathcal{F}_{l}(u_{h},\dots,u_{i+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{j+1})\in\mathfrak{R}[h+2].$
Thus

$\sum_{k=0}^{m}C_{k}^{l}\sum_{\sigma}\frac{\operatorname{sign}(\sigma)}{i!j!}{\bar{\partial}}^{m-k}|u_{h},\dots,u_{i+1},\sigma(u_{i}),\dots,\sigma(u_{1})|\,\,{\bar{\partial}}^{k}|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{j+1},\sigma(u^{\prime}_{j}),\dots,\sigma(u^{\prime}_{1})$
$={\bar{\partial}}^{m-l}\mathcal{F}_{l}(u_{h},\dots,u_{i+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{j+1})\in\mathfrak{R}[h+2].$

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

$0=\sum_{l=0}^{l_{0}}\sum_{j=0}^{l_{0}}b_{l,j}a_{k_{0}+j}{\bar{\partial}}^{m-l}\mathcal{F}_{l}(u_{h},\dots,u_{i+1};u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{j+1})=\text{right
hand side of Equation (\ref{eqn: relation4'})}.$

∎

### Standard monomials

Now we give the definition of standard monomials of $\mathcal{J}$.

######  Definition 2.7.

An ordered product $E_{1}E_{2}\cdots E_{m}$ of elements of $\mathcal{E}$ is
said to be standard if

  1. (1)

$E_{a}\leq E_{a+1}$, $1\leq a<m$,

  2. (2)

$E_{1}$ is the largest in $\mathcal{E}(||E_{1}||)$ under the order $\prec$,

  3. (3)

$E_{a+1}$ is the largest in $\mathcal{E}(||E_{a+1}||)$ such that $E_{a}\leq
E_{a+1}$.

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

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

By Definition 2.7, if $J_{1}J_{2}\cdots J_{m}$ is standard, the standard
monomial $E_{1}\cdots E_{m}\in\mathcal{S}\mathcal{M}(\mathcal{E})$
corresponding to $J_{1}\cdots J_{m}$ is unique and $E_{1}$ has the form

$|(u_{h},wt(E_{1})),(u_{h-1},0),\dots,(u_{1},0)|\in\mathcal{E}$

with $u_{i}<u_{i+1}$. Therefore 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.

We order $\mathcal{M}(\mathcal{J})$, the set of ordered products of elements
of $\mathcal{J}$, lexicographically. The following lemma will be proved later
in Section 6.

######  Lemma 2.8.

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

Recall that $\mathfrak{R}[h]$ denotes the ideal generated by $J\in\mathcal{J}$
with $sz(J)=h$, and $\mathfrak{R}_{h}=\mathfrak{R}/\mathfrak{R}[h+2]$. If
$h\geq p$, then $\mathcal{J}_{h}=\mathcal{J}$ and
$\mathfrak{R}_{h}=\mathfrak{R}$. By the above lemma, we immediately have

######  Lemma 2.9.

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

###### Proof.

We only need to show that any element of $\mathfrak{R}$ can be written as a
linear combination of standard monomials of $\mathcal{J}$ with integer
coefficients. If the lemma is not true, there must be a smallest element
$J\in\mathcal{M}(\mathcal{J})$, which cannot be written as a linear
combination of elements of $\mathcal{S}\mathcal{M}(\mathcal{J})$ with integer
coefficients. So $J$ is not standard. By Lemma 2.8, $J=\sum_{\alpha}\pm
J_{\alpha}$ with $J_{\alpha}\in\mathcal{M}(\mathcal{J})$ and $J_{\alpha}\prec
J$. Since $J_{\alpha}$ can be written as a linear combination of elements of
$\mathcal{S}\mathcal{M}(\mathcal{J})$ with integer coefficients, $J$ can also
be written as such a linear combination, which is a contradiction. ∎

##  3\. A canonical basis

### A ring homomorphism

Let

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

and $\mathfrak{B}=\mathbb{Z}[\mathcal{S}_{h}]$, the polynomial ring generated
by $\mathcal{S}_{h}$. Let $\partial$ be the derivation on $\mathfrak{B}$ given
by $\partial a^{(k)}_{ij}=(k+1)a^{(k+1)}_{ij}$, $\partial
b^{(k)}_{ij}=(k+1)b^{(k+1)}_{ij}$. We have a ring homomorphism

$\tilde{Q}_{h}:\mathfrak{R}\to\mathfrak{B},\quad\quad
x^{(k)}_{uv}\mapsto{\bar{\partial}}^{k}\sum_{i=1}^{h/2}(a^{(0)}_{u{2i-1}}a^{(0)}_{v2i}-a^{(0)}_{v{2i-1}}a^{(0)}_{u2i}).$

For any $J\in\mathcal{J}$ with $sz(J)>h$, we have $\tilde{Q}_{h}(J)=0$. So
$\tilde{Q}_{h}$ induces a ring homomorphism
$Q_{h}:\mathfrak{R}_{h}\to\mathfrak{B}$.

### Tableaux

Let $\tilde{\mathcal{S}}_{h}$=$\mathcal{S}_{h}\cup\\{*\\}$. We define an
ordering on the set $\tilde{\mathcal{S}}_{h}$:
for $a^{(k)}_{ij},a^{(k^{\prime})}_{i^{\prime}j^{\prime}}\in\mathcal{S}_{h}$,
$a^{(k)}_{ij}<*$ and $a^{(k)}_{ij}\leq
a^{(k^{\prime})}_{i^{\prime}j^{\prime}}$ if $kij\prec
k^{\prime}i^{\prime}j^{\prime}$.

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

(3.1) $\left|\begin{array}[]{c}y_{1,h},\cdots,y_{1,2},y_{1,1}\\\ \vdots\\\
y_{m,h},\cdots,y_{m,2},y_{m,1}\\\ \end{array}\right|.$

Here $y_{s,l}$ are some $a^{(k)}_{il}$ or $*$, every row of the tableau has
elements in $\mathcal{S}_{h}$ and $y_{s,j}\leq y_{s+1,j}.$ We use the tableau
(3.1) to represent a monomial in $\mathfrak{B}$, which is the product of
${a^{(k)}_{ij}}^{\prime}s$ in the tableau. It is easy to see that the
representation is a one-to-one correspondence between $\mathcal{T}$ and the
set of monomials of $\mathfrak{B}$. We associate to the tableau (3.1) the
word:

$y_{1,h}\cdots y_{1,1}y_{2,h}\cdots y_{2,1}\cdots y_{m,h}\cdots y_{m,1}$

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
$\mathcal{T}$.

For
$E_{i}=|(u^{i}_{h_{1}},k^{i}_{h_{1}}),\dots,(u^{i}_{2},k^{i}_{2}),(u^{i}_{1},k^{i}_{1})|\in\mathcal{E}$,
$1\leq i\leq m$, we use a tableau to represent $E_{1}\cdots
E_{m}\in\mathcal{S}\mathcal{M}(\mathcal{E})$,

(3.2)
$\left|\begin{array}[]{c}(u^{1}_{h_{1}},k^{1}_{h_{1}}),\cdots,(u^{1}_{2},k^{1}_{2}),(u^{1}_{1},k^{1}_{1})\\\
(u^{2}_{h_{2}},k^{2}_{h_{2}}),\cdots,(u^{2}_{2},k^{2}_{2}),(u^{2}_{1},k^{2}_{1})\\\
\vdots\\\
(u^{m}_{h_{m}},k^{m}_{h_{m}}),\cdots,(u^{m}_{2},k^{m}_{2}),(u^{m}_{1},k^{m}_{1})\\\
\end{array}\right|.$

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

$T(E_{1}\cdots
E_{m})=\left|\begin{array}[]{c}*,\cdots,*,a_{u^{1}_{h_{1}}h_{1}}^{(k^{1}_{h_{1}})},\cdots,a_{u^{1}_{1}1}^{(k^{1}_{1})}\\\
,\cdots,*,a_{u^{2}_{h_{2}}h_{2}}^{(k^{2}_{h_{2}})},\cdots,a_{u^{2}_{1}1}^{(k^{2}_{1})}\\\
\vdots\\\
,\cdots,*,a_{u^{m}_{h_{m}}h_{m}}^{(k^{m}_{h_{m}})},\cdots,a_{u^{m}_{1}1}^{(k^{m}_{1})}\end{array}\right|.$

Obviously, $T$ is an injective map and $T(E)\prec T(E^{\prime})$ if $E\prec
E^{\prime}$.

######  Lemma 3.1.

Let $J_{1}\cdots J_{m}\in\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$ and
$E_{1}\cdots E_{m}\in\mathcal{S}\mathcal{M}(\mathcal{E}_{h})$ be its
associated standard monomial. Assume the double tableau representing
$E_{1}\cdots E_{m}$ is (3.2). Then the leading monomial of $Q_{h}(J_{1}\cdots
J_{m})$ is represented by the double tableau $T(E_{1}E_{2}\cdots E_{m})$. Thus

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

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

###### Proof.

Let $W_{m}$ be the monomial corresponding to the double tableau $T(E_{1}\cdots
E_{m})$. Let

$M_{m}=a_{u^{m}_{h_{m}}h_{m}}^{(k^{m}_{h_{m}})}\cdots
a_{u^{m}_{1}1}^{(k^{m}_{1})}$

be the monomial corresponding to the double tableau $T(E_{m})$. Then
$W_{m}=W_{m-1}M_{m}$. Let $l_{m}=\frac{h_{m}}{2}$, by a direct calculation,

$Q_{h}(J_{m})=\sum\pm
a_{u^{m}_{\sigma(1)}2s_{1}-1}^{(k_{1})}a_{u^{m}_{\sigma(2)}2s_{1}}^{(k_{2})}a_{u^{m}_{\sigma(3)}2s_{2}-1}^{(k_{3})}a_{u^{m}_{\sigma(4)}2s_{2}}^{(k_{4})}\cdots
a_{u^{m}_{\sigma(h_{m}-1)}2s_{l_{m}}-1}^{(k_{h_{m}-1})}a_{u^{m}_{\sigma(h_{m})}2s_{l_{m}}}^{(k_{h_{m}})}.$

The summation is over all $k_{i}\geq 0$ with $\sum k_{i}=wt(E_{m})$, all
$s_{i}$ with $1\leq s_{1}<s_{2}<\cdots<s_{l_{m}}\leq h/2$ and all permutations
$\sigma$ of $1,2,\dots,h_{m}$.

We prove the lemma by induction on $m$. If $m=1$, the lemma is obviously true.
Assume it is true for $J_{1}\cdots J_{m-1}$. Then $Ld(Q_{h}(J_{1}\cdots
J_{m-1}))=W_{m-1}$, the monomial corresponding to $T(E_{1}\cdots E_{m-1})$,
and the coefficient of $W_{m-1}$ in $Q_{h}(J_{1}\cdots J_{m-1})$ is $\pm 1$.
$M_{m}$ is one of the monomials in $Q_{h}(J_{m})$ with coefficient $\pm 1$.
All of the monomials in $Q_{h}(J_{1}\cdots J_{m-1})$ except $W_{n-1}$ are less
than $W_{n-1}$, so any monomial in $Q_{h}(J_{1}\cdots J_{m-1})$ except
$W_{n-1}$ times any monomial in $Q_{h}(J_{m})$ is less than $W_{m-1}$. Since
$W_{m-1}\prec W_{m}$, the coefficient of $W_{m}$ in $Q_{h}(J_{1}\cdots J_{m})$
is not zero. Now

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

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

$W=W_{m-1}a_{u^{m}_{\sigma(1)}2s_{1}-1}^{(k_{1})}a_{u^{m}_{\sigma(2)}2s_{1}}^{(k_{2})}a_{u^{m}_{\sigma(3)}2s_{2}-1}^{(k_{3})}a_{u^{m}_{\sigma(4)}2s_{2}}^{(k_{4})}\cdots
a_{u^{m}_{\sigma(h_{m}-1)}2s_{l_{m}}-1}^{(k_{h_{m}-1})}a_{u^{m}_{\sigma(h_{m})}2s_{l_{m}}}^{(k_{h_{m}})}.$

If some $s_{i}$ is greater than $h_{m-1}/2$, then $W\prec W_{n-1}$. So
$s_{i}\leq h_{m-1}/2$. We must have $a_{u^{m}_{i}s_{i}}^{(k_{i})}\geq
a_{u^{m-1}_{s_{i}}s_{i}}^{(k^{m-1}_{s_{i}})}$; otherwise, $W\prec W_{m-1}$. If
there is some $h_{m-1}/2\geq s_{i}>l_{m}$, then $W\prec W_{m}$. So we can
assume $s_{i}=i$. Such monomials in $Q_{h}(J_{m})$ are in one-to-one
correspondence with $E^{\prime}_{m}\in\mathcal{E}(J_{m})$ such that
$E_{m-1}\leq E^{\prime}_{m}$. $E_{m}$ is the largest in $\mathcal{E}(J_{m})$
with $E_{m-1}\leq E_{m}$ since $E$ is standard, so $W_{m}$ is the leading term
of $Q_{h}(J_{1}\cdots J_{m})$. The coefficient of $W_{m}$ in
$Q_{h}(J_{1}\cdots J_{m})$ is $\pm 1$ since the coefficients of $W_{m-1}$ in
$Q_{h}(J_{1}\cdots J_{m-1})$ and $M_{m}$ in $Q_{h}(J_{m})$ are $\pm 1$. ∎

###### Proof of Theorem 1.2..

By Lemma 3.1, $Ld(Q_{h}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h})))$ are
linearly independent, so $\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$ are
linearly independent. By Lemma 2.9, $\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$
generates $\mathfrak{R}_{h}$. So $\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$ is
a $\mathbb{Z}$-basis of $\mathfrak{R}_{h}$. ∎

Let $\mathfrak{C}=Q_{h}(\mathfrak{R}_{h})$. It is the subring of
$\mathfrak{B}$ generated by
${\bar{\partial}}^{k}\sum_{i}^{h/2}(a^{(0)}_{u{2i-1}}a^{(0)}_{v2i}-a^{(0)}_{v{2i-1}}a^{(0)}_{u2i})$.

######  Theorem 3.2.

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

###### Proof.

By Lemma 3.1, $Ld(Q_{h}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h})))$ are
linearly independent. $\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$ is a
$\mathbb{Z}$-basis of $\mathfrak{R}_{h}$, so
$Q_{h}:\mathfrak{R}_{h}\to\mathfrak{B}$ is injective. ∎

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

######  Corollary 3.3.

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

##  4\. Application

In this section, we give the main application of our standard monomial basis,
which is the arc space analogue of Theorem 1.1.

### Arc spaces

Suppose that $X$ is a scheme of finite type over $K$. Its arc space (cf. [3])
$J_{\infty}(X)$ 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 $i:X\to Y$ is a morphism of schemes, we get a morphism of schemes
$i_{\infty}:J_{\infty}(X)\to J_{\infty}(Y)$. If $i$ is a closed immersion,
then $i_{\infty}$ is a closed immersion. If $i:X\to Y$ is an étale morphism,
then we have a Cartesian diagram

$\begin{array}[]{ccc}J_{\infty}(X)&\to&J_{\infty}(Y)\\\
\downarrow&&\downarrow\\\ X&\to&Y\end{array}$

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 be identified with the subalgebra
$K[x^{(0)}_{1},\dots,x^{(0)}_{n}]\subset K[x_{i}^{(k)}]$, 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

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

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

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

obtained by tensoring with $K$.

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

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

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

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

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

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

If $Y$ is the affine scheme $\text{Spec}\
K[y^{(0)}_{1},\dots,y^{(0)}_{m}]/(g_{1},\dots,g_{s})$, a morphism $P:X\to Y$
gives 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
$k\geq 0$.

### Arc space of the Pfaffian variety

$R^{K}=K[x^{(0)}_{ij}|\ 1\leq i,\ j\leq
p]/(x^{(0)}_{ij}-x^{(0)}_{ji},x^{(0)}_{ii})$ is the affine coordinate ring of
the space of $p\times p$ skew-symmetric matrices $SM_{p}(K)$. The Pfaffian
variety $Pf_{h}(K)$ is the subvariety of $SM_{p}(K)$ determined by the ideal
$R^{K}[h]$ generated by the Pfaffians of all diagonal $h$-minors. So the
affine coordinate ring of $J_{\infty}(Pf_{h}(K))$ is
$\mathfrak{R}^{K}_{h-2}=K[x^{(k)}_{ij}]/({\bar{\partial}}^{l}J|J\text{ is the
Pfaffian of $h$-minor})=\mathfrak{R}_{h-2}\otimes K$.

###### Proof of Corollary 1.3.

By Theorem 1.2, $\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$ is a
$\mathbb{Z}$-basis of $\mathfrak{R}_{h}$. So it is a $K$-basis of
$\mathfrak{R}^{K}_{h}$. ∎

Let $\mathfrak{B}^{K}=\mathfrak{B}\otimes K=K[\mathcal{S}_{h}]$. Let
$Q_{h}^{K}=Q_{h}\otimes Id:\mathfrak{R}_{h}\otimes K\to\mathfrak{B}\otimes K$.

######  Theorem 4.1.

$Q^{K}_{h}$ is injective. So $\mathfrak{R}_{h}^{K}$ is integral.

###### Proof.

By Lemma 3.1, $Ld(Q_{h}(\mathcal{S}\mathcal{M}(\mathcal{J}_{h})))$ are
linearly independent. By Corollary 1.3,
$\mathcal{S}\mathcal{M}(\mathcal{J}_{h})$ is a $K$-basis of
$\mathfrak{R}_{h}$, so $Q^{K}_{h}:\mathfrak{R}_{h}^{K}\to\mathfrak{B}^{K}$ is
injective. Since $\mathfrak{B}^{K}$ is integral, so as $\mathfrak{R}^{K}_{h}$.
∎

In general, if $\text{char}\ K=0$, the arc space of an integral scheme is
irreducible [4] but it may not be reduced. $Pf_{h}(K)$ is an example whose arc
space is integral.

### Principle $G$-bundles

Let $G$ be an algebraic group over $K$. If the $G$ acts morphically on an
algebraic variety $X$, then we say that $X$ is a $G$-variety. An affine
$G$-variety $X$ is a principle $G$-bundle in the the étale topology if, for
every $x\in X/\\!\\!/G$, there is an étale neighborhood $V\to X$ of $x$ such
that $V\times_{X/G}X\cong V\times G$ as $G$-varieties. The following result is
from [1] by P. Bardsley and R. W. Richardson,

######  Proposition 4.2.

Let $X$ be an affine $G$-variety. Then $X$ is a principal $G$-bundle in the
étale topology if and only if for every $x$ in $X$, the orbit $G\cdot x$ is
separable and the stabilizer $G_{x}$ is trivial.

The group structure $G\times G\to G$ induces the group structure on its arc
space

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

So $J_{\infty}(G)$ is an algebraic group. For a $G$-variety $X$, the action
$G\times X\to X$ induces the action of $J_{\infty}(G)$ on $J_{\infty}(X)$,

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

The quotient map $X\to X/\\!\\!/G$ induces morphisms $J_{\infty}(X)\to
J_{\infty}(X/\\!\\!/G)$ and

$\pi_{X}:J_{\infty}(X)/\\!\\!/J_{\infty}(G)\to J_{\infty}(X/\\!\\!/G).$

######  Proposition 4.3.

If $X$ is a principle $G$-bundle in the étale topology and $X/\\!\\!/G$ is
smooth, then $\pi_{X}$ is an isomorphism.

###### Proof.

For any étale morphism $V\to X/\\!\\!/G$ with $V\times_{X/\\!\\!/G}X\cong
V\times G$ as $G$-varieties, we have Cartesian diagrams

$\begin{array}[]{ccc}J_{\infty}(V)&\to&J_{\infty}(X/\\!\\!/G)\\\
\downarrow&&\downarrow\\\
V&\to&X/\\!\\!/G\end{array}\quad\quad\text{and}\quad\quad\begin{array}[]{ccc}J_{\infty}(V)\times
J_{\infty}(G)&\to&J_{\infty}(X)\\\ \downarrow&&\downarrow\\\ V\times
G&\to&X\\\ \downarrow&&\downarrow\\\ V&\to&X/\\!\\!/G\end{array}.$

So $J_{\infty}(V)\to J_{\infty}(X/\\!\\!/G)$ is an étale morphism and

$\displaystyle
J_{\infty}(V)\times_{J_{\infty}(X/\\!\\!/G)}J_{\infty}(X)/\\!\\!/J_{\infty}(G)$
$\displaystyle\cong$
$\displaystyle(J_{\infty}(V)\times_{J_{\infty}(X/\\!\\!/G)}J_{\infty}(X))/\\!\\!/J_{\infty}(G)$
$\displaystyle\cong$
$\displaystyle(V\times_{X/\\!\\!/G}J_{\infty}(X))/\\!\\!/J_{\infty}(G)$
$\displaystyle\cong$ $\displaystyle J_{\infty}(V)\times
J_{\infty}(G)/\\!\\!/J_{\infty}(G)$ $\displaystyle\cong$ $\displaystyle
J_{\infty}(V).$

If $\pi_{X}$ is not an isomorphism, there is an étale morphism $V\to
X/\\!\\!/G$ such that $V\times_{X/\\!\\!/G}X\cong V\times G$ as $G$-varieties
and
$\pi_{V}:J_{\infty}(V)\times_{J_{\infty}(X/\\!\\!/G)}J_{\infty}(X)/\\!\\!/J_{\infty}(G)\to
J_{\infty}(V)$ is not an isomorphism. But
$V\times_{X/\\!\\!/G}J_{\infty}(X/\\!\\!/G)\cong J_{\infty}(V)$, which is a
contradiction. ∎

### Invariants for the arc space of the symplectic group action

Let $G=Sp_{h}(K)$ be the symplectic group over $K$, $W=K^{\oplus h}$ its
standard representation, and $V=W^{\oplus p}$. The affine coordinate ring of
$V$ is $B^{K}=K[a^{(0)}_{il}|0\leq i,j\leq p,1\leq l\leq h]$. The affine
coordinate ring of $J_{\infty}(V)$ is $\mathfrak{B}^{K}=K[\mathcal{S}_{h}]$.
Therefore $J_{\infty}(G)$ acts on $\mathfrak{B}^{K}$.

Let $\mathfrak{C}^{K}$ be the image of $Q_{h}^{K}$. If $p\geq h$, let
$\Delta=Q^{K}_{h}(|h,\dots,1|)$. Let $\mathfrak{B}^{K}_{\Delta}$ and
$\mathfrak{C}^{K}_{\Delta}$ be the localization of $\mathfrak{B}^{K}$ and
$\mathfrak{C}^{K}$ at $\Delta$.

######  Lemma 4.4.

If $p\geq h$,

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

###### Proof.

Let $B^{K}_{\Delta}$ be the localization of $B^{K}$ at $\Delta$ and
$V_{\Delta}=\operatorname{Spec}B^{K}_{\Delta}$. By Theorem 1.1, the space of
invariants in $B^{K}$ is generated by $Q_{h}^{K}(x^{(0)}_{uv})$. So the affine
coordinate ring of $J_{\infty}(V_{\Delta}/\\!\\!/G)$ is isomorphic to
$\mathfrak{C}^{K}_{\Delta}$. To show the lemma, we only need to show that
$\pi_{V_{\Delta}}:J_{\infty}(V_{\Delta})/\\!\\!/J_{\infty}(G)\cong
J_{\infty}(V_{\Delta}/\\!\\!/G)$. By Proposition 4.2, $V_{\Delta}$ is a
principle bundle in the étale topology, and by Proposition 4.3,
$\pi_{V_{\Delta}}$ is an isomorphism. ∎

######  Theorem 4.5.

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

###### Proof.

Let $G_{\infty}=J_{\infty}(G)$. If $p\geq h$, regard $\mathfrak{B}^{K}$ and
$\mathfrak{C}^{K}_{\Delta}$ as subrings of $\mathfrak{B}^{K}_{\Delta}$. By
Lemma 4.4, we have

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

Now for any $f\in\mathfrak{B}^{K}\cap\mathfrak{C}^{K}_{\Delta}$. Then
$f=\frac{g}{\Delta^{n}}$, with $\Delta^{n}f=g\in\mathfrak{C}^{K}$. 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
$L(Q_{h}(J))=L(g)$. $J$ has the factor $|h,\dots,1|^{n}$, so $Q_{h}(J)$ has
the factor $\Delta^{n}$. Thus
$f-C_{0}\frac{Q_{h}(J)}{{\Delta}^{n}}\in\mathfrak{B}\cap\mathfrak{C}_{\Delta}$
with a lower leading monomial and
$\frac{Q_{h}(J)}{{\Delta}^{n}}\in\mathfrak{C}$. By induction on the leading
monomial of $f$, $f\in\mathfrak{C}^{K}$. So
$\mathfrak{B}^{K}\cap\mathfrak{C}^{K}_{\Delta}=\mathfrak{C}^{K}$, and
${\mathfrak{B}^{K}}^{G_{\infty}}=\mathfrak{C}^{K}$.

In general, let $\mathfrak{B}^{\prime}=K[a^{(k)}_{il}|\ 1\leq i\leq p+h,\
1\leq l\leq h,\ k\in\mathbb{Z}_{\geq 0}]$. Let $\mathfrak{C}^{\prime}$ be the
space of $G_{\infty}$-invariant elements in $\mathfrak{B}^{\prime}$. It is
generated by
$X^{(k)}_{uv}={\bar{\partial}}^{k}\sum_{i=1}^{h/2}(a^{(0)}_{u{2i-1}}a^{(0)}_{v2i}-a^{(0)}_{v{2i-1}}a^{(0)}_{u2i})$,
by the previous argument. Let $\mathcal{I}^{\prime}$ be the ideal of
$\mathfrak{B}^{\prime}$ generated by $a^{(k)}_{il}$ with $i>p$. Then
$\mathfrak{B}^{\prime}=\mathfrak{B}^{K}\oplus{\mathcal{I}^{\prime}}$.
$\mathfrak{B}^{K}$ and $\mathcal{I}^{\prime}$ are $G_{\infty}$-invariant
subspaces of $\mathfrak{B}^{\prime}$.

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

If $i>p$ or $j>p$, $X^{(k)}_{ij}\in{\mathcal{I}^{\prime}}^{G_{\infty}}$. So
${\mathfrak{B}^{\prime}}^{G_{\infty}}\cong\mathfrak{C}^{\prime}/{\mathcal{I}^{\prime}}^{G_{\infty}}$
is generated by $X^{(k)}_{ij}$, $1\leq i,j\leq p$. And
${\mathfrak{B}^{K}}^{G_{\infty}}=\mathfrak{C}$. ∎

###### Proof of Theorem 1.4.

By Theorem 4.5 and Theorem 4.1,
${\mathfrak{B}^{K}}^{J_{\infty}(Sp_{h}(K))}=\mathfrak{C}^{K}\cong\mathfrak{R}^{K}_{h}$.
The theorem follows from the definition of $R^{K}_{h}$ and the map
$Q_{h}^{K}:\mathfrak{R}_{h}^{K}:\to\mathfrak{B}^{K}$. ∎

###### Proof of Corollary 1.5.

The affine coordinate ring of $V/\\!\\!/G$ is isomorphic to
$R^{K}/R^{K}[h+2]$. So the affine coordinate ring of $J_{\infty}(V/\\!\\!/G)$
is isomorphic to $\mathfrak{R}^{K}_{h}$. $J_{\infty}(V)\to
J_{\infty}(V/\\!\\!/G)$ is given by
$Q_{h}^{K}:\mathfrak{R}^{K}_{h}\to\mathfrak{B}^{K}$. By Theorem 4.1,
$Q_{h}^{K}$ is injective and by Theorem 4.5,
${\mathfrak{B}^{K}}^{J_{\infty}(G)}=\mathfrak{C}^{K}$, the image of
$Q_{h}^{K}$. So $J_{\infty}(V)/\\!\\!/J_{\infty}(G)\cong
J_{\infty}(V/\\!\\!/G)$. ∎

##  5\. Some properties of standard monomials

By the definition of standard monomials, if $E_{1}E_{2}\cdots
E_{n}\in\mathcal{S}\mathcal{M}(\mathcal{E})$, then $E_{i+1}$ is the largest
element in $||\mathcal{E}(E_{i+1})||$ such that $E_{i}\leq E_{i+1}$. 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.

Let

$E=|(u_{h},k_{h}),\dots,(u_{1},k_{1})|\in\mathcal{E},$
$J^{\prime}={\bar{\partial}}^{n^{\prime}}|u^{\prime}_{h^{\prime}},\dots,u^{\prime}_{1}|\in\mathcal{J}.$

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

For $h^{\prime}\leq h$, let $\sigma$ be the permutation of
$\\{1,2,\dots,h^{\prime}\\}$ such that
$u_{\sigma_{L}(i)}<u_{\sigma_{L}(i+1)}$. Let $L(E,J^{\prime})$ be the smallest
non-negative integer $i_{0}$ such that $u^{\prime}_{i}\geq
u_{\sigma_{L}(i-i_{0})}$, $i_{0}<i\leq h^{\prime}$ Let

$E(h^{\prime})=|(u_{h^{\prime}},k_{h^{\prime}}),\dots,(u_{1},k_{1})|.$

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

The following lemma is obvious.

######  Lemma 5.1.

For
$J^{\prime\prime}={\bar{\partial}}^{k}|u^{\prime\prime}_{h^{\prime}},\dots,u^{\prime\prime}_{1}|\in\mathcal{J}$,
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
$L(E,J^{\prime\prime})\geq L(E,J^{\prime})-h^{\prime}+s$;

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

We say $J^{\prime}$ 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 a criterion for $J^{\prime}$ to be
greater than $E$.

######  Lemma 5.2.

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

###### Proof.

Let $i_{0}=L(E,J^{\prime})$ and $\sigma$ be the permutation of
$\\{1,2,\dots,h^{\prime}\\}$ such that $u_{\sigma_{L}(i)}<u_{\sigma_{L}(i+1)}$
.

If $wt(E(h^{\prime}))-wt(J^{\prime})\geq L(E,J^{\prime})$, let

$\tilde{u}^{\prime}_{\sigma(i)}=\left\\{\begin{array}[]{cc}u^{\prime}_{i+i_{0}},&\sigma(i)+i_{0}\leq
h^{\prime}\\\
u^{\prime}_{i+i_{0}-h_{b}},&i+i_{0}>h^{\prime}\end{array}\right.,\quad\quad
k^{\prime}_{\sigma(i)}=\left\\{\begin{array}[]{cc}k_{\sigma(i)},&i+i_{0}\leq
h^{\prime},i\neq h^{\prime}\\\ k_{\sigma(i)}+1,&i+i_{0}>h^{\prime},i\neq
h^{\prime}\end{array}\right.,$
$k^{\prime}_{\sigma(h^{\prime})}=wt(J^{\prime})-\sum_{i=1}^{h^{\prime}-1}k^{\prime}_{i}.$

Then

$k^{\prime}_{\sigma(h^{\prime})}=wt(J^{\prime})-wt(E(h^{\prime}))-i_{0}+k_{\sigma(h^{\prime})}+1-\delta_{i_{0}}^{0}\geq
k_{\sigma(h^{\prime})}+1-\delta_{i_{0}}^{0}.$
$(\tilde{u}^{\prime}_{\sigma(i)},k^{\prime}_{\sigma(i)})\geq(u_{\sigma(i)},k_{\sigma(i)}).$

So

$\tilde{E}^{\prime}=|(\tilde{u}^{\prime}_{h^{\prime}},k^{\prime}_{h^{\prime}}),\dots,(\tilde{u}^{\prime}_{2},k^{\prime}_{2}),(\tilde{u}^{\prime}_{1},k^{\prime}_{1})|$

is an element in $\mathcal{E}(J^{\prime})$ with $\tilde{E}^{\prime}\geq E$.

On the other hand, suppose $\tilde{E}^{\prime}\in\mathcal{E}(J^{\prime})$ with
$\tilde{E}^{\prime}\geq E$. Assume

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

We have $(\tilde{u}^{\prime}_{i},k^{\prime}_{i})\geq(u_{i},k_{i})$ i.e.,
$k^{\prime}_{i}>k_{i}$ or $k^{\prime}_{i}=k_{i},\tilde{u}^{\prime}_{i}\geq
u_{i}$. So

$\sum_{i=1}^{h^{\prime}}(k_{i}^{\prime}-k_{i})+\sharp\\{\tilde{u}^{\prime}_{i}\geq
u_{i},i|1\leq i\leq h^{\prime}\\}\geq h^{\prime},$

Let $i^{\prime}_{0}=h^{\prime}-\sharp\\{\tilde{u}^{\prime}_{i}\geq
u_{i},i|1\leq i\leq h^{\prime}\\}$. Then

$i^{\prime}_{0}\leq\sum_{i=1}^{h^{\prime}}(k_{i}^{\prime}-k_{i})=wt(J^{\prime})-wt(E(h^{\prime})).$

Here $\tilde{u}_{1}^{\prime},\dots,\tilde{u}^{\prime}_{h^{\prime}}$ is a
permutation of $u^{\prime}_{1},\dots,u^{\prime}_{h^{\prime}}$. By the
definition of $i^{\prime}_{0}$, it is easy to see that $u^{\prime}_{i}\geq
u_{\sigma(i-i^{\prime}_{0})}$, $i^{\prime}_{0}<i\leq h^{\prime}$. So
$i^{\prime}_{0}\geq L(E,J^{\prime})$. Thus

$wt(E(h^{\prime}))-wt(J^{\prime})\geq i_{0}^{\prime}\geq L(E,J^{\prime}).$

∎

######  Corollary 5.3.

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

###### Proof.

By Lemma 5.2, $J^{\prime}$ is greater than $E$ if and only if
$wt(E(h^{\prime}))-wt(J^{\prime})\geq L(E,J^{\prime})$ and
$||E(h^{\prime})||J^{\prime}$ is standard if and only if
$wt(E(h^{\prime}))-wt(J^{\prime})\geq
L(E(h^{\prime}),J^{\prime})=L(E,J^{\prime})$. ∎

### The property “largest”

Let

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

######  Lemma 5.4.

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

###### Proof.

Assume

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

For $s<h^{\prime}$, let $J_{s}$ be the smallest element in
$\mathcal{W}_{s}(E,J^{\prime})$. 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 $\mathcal{E}(J_{s})$ such that $E(s)\leq E_{s}$.

Assume $l$ is the largest number such that
$(u^{\prime}_{j},k^{\prime}_{j})=(u^{\prime}_{i_{j}},\tilde{k}_{j})$ for
$j<l\leq s+1$. If $l\leq s$, 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 $E^{\prime}$ and the minimality of $J_{s}$, we
must have
$(u^{\prime}_{i_{l}},\tilde{k}_{l})=(u^{\prime}_{l},k^{\prime}_{l})$, 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 $\mathcal{E}(J^{\prime})$ obtained by
replacing $(u^{\prime}_{l},k^{\prime}_{l})$ and
$(u^{\prime}_{i_{l}},k^{\prime}_{i_{l}})$ in $E^{\prime}$ 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 $E^{\prime}\prec E^{\prime\prime}$ and
$E(h^{\prime})\leq E^{\prime\prime}$. But $E^{\prime}\neq E^{\prime\prime}$ is
the largest element in $\mathcal{E}(||E^{\prime L}||)$ such that $E\leq
E^{\prime}$, which is 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 $E(s)\leq
E^{\prime}_{s}$ and $||E^{\prime}_{s}||\prec J_{s}$. This is impossible since
$J_{s}\neq||E^{\prime}_{s}||$ is the smallest element in
$\mathcal{W}_{s}(E,E^{\prime})$. 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 $E\leq E^{\prime}_{s}$.
But $E^{\prime}_{s}\neq E_{s}$ is the largest element in $\mathcal{E}(J_{s})$
such that $E\leq E^{\prime}_{s}$, a contradiction. Therefore
$E_{s}=E^{\prime}(s)$, and $||E^{\prime}(s)||$ is the smallest element in
$\mathcal{W}_{s}(E,J^{\prime})$. ∎

######  Corollary 5.5.

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

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

###### Proof.

Since $E\leq E^{\prime}$, $E\leq E^{\prime}(s)$. By Lemma 5.4,
$||E^{\prime}(s)||$ is the smallest element in
$\mathcal{W}_{s}(E,J^{\prime})$. By Lemma 5.2,
$L(E,||E^{\prime}(s)||)=wt(E^{\prime}(s))-wt(E(s)).$ ∎

######  Corollary 5.6.

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
$J\in\mathcal{W}_{s}(E,E^{\prime})$,

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

###### Proof.

Assume $J={\bar{\partial}}^{k}|u_{s},\dots,u_{1}|$ and
$||E^{\prime}(s)||={\bar{\partial}}^{l}|u^{\prime}_{s},\dots,u^{\prime}_{1}|$.
If $m=L(E,J)-L(E,||E^{\prime}(s)||)<0$, let
$J^{\prime\prime}={\bar{\partial}}^{l+m}|u_{s},\dots,u_{1}|$. By Lemma 5.2,
$J^{\prime\prime}\in W_{s}(E,J^{\prime})$. By Lemma 5.4, $||E^{\prime}(s)||$
is the smallest element in $\mathcal{W}_{s}(E,||E^{\prime}||)$. But
$wt(||E^{\prime}(s)||)>wt(J^{\prime\prime})$, a contradiction. ∎

######  Lemma 5.7.

Let $E_{i}=|(u^{i}_{h_{i}},k^{i}_{h_{i}}),\dots,(u^{i}_{1},k^{i}_{1})|$,
$i=a,b$. 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 $\sigma_{i}$ be permutations of $\\{1,\dots,h\\}$, such that
$u^{i}_{\sigma_{i}(1)}<u^{i}_{\sigma_{i}(2)}<\cdots<u^{i}_{\sigma_{i}(h)}$.
Let $u_{1}^{\prime},\dots,u^{\prime}_{h_{a}}$ be a permutation of
$u^{a}_{1},\dots,u^{a}_{h_{a}}$ such that
$u_{1}^{\prime}<u_{2}^{\prime}<\cdots<u^{\prime}_{h_{a}}$. Assume
$u^{\prime}_{i_{2}}=u^{a}_{\sigma(i_{1})}$ with $i_{2}>i_{1}$. Then for any

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

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

###### Proof.

Let $n=wt(E_{a}(h))$.
Let $i_{0}=R(E_{b},K)$, then $u^{\prime}_{t_{i}}\geq
u^{b}_{\sigma_{b}(i-i_{0})}$, for $s\geq i>i_{0}$.
Let $i^{\prime}_{0}=R(E_{b},||E_{a}(h)||)$, then $u^{a}_{\sigma_{a}(i)}\geq
u^{b}_{\sigma_{b}(i-i^{\prime}_{0})}$, for $h\geq i>i^{\prime}_{0}$.

We have $i_{1}>i^{\prime}_{0}$. Otherwise, $i_{1}\leq i^{\prime}_{0}$.
Replacing $u^{a}_{\sigma_{a}(i_{1})}$ in $||E^{a}(h)||$ by some
$u^{a}_{i}<u^{a}_{\sigma_{a}(i_{1})}$ with $i>h$, (such $u^{a}_{i}$ exists
since $i_{2}>i_{1}$), we get

$J={\bar{\partial}}^{n}|u_{\sigma^{\prime}_{a}(h)}^{a},\dots,u_{\sigma^{\prime}_{a}(i_{1}+1)}^{a}\,u^{a}_{i},u_{\sigma^{\prime}_{a}(i_{1}-1)}^{a},\dots,u^{a}_{\sigma^{\prime}_{a}(1)}|$

with $R(E_{b},J)\leq i^{\prime}_{0}.$ By Lemma 5.2, $J$ is greater than
$E_{b}$. It is impossible by Lemma 5.4 since $J\prec||E_{a}(h)||$ and $E_{a}$
is the largest element in $\mathcal{E}(||E_{a}||)$ such that $E_{b}\leq
E_{a}$.

If $s\geq i_{1}$, let

$J^{\prime}={\bar{\partial}}^{n}|u_{\sigma_{a}(h)}^{a},\dots,u_{\sigma_{a}(i_{1}+1)}^{a},u^{\prime}_{t_{s}},\dots,u^{\prime}_{t_{s-i_{1}+1}}|.$

If $L(E_{b},K)\leq L(E_{b},||E_{a}(h)||)+s-i_{1}$,

$u^{\prime}_{t_{i}}\geq u^{b}_{\sigma_{b}(i-i_{0})}\geq
u^{b}_{\sigma_{b}(i-i^{\prime}_{0}-s+i_{1})},\,\,\text{for }i\leq s.$

We have $R(E_{b},J^{\prime})\leq i^{\prime}_{0}.$ By Lemma 5.2, $J^{\prime}$
is greater than $E_{b}$. By Lemma 5.4, it is impossible since
$J^{\prime}\prec||E_{a}(h)||$ and $E_{a}$ is the largest element in
$\mathcal{E}(||E_{a}||)$ such that $E_{b}\leq E_{a}$. So when $s\geq i_{1}$,
$L(E_{b},K)\geq L(E_{b},||E_{a}||)+s-i_{1}$.

If $s<i_{1}$, let
$t^{\prime}_{1}<t^{\prime}_{2}<\cdots<t^{\prime}_{i_{1}}<i_{2}$ with
$\\{t_{1},\dots,t_{s}\\}\subset\\{t^{\prime}_{1},\dots,t^{\prime}_{i_{1}}\\}$.
Let

$K^{\prime}={\bar{\partial}}^{k}|u^{\prime}_{t^{\prime}_{1}},\dots,u^{\prime}_{t^{\prime}_{i_{1}}},u_{i_{1}+1},\dots,u_{h}|.$

By Lemma 5.1, we have

$L(E_{b},K)\geq L(E_{b},K^{\prime})+s-i_{1}\geq L(E_{b},||E_{a}||)+s-i_{1}.$

So for any $s>0$, we have $L(E_{b},K)>L(E_{b},||E_{a}||)+s-i_{1}$. ∎

The following lemmas are obvious.

######  Lemma 5.8.

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 5.9.

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 2.8

In this section we prove Lemma 2.8. By Lemma 5.8, we can assume the monomials
are expressed as ordered products $J_{1}J_{2}\cdots J_{b}$ with $J_{a}\prec
J_{a+1}$. For $\alpha\in\mathcal{M}(\mathcal{J})$, let

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

the space of linear combinations of elements preceding $\alpha$ in
$\mathcal{M}(\mathcal{J})$ with integer coefficients.

######  Lemma 6.1.

If $J_{1}J_{2}$ is not standard, $J_{1}J_{2}\in\mathfrak{R}(J_{1})$.

###### Proof.

Assume
$J_{i}={\bar{\partial}}^{n_{i}}|u^{i}_{h_{i}},\dots,u^{i}_{2},u^{2}_{1}|$, for
$i=1,2$.
Let $E_{1}=|(u^{1}_{h_{1}},n_{1}),\dots,(u^{1}_{1},0)|$. Let
$i_{0}=L(E_{1},J_{2})$. If $i_{0}\neq 0$, there is $i_{0}\leq i_{1}\leq
h_{2}$, such that $u^{2}_{i_{1}}<u^{1}_{i_{1}-i_{0}+1}$. If $i_{0}=0$, let
$i_{1}=0$. Let $m=n_{1}+n_{2}$ and $a_{n_{2}-l}=\delta^{0}_{l}$, $0\leq l\leq
l_{0}-1$. By Lemma 2.5, there are integers $a_{k}$ such that

(6.1) $\sum\epsilon
a_{k}{\bar{\partial}}^{m-k}|\underline{u^{1}_{h_{1}},\dots,u^{1}_{i_{1}-i_{0}+1}},u^{1}_{i_{1}-i_{0}},\dots,u^{1}_{1}|{\bar{\partial}}^{k}|u^{2}_{h_{2}},\dots,u^{2}_{i_{1}+1},\underline{u^{2}_{i_{1}},\dots,u^{2}_{1}}|\in\mathfrak{R}[h_{1}+1].$

  1. (1)

If $h_{1}=h_{2}$, then $n_{1}\leq n_{2}$. Since $J_{1}J_{2}$ is not standard,
$J_{2}$ is not greater than $E_{1}$. By Lemma 5.2, $i_{0}>n_{2}-n_{1}\geq 0$.
$J_{1}J_{2}\in\mathfrak{R}(J_{1})$ since in Equation (6.1):

     * •

All the terms with $k=n_{2}$ are preceding $J_{1}$ 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 $k<n_{1}$;

     * •

The terms in $\mathfrak{R}[h_{1}+2]$ are preceding $J_{1}$ since they have
bigger sizes.

  2. (2)

If $h_{1}>h_{2}$. Since $J_{1}J_{2}$ is not standard, by Lemma 5.2,
$i_{0}>n_{2}$. $J_{1}J_{2}\in\mathfrak{R}(J_{1})$ since in Equation (6.1):

     * •

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

     * •

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

     * •

The terms in $\mathfrak{R}[h_{1}+2]$ are preceding $J_{1}$ since they have
bigger sizes.

∎

###### Proof of Lemma 2.8.

We prove the lemma by induction on $b$.
If $b=1$, $J_{1}$ is standard.
If $b=2$, by Lemma 6.1, the lemma is true.
For $b\geq 3$, assume the lemma is true for $b-1$. We can assume $J_{1}\cdots
J_{b-1}$ is standard by induction and Lemma 5.9. Let $E_{1}\cdots
E_{b-1}\in\mathcal{S}\mathcal{M}(\mathcal{E})$ be the standard ordered product
of elements of $\mathcal{E}$ corresponding to $J_{1}\cdots J_{b-1}$. If
$J_{1}\cdots J_{b}$ is not standard, then $J_{b}$ is not greater than
$E_{b-1}$. By Lemma 6.2 (below), $J_{b-1}J_{b}=\sum K_{i}f_{i}$ with
$K_{i}\in\mathcal{J}$, $f_{i}\in\mathfrak{R}$ such that $K_{i}$ is either
smaller than $J_{b-1}$ or $K_{i}$ is not greater than $E_{b-2}$. If $K_{i}$ is
smaller than $J_{b-1}$, then $J_{1}\cdots
J_{b-1}K_{i}f\in\mathfrak{R}(J_{1}\cdots J_{b-1})$. If $K_{i}$ is not greater
than $E_{b-2}$, $J_{1}\cdots J_{b-2}K_{i}$ is not standard, so it is in
$\mathfrak{R}(J_{1}\cdots J_{b-2})$ by induction. Then $J_{1}\cdots
J_{b-2}K_{i}f\in\mathfrak{R}(J_{1}\cdots J_{b-1})$. So $J_{1}\cdots J_{b}=\sum
J_{1}\cdots J_{b-2}K_{i}f_{i}\in\mathfrak{R}(J_{1}\cdots J_{b-1})$. ∎

######  Lemma 6.2.

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 $E$.

###### Proof.

Assume
$J_{a}={\bar{\partial}}^{n_{a}}|u^{\prime}_{h_{a}},\dots,u^{\prime}_{1}|$ and
$J_{b}={\bar{\partial}}^{n_{b}}|u^{b}_{h_{b}},\dots,u^{b}_{1}|$. $J_{a}\prec
J_{b}$, so $h_{a}\geq h_{b}$. Assume
$||E_{a}(h_{b})||={\bar{\partial}}^{m_{a}}|u^{a}_{h_{b}},\dots,u^{a}_{1}|$.
Let $m=n_{b}+n_{a}$. Let $i_{0}=L(E_{a},J_{b})$. If $i_{0}\neq 0$, there is
$i_{0}\leq i_{1}\leq h_{b}$, such that $u^{b}_{i_{1}}<u^{a}_{i_{1}-i_{0}+1}$.
If $i_{0}=0$, let $i_{1}=0$. Since $J_{b}$ is not greater than $E_{a}$, by
Lemma 5.2,

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

By definition, $\\{u^{a}_{h_{b}},\dots,u^{a}_{1}\\}$ is a subset of
$\\{u^{\prime}_{h_{a}},\dots,u^{\prime}_{1}\\}$ with
$u^{\prime}_{i}<u^{\prime}_{i+1}$ and $u^{a}_{i}<u^{a}_{i+1}$. If we assume
$u^{\prime}_{i_{2}}=u^{a}_{i_{1}-i_{0}+1}$, we have $i_{2}\geq i_{1}-i_{0}+1$.

Now we prove the lemma. The proof is quite long and it is divided into three
cases.

Case 1: $h_{a}=h_{b}$. Let $a_{n_{b}-l}=\delta^{0}_{l}$ for $0\leq l\leq
i_{0}-1$. By Lemma 2.5, there are integers $a_{k}$, such that

(6.3) $\sum\epsilon
a_{k}{\bar{\partial}}^{m-k}|\underline{u^{\prime}_{h_{a}},\dots,u^{\prime}_{i_{1}-i_{0}+1}},u^{\prime}_{i_{1}-i_{0}},\dots,u^{\prime}_{1}|\,\,{\bar{\partial}}^{k}|u^{b}_{h_{b}},\dots,u^{b}_{i_{1}+1},\underline{u^{b}_{i_{1}},\dots,u^{b}_{1}}|\in\mathfrak{R}[h_{a}+2].$

$J_{a}J_{b}\in\mathfrak{R}(J_{a})$ since in the above equation,

  * •

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

  * •

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

  * •

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

  * •

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

  * •

The terms in $\mathfrak{R}[h_{a}+2]$ are preceding $J_{a}$ since they have
bigger sizes.

Case 2: $h_{b}<h_{a}$ and $n_{b}<m_{a}$.

By Lemma 2.5,

(6.4) $\sum_{0\leq
i<h_{b}}\sum_{\sigma}\frac{(-1)^{i}\operatorname{sign}(\sigma)}{i!(h_{b}-i)}\sum\epsilon
a^{i}_{k}$
${\bar{\partial}}^{m-k}|\underline{u^{\prime}_{h_{a}},\dots,u^{\prime}_{i+1}},u^{b}_{\sigma{(i)}},\dots,u_{\sigma(2)}^{b},u^{b}_{\sigma(1)}|\,\,{\bar{\partial}}^{k}|\underline{u_{\sigma(h_{b})}^{b},\dots,u^{b}_{\sigma{(i+1)}},u^{\prime}_{i},\dots,u^{\prime}_{1}}|\in\mathfrak{R}[h_{a}+2].$

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

  * •

The terms in $\mathfrak{R}[h_{a}+2]$ are preceding $J_{a}$ since they have
bigger sizes.

  * •

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

  * •

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

$K_{0}={\bar{\partial}}^{n_{b}}|u^{\prime}_{i_{h_{b}}},\dots,u^{\prime}_{i_{2}},u^{\prime}_{i_{1}}|\in\mathcal{J}.$

And all the other terms are cancelled. By Corollary 5.5 and 5.6,

$L(E_{b-2},K_{0})\geq L(E_{b-2},E_{a}(h_{b}))=m_{a}-m_{b-2}>n_{b}-m_{b-2}.$

By Lemma 5.2, $K_{0}$ is not greater than $E_{b-2}$.

  * •

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

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

By Lemma 5.1,

$L(E_{b-2},K_{1})\geq L(E_{b-2},K_{0})-(h_{b}-i).$

So

$L(E_{b-2},K_{1})>n_{b}-m_{b-2}-(h_{b}-i)\geq k-m_{b-2}.$

By Lemma 5.2, $K_{1}$ is not greater than $E_{b-2}$.

Case 3: $h_{a}>h_{b}$ and $n_{b}\geq m_{a}$. If $i_{0}=0$, then $J_{b}$ is
greater than $E_{a}$ and $J_{1}\cdots J_{b}$ is standard. So $i_{0}>0$. By
Lemma 2.5, we have

(6.5)
$\sum_{\min\\{i_{1},i_{2}\\}>s}\sum_{\sigma}\frac{(-1)^{s}\operatorname{sign}(\sigma)}{s!(i_{2}-1-s)!}\sum_{k=0}^{m}\epsilon
a^{s}_{k}$
${\bar{\partial}}^{m-k}|\underline{u^{\prime}_{h},\dots,u^{\prime}_{i_{2}},u^{\prime}_{\sigma(i_{2}-1)},\dots,u^{\prime}_{\sigma(s+1)},u_{s}^{b},\dots,u^{b}_{1}}|\,\,{\bar{\partial}}^{k}|u^{b}_{h_{b}},\dots,u^{b}_{i_{1}+1},\underline{u^{b}_{i_{1}},\dots,u^{b}_{s+1}},u^{\prime}_{\sigma(s)},\dots,u^{\prime}_{\sigma(1)}|$
$\in\mathfrak{R}[h_{a}+1].$

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

(6.6) $\sum_{\begin{subarray}{c}i\geq i_{1}>s\\\
i_{2}>s\end{subarray}}\sum_{\sigma,\sigma_{1}}\frac{(-1)^{i+s+t}\operatorname{sign}(\sigma)\operatorname{sign}(\sigma_{1})}{(i-i_{1})!(h_{b}-i)!s!(i_{2}-1-s)!}\sum_{k=0}^{m}\epsilon
a^{i,s}_{k}$
${\bar{\partial}}^{n-k}|u^{b}_{\sigma(h_{b})}\dots\underline{u^{b}_{\sigma(i)}\dots
u^{b}_{\sigma(i_{1}+1)},u^{b}_{i_{1}}\dots u^{b}_{1},u^{\prime}_{h_{a}}\dots
u^{\prime}_{h_{b}+1}}|\,{\bar{\partial}}^{k}|\underline{u^{\prime}_{h_{b}+1},\dots
u^{\prime}_{i_{2}},u^{\prime}_{\sigma_{1}(i_{2}-1)},\dots
u^{\prime}_{\sigma_{1}(s+1)}},u^{\prime}_{\sigma_{1}(s)},\dots
u^{\prime}_{\sigma_{1}(1)}|$ $\in\mathfrak{R}[h_{a}+1].$

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

  * •

We use Equation (6.5) when $i_{2}>i_{1}-i_{0}+1$;

  * •

We use Equation (6.6) when $i_{2}=i_{1}-i_{0}+1$.

In the above relations:

  1. (1)

The terms in $\mathfrak{R}[h_{a}+2]$ are preceding $J_{a}$ since they have
bigger sizes.

  2. (2)

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

  3. (3)

The terms with $k=n_{b}$ are $J_{a}J_{b}$, the terms with the upper sequence
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 $u^{b}_{k}$) and the
terms with the lower sequences

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

And all of the other terms cancel. By Lemma 5.7,

$L(E_{b-2},K_{0})>L(E_{b-2},||E_{a}(h)||)+i_{1}-(i_{1}-i_{0}+1);$

By the above inequality,

$\displaystyle L(E_{b-2},K_{0})$ $\displaystyle\geq$ $\displaystyle
L(E_{b-2},||E_{a}(h_{b})||)+i_{0}$ $\displaystyle(\text{by corollary
\ref{cor:lrnumber1}})\quad\quad$ $\displaystyle=$ $\displaystyle
wt(E_{a}(h_{b}))-wt(E_{b-2})+i_{0}$ $\displaystyle(\text{by Equation
(\ref{eqn:numberl0}})\quad\quad$ $\displaystyle>$ $\displaystyle
wt(E_{b})-wt(E_{b-2}(h_{b})).$

By Lemma 5.2, $K_{0}$ is not greater than $E_{b-2}$.

  4. (4)

When $i_{2}>i_{1}-i_{0}+1$, the terms with $k<n_{b}$ in Equation (6.5)
vanishes unless $i_{1}-s\leq n_{b}-k$. In this case the lower sequences of the
terms are

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

The underlined $u$ in $K_{1}$ can be any underlined $u$ in Equation (6.5). By
Lemma 5.7

$L(E_{b-2},K_{1})>L(E_{b-2},E_{a}(h_{b}))+s-(i_{1}-i_{0}+1);$ $\displaystyle
L(E_{b-2},K_{1})$ $\displaystyle\geq$ $\displaystyle
L(E_{b-2},E_{a}(h_{b}))+k-n_{b}+i_{0}$ $\displaystyle(\text{by Corollary
\ref{cor:lrnumber1}})$ $\displaystyle=$ $\displaystyle
wt(E_{a}(h_{b}))-wt(E_{b-2}(h_{b}))+k-n_{b}+i_{0}$ $\displaystyle(\text{by
Equation (\ref{eqn:numberl0}})\,$ $\displaystyle>$ $\displaystyle
k-wt(E_{b-2}(h_{b})).$

By Lemma 5.2, $K_{1}$ is not greater than $E_{b-2}$.

  5. (5)

When $i_{2}=i_{1}-i_{0}+1$, the terms with $k<n_{b}$ are the terms in the
Equation (6.6), such that $i-s\leq n_{b}-k$. In this case, the lower sequences
of the terms are

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

The underlined $u$ in $K_{1}$ can be any underlined $u$ in Equation (6.6). Let

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

Here $i_{2}\leq k_{i_{2}}<k_{i_{2}+1}<\cdots<k_{h_{b}}\leq h_{a}$. By Lemma
5.1, there is some $K^{\prime}_{1}$ such that

(6.7) $\displaystyle L(E_{b-2},K_{1})\geq
L(E_{b-2},K^{\prime}_{1})-(i_{2}-1-s)-(i-i_{1}),$

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

(6.8) $L(E_{b-2},K_{1}^{\prime})\geq L(E_{b-2},||E_{a}(h_{b})||).$

By Equation (6.7, 6.8),

$\displaystyle L(E_{b-2},K_{1})$ $\displaystyle\geq$ $\displaystyle
L(E_{b-2},||E_{a}(h_{b})||)+s-(i-i_{0})$ $\displaystyle(\text{by Corollary
\ref{cor:lrnumber1}})$ $\displaystyle\geq$ $\displaystyle
wt(E_{a}(h_{b}))-wt(E_{b-2}(h_{b}))+k-n_{b}+i_{0}$ $\displaystyle(\text{by
Equation \ref{eqn:numberl0}})$ $\displaystyle>$ $\displaystyle
k-wt(E_{b-2}(h_{b})).$

By Lemma 5.2, $K_{1}$ is not greater than $E_{b-2}$.

∎

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