# Proof of Kac and Rudakov’s Conjecture on Generalized Verma Module over Lie
Superalgebra $\mbox{E}(5,10)$111Research supported by CSC

Yufeng Zhao

LMAM, School of Mathematical Sciences
Peking University, Beijing, 100871, P. R. China

###### Abstract

The exceptional infinite-dimensional linearly compact simple Lie superalgebra
$\mbox{E}(5,10)$, which Kac believes, is the algebra of symmetries of the
$\mbox{SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and
Rudakov’s conjecture about the classification of all the degenerate
generalized Verma module over $\mbox{E}(5,10)$. Also, we work out all the
nontrivial singular vectors degree by degree. It is a potential that the
representation theory of $\mbox{E}(5,10)$ will shed new light on various
features of the the $\mbox{SU}_{5}$ Grand unified model.

##  1 Introduction

A linearly compact infinite-dimensional Lie algebra is a topological Lie
algebra whose underlying space is a topological space isomorphic to the space
of formal power series over complex field in finite number of variables with
formal topology. Cartan’s list of linearly compact infinite-dimensional simple
Lie algebras consists of four series: the Lie algebra of all complex vector
fields and its subalgebras of divergence 0 vector fields, symplectic vector
fields and contact vector fields.

Kac proved the “super” version of this result. In other words, he classified
linearly compact infinite-dimensional Lie superalgebras [K1]. There turn out
to be 10 families and 5 exceptions, which are called
$\mbox{E}(1,6),\mbox{E}(3,6),\mbox{E}(3,8),\mbox{E}(4,4)$ and
$\mbox{E}(5,10)$. Many of the families are straightforward “super”
generalizations of the 4 families of linearly compact infinite-dimensional
simple Lie algebras. Some are stranger. Most important for us today are the 5
exceptions discovered by Irina Shchepochkina [Sh].

The representation theory of $E(3,6)$ and $E(3,8)$ was developed by Kac and
Rudakov [KR1-KR3], and some further observations were made on its connections
to the Standard Model [K2]. It was found quite remarkable that the
$\mbox{SU}_{5}$ Grand unified model of Georgi-Glashow combines the left
multiplets of fundamental fermions in precisely the negative part of the
consistent gradation of $\mbox{E}(5,10)$. This is perhaps an indication of the
possibility that an extension from $\mbox{su}_{5}$ to algebra of internal
symmetries may resolve the difficulties with the proton decay. It is a
potential that the representation theory of $\mbox{E}(5,10)$ will shed new
light on various features of the the $\mbox{SU}_{5}$ Grand unified model.

As to the representation theory of $E(5,10)$, Kac and Rudakov formulate an
conjecture [KR3], which can be stated as follows. The Lie superalgebra
$L=E(5,10)$ carries a unique consistent irreducible $\mathbb{Z}$-gradation
$L=\bigoplus\limits_{j\geq-2}L_{j}$, where $L_{0}$ is isomorphic to simple Lie
algebra $sl_{5}$. Given $L_{0}$\- module $V$, we extend it to a $L$ module by
letting $L_{+}$ acts trivially, and define the induced module

$M(V)=U(L)\otimes_{U(L_{0})}V\cong U(L_{-})V.$

If $V$ is finite-dimensional irreducible $L_{0}$ -module, the $L$ module
$M(V)$ is called a generalized Verma module  associated to $V$, and it is
called degenerate  if it is not irreducible.

We denote by $\mbox{V}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$ the
finite-dimensional irreducible $L_{0}$ -module with highest weight
$\sum\limits_{i=1}^{4}\lambda_{i}\omega_{i}$, where
$\omega_{1},\omega_{2},\omega_{3},\omega$ are the fundamental weights for
$sl_{5}$. Let

$M=M(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=M(V(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}))$

denote the corresponding generalized Verma module over $\mbox{E}(5,10)$.
Denote by $\mathbb{N}$ the additive semigroup of nonnegative integers.

Kac and Rudakov’s Conjecture The following is a complete list of degenerate
Verma modules over $\mbox{E}(5,10)$:

$\mbox{M}(m,n,0,0);\mbox{M}(m,0,0,n);\mbox{M}(0,0,m,n)\ (m,n\in\mathbb{N}).$

In this paper, we give a proof of this conjecture and work out all the
nontrivial singular vectors for any generalized Verma module over
$\mbox{E}(5,10)$. The first key point of our proof is investigating that there
exists a grading on the generalized Verma module, through which we find that
any singular vector is controlled by its leading term via an exponential-like
differential operator, where the leading term lies in certain tensor product
module of $sl_{5}$; the second one is the observation of an irreducible tensor
operator of rank $\omega_{1}+\omega_{2}$ for simple Lie algebra $sl_{5}$,
which plays the center role in our calculation of all the singular vectors.

The paper is organized as follows: In section 2, we recall Kac’s geometric
construction of Lie superalgebra $\mbox{E}(5,10)$ and the KR conjecture. In
Section 3, we provide some techniques concerning the irreducible tensor
operators and tensor module decomposition theory of simple Lie algebra. In
Section 4, we prove that all the nontrivial singular vectors are of degree
less than or equal to four. Also, the leading term of any singular vector must
lie in one of the tensor decomposition of four tensor product module of
$sl_{5}$ (cf. Theorem 4.6). In Section 5, we work out all the nontrivial
singular vectors degree by degree (cf. Theorem 5.3, Theorem 5.4, Theorem 5.5,
Theorem 5.6).

##  2 Lie superalgebra $\mbox{E}(5,10)$ and KR conjecture

In this section, we recall Kac’s geometric construction of Lie superalgebra
$\mbox{E}(5,10)$ and KR Conjecture which are stated in [KR3].

For two integers $m<n$, we denote $\overline{m,n}=\\{m,m+1,\cdots,n\\}$. Let

$\mbox{W}_{n}=\\{\sum\limits_{i=1}^{n}p_{i}(x)\partial_{i}\ |\
p_{i}(x)\in\mathbb{C}[[x_{1},\cdots,x_{n}]],\partial_{i}=\partial_{x_{i}}\\}$
$None$

denote the Lie algebra of formal vector fields in $n$ indeterminates;

$\mbox{S}_{n}=\\{D=\sum\limits_{i=1}^{n}p_{i}\partial_{i}\ |\
\mbox{div}D=\sum\limits_{i=1}^{n}\partial_{i}(p_{i})=0\\}$ $None$

denote the Lie subalgebra of divergenceless formal vector fields;
$\Omega^{k}(n)$ denote the associative algebra of formal differential forms of
degree k in $n$ indeterminates, $\Omega^{k}_{\mbox{cl}}(n)$ denote the
subspace of closed forms.

The exceptional infinite-dimensional linearly compact Lie superalgebra
$\mbox{E}(5,10)=\mbox{E}(5,10)_{\underline{0}}+\mbox{E}(5,10)_{\underline{1}}$
is constructed as follows:

$\mbox{E}(5,10)_{\underline{0}}=S_{5},\
\mbox{E}(5,10)_{\underline{1}}=\Omega_{\mbox{cl}}^{2}(5),$ $None$

where $\mbox{E}(5,10)_{\underline{0}}$ acts on
$\mbox{E}(5,10)_{\underline{1}}$ via the Lie derivative,

$[\omega_{2},\omega_{2}^{\prime}]=\omega_{2}\wedge\omega_{2}^{\prime}\in\Omega_{\mbox{cl}}^{4}(5)=S_{5}$
$None$

for $\omega_{2},\omega_{2}^{\prime}\in\mbox{E}(5,10)_{\underline{1}}$.

We use for the odd elements of $\mbox{E}(5,10)$ the notation
$d_{ij}=dx_{i}\wedge dx_{j}(i,j\in\overline{1,5})$; recall that we have the
following commutation relation ($f,g\in C[[x_{1},\cdots,x_{5}]]$ ):

$[fd_{jk};gd_{lm}]=\varepsilon_{ijklm}fg\partial_{i};$ $None$

where

$\varepsilon_{ijklm}=\left\\{\begin{array}[]{ll}\mbox{the \ sign \ of \ the \
permutation}\ $(ijklm)$,&\mbox{ if \ all \ indices \ $ijklm$ \ are \
distinct},\\\ 0,&\mbox{otherwise}.\end{array}\right.$ $None$

And the Lie superalgebra $\mbox{L}=\mbox{E}(5,10)$ carries a unique consistent
irreducible $\mathbb{Z}$-gradation
$\mbox{L}=\bigoplus\limits_{j\geq-2}\mbox{L}_{j}$. It is defined by:

$degx_{i}=2=-\partial_{i},degd_{ij}=-1$ $None$

One has: $\mbox{L}_{0}\simeq\mbox{sl}_{5}$ and the $\mbox{L}_{0}$-modules
occurring in the negative part are:

$\mbox{L}_{-1}=\mbox{Span}_{\mathbb{C}}\\{d_{ij}\ |\
i,j\in\overline{1,5}\\}\simeq\Lambda^{2}\mathbb{C}^{5},$
$\mbox{L}_{-2}=\mbox{Span}_{\mathbb{C}}\\{\partial_{i}\ |\
i\in\overline{1,5}\\}\simeq\mathbb{C}^{5*}$ $None$

Recall also that $\mbox{L}_{1}$ consist of closed 2-forms with linear
coefficients, that $\mbox{L}_{1}$ is an irreducible $\mbox{L}_{0}$-module and
$\mbox{L}_{j}=[\mbox{L}_{1}[\cdots]]=\mbox{L}_{1}^{j}$ for $j\geq 1$. We take
for the Borel subalgebra of $\mbox{L}_{0}\simeq\mbox{sl}_{5}$the subalgebra of
the vector fields

$\mbox{Span}\\{x_{i}\partial_{j}(1\leq i\leq j\leq 5),\
x_{i}\partial_{i}-x_{i+1}\partial_{i+1}(i\in\overline{1,4})\\}.$ $None$

Given $\mbox{L}_{0}$ module $V$, we extend it to a L module by letting
$\mbox{L}_{+}$ acts trivially, and define the induced module

$M(V)=U(\mbox{L})\otimes_{U(\mbox{L}_{0})}V\cong U(\mbox{L}_{-})V.$ $None$

We denote by $\mbox{V}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$ the
finite-dimensional irreducible $\mbox{L}_{0}$ -module with highest weight
$\sum\limits_{i=1}^{4}\lambda_{i}\omega_{i}$, where $\omega_{i}$
($i\in\overline{1,4}$) are the fundamental weights for $sl_{5}$. Let

$M=M(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=M(V(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}))$
$None$

denote the corresponding generalized Verma module over $\mbox{E}(5,10)$.

Definition 2.1 If $\xi\in M$ satisfies :

$(x_{i}\partial_{x_{i+1}}).\xi=0(i\in\overline{1,4}),$ $None$
$x_{5}d_{45}.\xi=0,$ $None$

then we call $\xi$ a singular vector for generalized verma module $M$ of
$E(5,10)$.

The aim of the following sections is to determine all the nontrivial singular
vectors for $\mbox{E}(5,10)$-module $M$.

##  3 Preliminary

In this section, we give some preparatory techniques about the irreducible
tensor operators and the decomposition of tensor product module of simple Lie
algebra.

Following the notations of Humphreys [H], let $H$ be the Cartan subalgebra of
simple Lie algebra $L$, and let $\Delta=\\{\alpha_{1},\cdots,\alpha_{l}\\}$ be
a base for the root system $\phi$ of $H^{*}$. The corresponding fundamental
dominant weights $\\{\omega_{1},\cdots,\omega_{l}\\}$ are defined from the
root system via the form $<\cdot,\cdot>$ given by:

$<\omega_{i},\alpha_{j}>\equiv\frac{2(\omega_{i},\alpha_{j})}{(\alpha_{j},\alpha_{j})}=\delta_{ij},$
$None$

where $(\cdot,\cdot)$ denotes the inner product induced on $H^{*}$ by the
Killing form on $H$. Consider a basis
$\\{h_{1},\cdots,h_{l},x_{\alpha},\alpha\in\phi\\}$ of $L$ where
$h_{1},\cdots,h_{l}$ is a basis for $H$ and $x_{\alpha}$ is a nonzero element
of the root space $L_{\alpha}$. The dual basis may therefore be written
$\\{h^{1},\cdots,h^{l},x^{\alpha},\alpha\in\phi\\}$ where $x^{\alpha}$ is the
unique element of $L_{-\alpha}$ which is dual to $x_{\alpha}$ under the
Killing form of $L$. Write the universal Casimir element in the form:

$c_{L}=\sum\limits_{i=1}^{l}h_{i}h^{i}+\sum\limits_{\alpha\in\phi}x_{\alpha}x^{\alpha}.$
$None$

Let $V(\mu)$ be an irreducible highest weight module over $L$ and let
$\pi_{\mu}$ be the representation afforded by $V(\mu)$. Choose an ordeded
basis $\\{e_{1},\cdots,e_{d}\\}$ of $V(\mu)$, let $\pi_{\mu}(x)$ denote the
matrix representing $x\in L$ on $V(\mu)$ with respect to this basis.

Definition 3.1 We call a collection of linear operators $\\{T_{i}:V\rightarrow
W\ |\ i\in\overline{1,d}\\}$ an irreducible tensor operator of rank $\mu$ if
these components transform according to the rule:

$[x,T_{i}]=\pi_{W}(x)T_{i}-T_{i}\pi_{V}(x)=\sum\limits_{j=1}^{d}\pi_{\mu}(x)_{ji}T_{j},\
x\in L,$ $None$

where $V$,$W$ are (possibly infinite dimensional) $L$-modules and $\pi_{V}$
(resp. $\pi_{W}$) is the representation afforded by $V$ (resp. $W$).

Then we can define the following intertwining operator between $L$-modules
$V(\mu)\otimes V$ and $W$:

$T:V(\mu)\otimes V\rightarrow W,\ T(e_{i}\otimes v)=T_{i}(v),\
i\in\overline{1,d},\ v\in V.$ $None$

In other words, $T\in\mbox{Hom}_{L}(V(\mu)\otimes V,W)$ is an element of the
set of all operators from $V(\mu)\otimes V$ to $W$ commuting with the action
of $L$.

Remark 3.2 In Section 5, we find an irreducible tensor operator of rank
$\omega_{1}+\omega_{2}$ for simple Lie algebra $sl_{5}$, which play the center
role in our determining all the singular vectors.

In the following two Lemmas, we record some well-known facts concerning the
decomposition of tensor modules:

Lemma 3.3 (1) (cf. [H]) The $\alpha$\- string through any weight $\nu$ of
$V(\mu)$ is of length $<\nu,\alpha>$, for $\alpha\in\phi$.

(2) (cf. [EG]) Denote $\mu_{1},\cdots,\mu_{m}$ the weights occurring in
$V(\mu)$ with multiplicities $n_{1},\cdots,n_{m}$ respectively. For each
$i\in\overline{1,m}$, let $V_{i}(\mu)$ denote the space of weight vectors of
weight $\mu_{i}$. The decomposition of the tensor product module
$V(\mu)\otimes V(\lambda)$ is written:

$V(\mu)\otimes
V(\lambda)=\sum\limits_{i=1}^{m}m(\lambda+\mu_{i}:\mu\otimes\lambda)V(\lambda+\mu_{i}),\
\lambda+\mu_{i}\in\Lambda^{+},$ $None$

where the multiplicities are given by

$m(\lambda+\mu_{i}:\mu\otimes\lambda)=dimV_{i,\lambda}(\mu),$ $None$
$V_{i,\lambda}(\mu)=\\{v\in V_{i}(\mu)\ |\
e_{j}^{<\lambda+\delta,\alpha_{j}>}v=0,j\in\overline{1,l}.\\}$ $None$

(3) (cf. [EG]) Assume $\\{e_{i,j}\ |\
j\in\overline{1,m(\lambda+\mu_{i}:\mu\otimes\lambda)}\\}$ is a basis for the
space $V_{i,\lambda}(\mu)$ and $v_{\lambda}$ is the maximal weight vector of
$V(\lambda)$. A full set of independent maximal weight states of weight
$\lambda+\mu_{i}$ is given by the vectors:

$\\{P_{i}(e_{i,j}\otimes v_{\lambda}),\
j\in\overline{1,m(\lambda+\mu_{i}:\mu\otimes\lambda)}\\},$

where

$P_{i}=\prod\limits_{\mu_{i}<\sigma\leq\mu}\frac{\tilde{c_{L}}-\chi_{\sigma+\lambda}(\tilde{c_{L}})}{\chi_{\mu_{i}+\lambda}(\tilde{c_{L}})-\chi_{\sigma+\lambda}(\tilde{c_{L}})},$
$\chi_{\sigma+\lambda}(\tilde{c_{L}})=\frac{(\sigma+\lambda,\sigma+\lambda+2\delta)-(\mu,\mu+2\delta)-(\lambda,\lambda+2\delta)}{2},$
$\tilde{c_{L}}=\sum\limits_{i=1}^{l}\pi_{\mu}(h_{i})\otimes\pi_{\lambda}(h^{i})+\sum\limits_{\alpha\in\phi}\pi_{\mu}(x_{\alpha})\otimes\pi_{\lambda}(x^{\alpha}).$
$None$

(4) (cf. [MS]) The tensor product module $V(\mu)\otimes V(\lambda)$ is a
cyclic module which is cyclically generated by the vector $v^{\mu}\otimes
v_{\lambda}$, where $v^{\mu}$ is the lowest weight vector for $V(\mu)$ and
$v_{\lambda}$ is the highest weight vector for $V(\lambda)$.

For $\overrightarrow{a}=(a_{1},\cdots,a_{n-1})\in\mathbb{N}^{n-1}$ and
$0<k\in\mathbb{N}$, we denote

$\overrightarrow{a}^{*}=(a_{n-1},a_{n-2},\cdots,a_{1}),$ $None$
$I(\overrightarrow{a},k)=\\{(a_{1}+c_{1}-c_{2},a_{2}+c_{2}-c_{3},\cdots,a_{n-1}+c_{n-1}-c_{n})\
|\ c_{i}\in\mathbb{N}\ $ $\mbox{such \ that}\ \sum\limits_{i=1}^{n}c_{i}=k\
\mbox{and}\ c_{s+1}\leq a_{s}\mbox{for}\ s\in\overline{1,n-1}\\}.$ $None$

Set

$\omega_{\overrightarrow{a}}=\sum\limits_{i=1}^{n-1}a_{i}\omega_{i},\ \
\mbox{for}\ \overrightarrow{a}\in\mathbb{N}^{n-1},$ $None$

Lemma 3.4 (Pieri’s formula cf. [FH]) (1) For any
$\overrightarrow{a}\in\mathbb{N}^{n-1}$, the tensor product of $sl_{n}$-module
$V(\omega_{\overrightarrow{a}})$ with $V(k\omega_{1})$ decomposes into a
direct sum:

$V(\omega_{\overrightarrow{a}})\otimes
V(k\omega_{1})=\bigoplus_{\overrightarrow{b}\in
I(\overrightarrow{a},k)}V(\omega_{\overrightarrow{b}}).$ $None$

(2) For $sl_{n}$, we have
$V(\omega_{\overrightarrow{a}})^{*}=V(\omega_{\overrightarrow{a}^{*}})$ and

$V(\omega_{\overrightarrow{a}})\otimes
V(k\omega_{n-1})=\bigoplus_{\overrightarrow{b}\in
I(\overrightarrow{a}^{*},k)}V(\omega_{\overrightarrow{b}^{*}}).$ $None$

In the rest of this section, we will concentrate on some special wedge and
tensor modules for $sl_{5}$. Take
$\\{h_{i}=E_{i,i}-E_{i+1,i+1}(i\in\overline{1,4}),E_{ij}(1\leq i\neq j\leq
5)\\}$ as a basis for Lie algebra $sl_{5}$. Then
$\\{\frac{h_{i}^{*}}{10}(i\in\overline{1,4}),\frac{E_{ji}}{10}(1\leq i\neq
j\leq 5)\\}$ is its dual basis via the Killing form, where

$h_{1}^{*}=\frac{4}{5}h_{1}+\frac{3}{5}h_{2}+\frac{2}{5}h_{3}+\frac{1}{5}h_{4},\
h_{2}^{*}=\frac{3}{5}h_{1}+\frac{6}{5}h_{2}+\frac{4}{5}h_{3}+\frac{2}{5}h_{4},$
$h_{3}^{*}=\frac{2}{5}h_{1}+\frac{4}{5}h_{2}+\frac{6}{5}h_{3}+\frac{3}{5}h_{4},\
h_{4}^{*}=\frac{1}{5}h_{1}+\frac{2}{5}h_{2}+\frac{3}{5}h_{3}+\frac{4}{5}h_{4}.$
$None$

And the Casimir operator $c$ of the universal enveloping algebra of $sl_{5}$
is

$c=\frac{1}{10}(\sum\limits_{i=1}^{4}h_{i}h_{i}^{*}+\sum\limits_{i\neq
j\in\overline{1,5}}E_{i,j}.E_{j,i}).$ $None$

Relative to the ordered basis $\omega_{1},\omega_{2},\omega_{3},\omega_{4}$,
the coordinates of the simple roots $\alpha_{i}(i\in\overline{1,4})$ are:

$\alpha_{1}=(2,-1,0,0),\ \alpha_{2}=(-1,2,-1,0),\ \alpha_{3}=(0,-1,2,-1),\
\alpha_{4}=(0,0,-1,2).$ $None$

And the killing form for the simple root $\alpha_{i}(i\in\overline{1,4})$ are:

$(\alpha_{i},\alpha_{j})=\left\\{\begin{array}[]{lll}0,&|i-j|>1,\\\
\frac{1}{5},&i=j,\\\ -\frac{1}{10},&|i-j|=1.\end{array}\right.$ $None$

Lemma 3.5  Assume $L=sl_{5}$, $C_{L}=c$ and
$\sigma=\mu-\sum\limits_{i=1}^{4}k_{i}\alpha_{i}$ in Lemma 3.4. Then
$\chi_{\sigma+\lambda}(\tilde{c})$ in (3.6) is explicitly given by:

$\displaystyle\chi_{\sigma+\lambda}(\tilde{c})=\frac{\lambda_{1}(4\mu_{1}+3\mu_{2}+2\mu_{3}+\mu_{4})}{50}+\frac{\lambda_{2}(3\mu_{1}+6\mu_{2}+4\mu_{3}+2\mu_{4})}{50}+\frac{\lambda_{3}(2\mu_{1}+4\mu_{2}+6\mu_{3}+3\mu_{4})}{50}$
$\displaystyle+\frac{\lambda_{4}(\mu_{1}+2\mu_{2}+3\mu_{3}+4\mu_{4})}{50}+\frac{\sum\limits_{i=1}^{4}k_{i}^{2}-k_{1}k_{2}-k_{2}k_{3}-k_{3}k_{4}-\sum\limits_{i=1}^{4}k_{i}-k_{i}(\lambda_{i}+\mu_{i})}{10}.\hskip
56.9055pt(3.18)$

From (2.7) and (2.9), we know that $L_{0}\simeq sl_{5}$. And $L_{0}$-module
$L_{-1}$ is isomorphic to fundamental module $V(\omega_{2})=W$. The set of its
weights and the basis for the corresponding weight space are tabulated in
Table 1. The $L_{0}$-module $L_{1}$ is isomorphic to highest weight module
$V(\omega_{1}+\omega_{2})$ with lowest weight vector $x_{5}d_{45}$ (cf. Table
9).

Lemma 3.6  The wedge module $\Lambda^{k}W$ ($k\in\overline{1,10}$) for
$sl_{5}$ are decomposed multiplicity freely into irreducible components, which
are listed in Table 2.

Proof By Weyl’s dimension formula, we get:
$\mbox{dim}V(\omega_{1}+\omega_{3})=45$, $\mbox{dim}V(2\omega_{3})=50$,
$\mbox{dim}V(2\omega_{1}+\omega_{4})=70$, $\mbox{dim}V(3\omega_{1})=35$,
$\mbox{dim}V(\omega_{1}+\omega_{3}+\omega_{4})=175$,
$\mbox{dim}V(2\omega_{1}+\omega_{3})=126$,
$\mbox{dim}V(\omega_{2}+2\omega_{4})=126$. Since $\mbox{dim}W=10$,
$\mbox{dim}\wedge^{k}W=C_{10}^{k}$. Thus the decomposition follows through
comparing the dimensions of both sides. $\Box$

Lemma 3.7 The tensor module $V(k\omega_{4})\otimes\Lambda^{n}W$
($k\in\mathbb{N},\ n\in\overline{1,10}$) for $sl_{5}$ are decomposed into
irreducible components, which are listed in Table 3.

For any highest weight module $V(\mu)$ of simple Lie algebra $sl_{5}$, denote
the set of its weights by $\Pi(\mu)$, which are listed by
$\\{\overrightarrow{w}_{j}^{\mu}\ |\ j\in\overline{1,|\Pi(\mu)|}\\}$. Let
$\\{v_{j,k}^{\mu}\ |\
j\in\overline{1,|\Pi(\mu)|},k\in\overline{1,\mbox{mult}(\overrightarrow{w}_{j}^{\mu})}\\}$
be the Verma basis for the weight space of weight
$\overrightarrow{w}_{j}^{\mu}$, where
$\mbox{mult}(\overrightarrow{w}_{j}^{\mu})$ denotes the multiplicity of the
weight $\overrightarrow{w}_{j}^{\mu}$.

Lemma 3.8 For
$\mu\in\\{\omega_{1}+\omega_{3},2\omega_{1}+\omega_{4},3\omega_{1}\\}$, the
set $\Pi(\mu)$ of weights for $V(\mu)$ and their corresponding Verma bases for
every weight space are listed in Table 5-Table 8 in the Appendix.

Proof Assume $\mu=\sum\limits_{i=1}^{4}m_{i}\omega_{i}$. The set $\Pi(\mu)$ is
obtained by the algorithm from [W]. The Verma bases for the weight space with
weight $\mu-\sum\limits_{i=1}^{4}k_{i}\alpha_{i}$ are (cf. [LMNP], [RS]):

$(f_{1}^{a_{10}}f_{2}^{a_{9}}f_{3}^{a_{8}}f_{4}^{a_{7}})(f_{1}^{a_{6}}f_{2}^{a_{5}}f_{3}^{a_{4}})(f_{1}^{a_{3}}f_{2}^{a_{2}})f_{1}^{a_{1}}v_{\mu},$
$None$

where

$a_{10}+a_{6}+a_{3}+a_{1}=k_{1},a_{9}+a_{5}+a_{2}=k_{2},a_{8}+a_{4}=k_{3},a_{7}=k_{4},$
$0\leq a_{1}\leq m_{1},0\leq a_{2}\leq m_{2}+a_{1},0\leq
a_{3}\leq\mbox{min}(m_{2},a_{2}),$ $0\leq a_{4}\leq m_{3}+a_{2},0\leq
a_{5}\leq\mbox{min}(m_{3}+a_{3},a_{4}),0\leq
a_{6}\leq\mbox{min}(m_{3},a_{5}),$ $0\leq a_{7}\leq m_{4}+a_{4},0\leq
a_{8}\leq\mbox{min}(m_{4}+a_{5},a_{7}),0\leq
a_{9}\leq\mbox{min}(a_{4}+a_{6},a_{8}),0\leq
a_{10}\leq\mbox{min}(m_{4},a_{9}).$ $None$

$\Box$

Remark 3.9 The coordinates of the weights appearing in Table1, Table5-Table9
are with respect to the ordered basis
$\omega_{1},\omega_{2},\omega_{3},\omega_{4}$. The basis of every weight space
appearing in these tables are Verma basis.

##  4 Singular vectors for GVM of $\mbox{E}(5,10)$

In Section 4.1, we analyze the detailed structure of the generalized Verma
module $M$ over $\mbox{E}(5,10)$. It turns out that there is a grading on $M$
and each graded subspace is a finite dimensional $sl_{5}$\- module (cf.
Equation (4.6) and (4.7)). Moreover, any singular vector for $M$ is controlled
by its leading term through an exponential-like differential operator (cf.
Equation (4.27) ). In section 4.2, we inductively prove that any leading term
must satisfy three equations, i.e. (4.28), (4.35) and (4.37). Based on the
Lemmas in Section 3, we simplify these three differential equations and prove
that any singular vector is of degree less than or equal to four. Also, the
leading term of any singular vector must lie in one of the tensor
decomposition of four tensor product module for $sl_{5}$ (cf. Theorem 4.6 ).

###  4.1 Gradation for GVM

Set

$T=\\{0,1\\},\ \
T^{\prime}=\\{(45),(35),(25),(15),(34),(24),(14),(23),(13),(12)\\}.$ $None$

Define order $``\prec^{\prime\prime}$ on the set $T^{\prime}$ by:

$(45)\prec(35)\prec(25)\prec(15)\prec(34)\prec(24)\prec(14)\prec(23)\prec(13)\prec(12).$
$None$

For
$\underline{n}=(n_{12},n_{13},n_{14},n_{24},n_{34},n_{15},n_{25},n_{35},n_{45})\in
T^{10}$ and $\underline{m}\in\mathbb{N}^{5}$, we take the following notations:

$\underline{n}\pm\varepsilon_{ij}=(n_{12},\cdots,n_{ij}\pm 1,\cdots,n_{45}),\
\underline{m}\pm\varepsilon_{i}=(m_{1},\cdots,m_{i}\pm 1,\cdots,m_{5}).$
$None$

Let

$d^{\underline{n}}=d_{12}^{n_{12}}d_{13}^{n_{13}}d_{23}^{n_{23}}d_{14}^{n_{14}}d_{24}^{n_{24}}d_{34}^{n_{34}}d_{15}^{n_{15}}d_{25}^{n_{25}}d_{35}^{n_{35}}d_{45}^{n_{45}},$
$None$
$\partial^{\underline{m}}=\partial_{1}^{m_{1}}\partial_{2}^{m_{2}}\partial_{3}^{m_{3}}\partial_{4}^{m_{4}}\partial_{5}^{m_{5}}.$
$None$

Then the induced module $M$ is spanned by
$\\{\partial^{\underline{m}}d^{\underline{n}}v_{\nu}\ |\ \underline{n}\in
T^{10},\underline{m}\in\mathbb{N}^{5},\nu\in\Pi(\lambda)\\}$. Define

$\partial^{m}\wedge^{n}V=\mbox{Span}\\{\partial^{\underline{m}}d^{\underline{n}}v\
|\ |\underline{m}|=m,|\underline{n}|=n\\},\
M_{k}=\mbox{Span}\\{\partial^{\underline{m}}d^{\underline{n}}v\ |\ 2m+n=k\\}.$
$None$

Then

$M=\bigoplus\limits_{k\in\mathbb{N}}M_{k}.$ $None$

Definition 4.1 We say any nonzero vector of $M_{k}$ is of degree $k$.

The equations

$[x_{i}\partial_{x_{j}},d_{kl}]=\delta_{j,k}d_{il}-\delta_{jl}d_{ik},[x_{5}d_{45},d_{12}]=x_{5}\partial_{x_{3}},\
[x_{5}d_{45},d_{13}]=-x_{5}\partial_{x_{2}},\
[x_{5}d_{45},d_{23}]=x_{5}\partial_{x_{1}},$
$[x_{5}d_{45},d_{i4}]=0(i\in\overline{1,3}),\
[x_{5}d_{45},d_{i5}]=0(i\in\overline{1,4}).$ $None$

yield

$L_{0}.\partial^{m}\wedge^{n}V\subseteq\partial^{m}\wedge^{n}V+\partial^{m+1}\wedge^{n-2}V,\
x_{5}d_{45}.M_{k}\subseteq M_{k-1}.$ $None$

That is to say, every graded vector subspace $M_{k}$ is an $sl_{5}$-module and
every singular vector for $\mbox{E}(5,10)$-module $M$ is in a certain graded
subspace $M_{k}$.

In the following of this section, we consider the maximal vectors for
$sl_{5}$-module $M_{k}$. On any linear vector space $\partial^{m}\wedge^{n}V$,
we define the following linear operators:

$(-1)^{|ij|}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto(-1)^{\sum\limits_{(kl)\prec(ij)}n_{kl}}\partial^{\underline{m}}d^{\underline{n}}v,$
$(-1)^{|ij,kl|}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto(-1)^{\sum\limits_{(kl)\prec(pq)\prec(ij)}n_{pq}}\partial^{\underline{m}}d^{\underline{n}}v,$
$y_{ij}\partial_{y_{kl}}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto
n_{kl}\partial^{\underline{m}}d^{\underline{n}+\varepsilon_{ij}-\varepsilon_{kl}}v,$
$z_{i}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m+1}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto\partial^{\underline{m}+\varepsilon_{i}}d^{\underline{n}}v,$
$\partial_{z_{i}}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m-1}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto
m_{i}\partial^{\underline{m}-\varepsilon_{i}}d^{\underline{n}}v,$
$E_{i,j}:\partial^{m}\wedge^{n}V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}d^{\underline{n}}v\mapsto\partial^{\underline{m}}d^{\underline{n}}(E_{i,j}.v).$
$None$

Set

$(x_{i}\partial_{x_{j}})_{0}^{{}^{\prime}}=\sum\limits_{k\in\overline{1,5},k\neq
i,j}(-1)^{|ki,kj|}y_{ki}\partial_{y_{kj}},\
(x_{i}\partial_{x_{j}})_{0}=-z_{j}\partial_{z_{i}}+(x_{i}\partial_{x_{j}})_{0}^{{}^{\prime}}+E_{i,j}(i\neq
j);$ $None$
$(x_{3}\partial_{x_{4}})_{-2}=z_{5}\partial_{y_{14}}\partial_{y_{24}},$ $None$
$(x_{4}\partial_{x_{5}})_{-2}=z_{1}\partial_{y_{25}}\partial_{y_{35}}+(-1)^{1+|15,35|}z_{2}\partial_{y_{15}}\partial_{y_{35}}+z_{3}\partial_{y_{15}}\partial_{y_{25}}.$
$None$

Using these settings, we could formulate the equation (2.12) in the following
explicit form:

$x_{1}\partial_{x_{2}}=(x_{1}\partial_{x_{2}})_{0},\quad
x_{2}\partial_{x_{3}}=(x_{2}\partial_{x_{3}})_{0},$
$x_{3}\partial_{x_{4}}=(x_{3}\partial_{x_{4}})_{0}+(x_{3}\partial_{x_{4}})_{-2},\
x_{4}\partial_{x_{5}}=(x_{4}\partial_{x_{5}})_{0}+(x_{4}\partial_{x_{5}})_{-2}.$
$None$

According to the Cartan subalgebra of $L_{0}$, $M$ can be decomposed to the
following direct sum of subspaces:

$M=\bigoplus\limits_{\mu\in\Gamma}M^{\mu},\
M^{\mu}=\mbox{Span}\\{\partial^{\underline{m}}d^{\underline{n}}v_{\nu}\ |\
(x_{i}\partial_{x_{i}}-x_{i+1}\partial_{x_{i+1}}).\partial^{\underline{m}}d^{\underline{n}}v_{\nu}=\mu_{i}\partial^{\underline{m}}d^{\underline{n}}v_{\nu}\\},$
$None$

where

$\mu_{i}=m_{i+1}-m_{i}+t_{i}(\underline{n})+\nu_{i},$
$t_{1}(\underline{n})=n_{13}+n_{14}+n_{15}-n_{23}-n_{24}-n_{25},\
t_{2}(\underline{n})=n_{12}+n_{24}+n_{25}-n_{13}-n_{34}-n_{35},$
$t_{3}(\underline{n})=n_{13}+n_{23}+n_{35}-n_{14}-n_{24}-n_{45},\
t_{4}(\underline{n})=n_{14}+n_{24}+n_{34}-n_{15}-n_{25}-n_{35}.$ $None$

For any vectors $v\in M^{\mu}$, we say that it is of weight $\mu$ and denote
$\mbox{wt}(v)=\mu,\ |\mbox{wt}(v)|=|\mu|=\sum\limits_{i=1}^{4}\mu_{i}.$

Proposition 4.2 The differential operators $(x_{i}\partial_{x_{j}})_{0}(1\leq
i\neq j\leq 5)$ and
$x_{i}\partial_{x_{i}}-x_{i+1}\partial_{x_{i+1}}(i\in\overline{1,4})$ give
every vector space $\partial^{m}\wedge^{n}V$ an $sl_{5}$-module structure,
which is isomorphic to tensor module $V(m\omega_{4})\otimes\wedge^{n}W\otimes
V$ for $sl_{5}$.

Proof The module isomorphism is given by:

$\phi:V(m\omega_{4})\otimes\wedge^{n}W\otimes
V\rightarrow\partial^{m}\wedge^{n}V;\partial^{\underline{m}}\otimes(d_{i_{1}j_{1}}\wedge\cdots\wedge
d_{i_{n}j_{n}})\otimes v\mapsto\partial^{\underline{m}}d_{i_{1}j_{1}}\cdots
d_{i_{n}j_{n}}v.$ $None$

$\Box$

Denote

$\Gamma_{k}=\\{(m,n)\in\mathbb{N}^{2}\ |\ 2m+n=k\\}$ $None$

For any $(m,n)\in\Gamma_{k}$, let

$\Gamma_{k}^{(m,n)}=\\{(m^{\prime},n^{\prime})\in\Gamma_{k}\ |\ m^{\prime}\geq
m\\}$ $None$

Assume $\xi\in M_{k}$ is any $E(5,10)$ singular vector. Then there exists
$(m,n)\in\Gamma_{k}$ such that

$\xi\in\bigoplus\limits_{(m^{\prime}n^{\prime})\in\Gamma_{k}^{(m,n)}}\partial^{m^{\prime}}\wedge^{n^{\prime}}V$
$None$

For emphasis, we write

$\xi=\xi^{m,n}=\sum\limits_{(m^{\prime}n^{\prime})\in\Gamma_{k}^{(m,n)}}\xi_{m^{\prime},n^{\prime}}$
$None$

We say that $\xi_{m,n}$ is the leading term  of $\xi^{m,n}$. It follows from
(4.14) that $\xi^{m,n}$ must satisfy the following equations inductively:

$(x_{i}\partial_{x_{i+1}})_{0}.\xi_{m,n}=0(i\in\overline{1,4}),$
$(x_{i}\partial_{x_{i+1}})_{-2}.\xi_{m^{\prime},n^{\prime}}+(x_{i}\partial_{x_{i+1}})_{0}.\xi_{m^{\prime}+1,n^{\prime}-2}=0,\
i\in\overline{1,4},\ (m^{\prime},n^{\prime})\in\Gamma_{k}^{(m,n)}.$ $None$

Remark 4.3 From Proposition 4.2 and (4.22), we derive that the leading term
$\xi_{m,n}$ of any singular vector $\xi=\xi^{m,n}$ is also a singular vector
of the tensor product module
$V(m\omega_{4})\otimes\wedge^{n}V(\omega_{2})\otimes V(\lambda)$ for simple
Lie algebra $sl_{5}$. In the following, we will point out that any singular
vector $\xi=\xi^{m,n}$ is completely controlled by its leading term
$\xi_{m,n}$ through certain exponential-like differential operator.

Set

$P=\sum\limits_{(kl)\prec(ij)\in
T^{\prime},m\in\overline{1,5}}\varepsilon_{mijkl}(-1)^{|ij,kl|}z_{m}\partial_{y_{ij}}\partial_{y_{kl}},$
$None$

where $\varepsilon_{mijkl}$ is defined in (2.6). The operator $P$ is checked
to satisfy the following equations:

$[(x_{3}\partial_{x_{4}})_{-2},P]=0,[(x_{4}\partial_{x_{5}})_{-2},P]=0,\
[(x_{1}\partial_{x_{2}})_{0},P]=0,[(x_{2}\partial_{x_{3}})_{0},P]=0,$
$[(x_{3}\partial_{x_{4}})_{0},P]=2(x_{3}\partial_{x_{4}})_{-2},\
[(x_{4}\partial_{x_{5}})_{0},P]=2(x_{4}\partial_{x_{5}})_{-2}.$ $None$

Inductively,

$[(x_{i}\partial_{x_{i+1}})_{0},P^{k}]=[x_{i}\partial_{x_{i+1}},P^{k}]=2kP^{k-1}(x_{i}\partial_{x_{i+1}})_{-2},k\in\mathbb{N}.$
$None$

It implies

$x_{i}\partial_{x_{i+1}}.e^{-\frac{1}{2}P}\xi_{m,n}=0,\quad(x_{i}\partial_{x_{i+1}})_{0}.e^{\frac{1}{2}P}\xi^{m,n}=0.$
$None$

Thus we prove the following formula:

Proposition 4.3 Assume
$\xi^{m,n}=\sum\limits_{(m^{\prime}n^{\prime})\in\Gamma_{k}^{(m,n)}}\xi_{m^{\prime},n^{\prime}}\in
M_{k}$ is any singular vector for $E(5,10)$-module $M$, then

$\xi^{m,n}=e^{-\frac{1}{2}P}\xi_{m,n}.$ $None$

###  4.2 Singular vectors for GVM

In this section, we continue the discussion concerning the equation (2.13) in
Definition 2.1. Recall the notations in (4.10), set

$(x_{5}d_{45})_{1}=(-1)^{1+|45|}\partial_{z_{5}}y_{45},$ $None$
$\displaystyle(x_{5}d_{45})_{-1}$ $\displaystyle=$ $\displaystyle-
z_{3}\partial_{z_{5}}\partial_{y_{12}}+(-1)^{|13|}z_{2}\partial_{z_{5}}\partial_{y_{13}}+(-1)^{1+|23|}z_{1}\partial_{z_{5}}\partial_{y_{23}}$
$\displaystyle+$
$\displaystyle\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}$
$\displaystyle+$
$\displaystyle(-1)^{|13,15|}y_{15}\partial_{y_{12}}\partial_{y_{13}}+(-1)^{|23,25|}y_{25}\partial_{y_{12}}\partial_{y_{23}}+(-1)^{|13|+|23,35|}y_{35}\partial_{y_{13}}\partial_{y_{23}}$
$\displaystyle+$
$\displaystyle(-1)^{1+|34,45|}y_{45}\partial_{y_{12}}\partial_{y_{34}}+(-1)^{1+|23|+|14,45|}y_{45}\partial_{y_{23}}\partial_{y_{14}}+(-1)^{|13|+|24,45|}y_{45}\partial_{y_{13}}\partial_{y_{24}},$
$None$ $\displaystyle(x_{5}d_{45})_{-3}$ $\displaystyle=$
$\displaystyle(-1)^{|23,34|}z_{1}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{34}}+(-1)^{1+|23,24|+|13|}z_{1}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{24}}$
$\displaystyle+$
$\displaystyle(-1)^{|13|}z_{2}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{14}}+(-1)^{1+|13,34|}z_{2}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{34}}$
$\displaystyle+$
$\displaystyle(-1)^{|13,24|}z_{3}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{24}}-z_{3}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{14}}-z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}}.\hskip
108.12054pt(4.30)$

It follows from the equation (4.8) that

$x_{5}d_{45}.\partial^{m}\wedge^{n}V\subseteq\partial^{m-1}\wedge^{n+1}V+\partial^{m}\wedge^{n-1}V+\partial^{m+1}\wedge^{n-3}V,\
x_{5}d_{45}.M_{k}\subseteq M_{k-1},$
$x_{5}d_{45}=(x_{5}d_{45})_{1}+(x_{5}d_{45})_{-1}+(x_{5}d_{45})_{-3}.$ $None$

Furthermore, $\xi^{m,n}$ must satisfy the following equations inductively:

$(x_{5}d_{45})_{1}.\xi_{m,n}=0,\
(x_{5}d_{45})_{-1}.\xi_{m,n}+(x_{5}d_{45})_{1}.\xi_{m+1,n-2}=0,$
$(x_{5}d_{45})_{-3}.\xi_{m,n}+(x_{5}d_{45})_{-1}.\xi_{m+1,n-2}+(x_{5}d_{45})_{1}.\xi_{m+2,n-4}=0,$
$(x_{5}d_{45})_{-3}.\xi_{m^{\prime},n^{\prime}}+(x_{5}d_{45})_{-1}.\xi_{m^{\prime}+1,n^{\prime}-2}+(x_{5}d_{45})_{1}.\xi_{m^{\prime}+2,n^{\prime}-4}=0,\
\mbox{ for \ any}\ (m^{\prime},n^{\prime})\in\Gamma_{k}^{(m,n)}.$ $None$

Applying (4.27), the leading term $\xi_{m,n}$ should be killed by the
following three operators:

$(x_{5}d_{45})_{1},\ (x_{5}d_{45})_{-1}+(x_{5}d_{45})_{1}(-\frac{1}{2}P),\
(x_{5}d_{45})_{-3}+(x_{5}d_{45})_{-1}(-\frac{1}{2}P)+(x_{5}d_{45})_{1}(\frac{1}{8}P^{2}).$
$None$

We can reduce the last two differential operators to be of more explicit
forms. Indeed, the following relations are easily checked:

$\displaystyle[(x_{5}d_{45})_{1},P]$ $\displaystyle=$ $\displaystyle-
z_{3}\partial_{z_{5}}\partial_{y_{12}}+(-1)^{|13|}z_{2}\partial_{z_{5}}\partial_{y_{13}}+(-1)^{1+|23|}z_{1}\partial_{z_{5}}\partial_{y_{23}}$
$\displaystyle+$
$\displaystyle(-1)^{1+|34,45|}y_{45}\partial_{y_{12}}\partial_{y_{34}}+(-1)^{1+|23|+|14,45|}y_{45}\partial_{y_{23}}\partial_{y_{14}}+(-1)^{|13|+|24,45|}y_{45}\partial_{y_{13}}\partial_{y_{24}},$
$[[(x_{5}d_{45})_{1},P],P]=2(x_{5}d_{45})_{-3}+2z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}},\
[(x_{5}d_{45})_{-1},P]=3(x_{5}d_{45})_{-3}.$ $None$

Therefore,

$\displaystyle[(x_{5}d_{45})_{-1}+(x_{5}d_{45})_{1}(-\frac{1}{2}P)]\xi_{m,n}$
$\displaystyle=\\{(x_{5}d_{45})_{-1}-\frac{1}{2}P(x_{5}d_{45})_{1}-\frac{1}{2}[(x_{5}d_{45})_{1},P]\\}\xi_{m,n}$
$\displaystyle\stackrel{{\scriptstyle\mbox{by}(4.34)}}{{=}}[-\frac{1}{2}z_{3}\partial_{z_{5}}\partial_{y_{12}}+\frac{1}{2}(-1)^{|13|}z_{2}\partial_{z_{5}}\partial_{y_{13}}+(-1)^{1+|23|}\frac{1}{2}z_{1}\partial_{z_{5}}\partial_{y_{23}}$
$\displaystyle+\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}$
$\displaystyle+(-1)^{|13,15|}y_{15}\partial_{y_{12}}\partial_{y_{13}}+(-1)^{|23,25|}y_{25}\partial_{y_{12}}\partial_{y_{23}}$
$\displaystyle+(-1)^{|13|+|23,35|}y_{35}\partial_{y_{13}}\partial_{y_{23}}+\frac{1}{2}(-1)^{1+|34,45|}y_{45}\partial_{y_{12}}\partial_{y_{34}}$
$\displaystyle+\frac{1}{2}(-1)^{1+|23|+|14,45|}y_{45}\partial_{y_{23}}\partial_{y_{14}}+\frac{1}{2}(-1)^{|13|+|24,45|}y_{45}\partial_{y_{13}}\partial_{y_{24}}]\xi_{m,n}.$
$\displaystyle\stackrel{{\scriptstyle\mbox{by}(4.11)}}{{=}}[(E_{53}+\frac{1}{2}(x_{5}\partial_{x_{3}})_{0}^{\prime}-\frac{1}{2}z_{3}\partial_{z_{5}})\partial_{y_{12}}+(E_{52}+\frac{1}{2}(x_{5}\partial_{x_{2}})_{0}^{\prime}-\frac{1}{2}z_{2}\partial_{z_{5}})(-1)^{1+|13|}\partial_{y_{13}}$
$\displaystyle+(E_{51}+\frac{1}{2}(x_{5}\partial_{x_{1}})_{0}^{\prime}-\frac{1}{2}z_{1}\partial_{z_{5}})(-1)^{|23|}\partial_{y_{23}}]\xi_{m,n}\hskip
204.85974pt(4.35)$

Hence,

$P(x_{5}d_{45})_{-1}\xi_{m,n}=\frac{1}{2}P(x_{5}d_{45})_{1}P\xi_{m,n}=\frac{1}{2}P[(x_{5}d_{45})_{1},P]\xi_{m,n}.$
$None$

Furthermore, (4.34) and (4.35) imply that

$\displaystyle[(x_{5}d_{45})_{-3}+(x_{5}d_{45})_{-1}(-\frac{1}{2}P)+(x_{5}d_{45})_{1}(\frac{1}{8}P^{2})]\xi_{m,n}$
$\displaystyle=\\{(x_{5}d_{45})_{-3}-\frac{3}{2}(x_{5}d_{45})_{-3}-\frac{1}{2}P(x_{5}d_{45})_{-1}$
$\displaystyle+\frac{1}{8}[[(x_{5}d_{45})_{1},P],P]+\frac{1}{8}P[(x_{5}d_{45})_{1},P]+\frac{1}{8}P(x_{5}d_{45})_{1}P\\}\xi_{m,n}$
$\displaystyle\stackrel{{\scriptstyle\mbox{by}(4.34),(4.35),(4.36)}}{{=}}[-\frac{1}{4}(x_{5}d_{45})_{-3}+\frac{1}{4}z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}}]\xi_{m,n}$
$\displaystyle=\frac{1}{4}[z_{1}(-1)^{|13|+|23,24|}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{24}}-z_{1}(-1)^{|23,34|}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{34}}$
$\displaystyle+z_{2}(-1)^{|13,34|}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{34}}-z_{2}(-1)^{|13|}\partial_{y_{13}}\partial_{y_{23}}\partial_{y_{14}}$
$\displaystyle+z_{3}\partial_{y_{12}}\partial_{y_{23}}\partial_{y_{14}}-z_{3}(-1)^{|13,24|}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{24}}+2z_{4}\partial_{y_{12}}\partial_{y_{13}}\partial_{y_{23}}]\xi_{m,n}.\hskip
119.50148pt(4.37)$

Denote the set of all the highest weight vectors for tensor modules
$V(m\omega_{4})\bigotimes\wedge^{n}W$ by

$S_{m,n}=\\{e_{m,n}^{1},\cdots,e_{m,n}^{\nu(m,n)}\\}$ $None$

By Lemma 3.3, any singular vector $\xi_{m,n}$ of the tensor product module
$V(m\omega_{4})\bigotimes\bigwedge^{n}V(\omega_{2})\bigotimes V(\lambda)$ for
$sl_{5}$ can be written by the following form:

$\xi_{m,n}=e_{m,n}^{i}\otimes v_{\vartheta}+\cdots.$ $None$

We consider the set

$S_{m,n}^{\prime}=\\{e_{m,n}^{i}\in S_{m,n}\ |\
(-1)^{1+|45|}\partial_{z_{5}}y_{45}.\phi(e_{m,n}^{i}\otimes v)=0,$
$[(x_{5}d_{45})_{-3}+(x_{5}d_{45})_{-1}(-\frac{1}{2}P)+(x_{5}d_{45})_{1}(\frac{1}{8}P^{2})].\phi(e_{m,n}^{i}\otimes
v)=0,\ \forall\ v\in V(\lambda)\\}.$ $None$

Proposition 4.4  All the non empty set of $S_{m,n}^{\prime}$ are listed in the
following:

$S_{0,0}^{\prime}=\\{1\\},S_{0,1}^{\prime}=\\{d_{12}\\},S_{0,2}^{\prime}=\\{d_{12}\wedge
d_{13}\\},S_{0,3}^{\prime}=\\{d_{12}\wedge d_{13}\wedge
d_{14}\\},S_{0,4}^{\prime}=\\{d_{12}\wedge d_{13}\wedge d_{14}\wedge
d_{15}\\}.$

Proof Let $V(\mu)$ be any highest weight module appearing in the decomposition
of the $sl_{5}$ wedge module $\wedge^{n}V(\omega_{2})$ (cf. Table 2). And the
highest weights appearing in the decomposition of $V(m\omega_{4})\otimes
V(\mu)$ are listed in Table 3. By Lemma 3.3, the maximal vector in the tensor
module $V(m\omega_{4})\otimes V(\mu)$ is written as:

$\partial_{5}^{m}\otimes
l_{\mu}+\sum\limits_{\underline{q}\in\mathbb{N}^{5}}\partial^{\underline{q}}\otimes
v_{\underline{q}},$ $None$

where $l_{\mu}$ satisfies
$E_{12}l_{\mu}=0,E_{23}l_{\mu}=0,E_{34}l_{\mu}=0,E_{45}^{m+1}l_{\mu}=0$. By
detailed calculation, we get all the $l_{\mu}$, which are listed in Table-4. A
straightforward but messy check case by case shows that the assertion holds.
$\Box$

Remark 4.5 For the 10-tuple $d_{12}\wedge d_{13}\wedge\cdots\wedge d_{45}$, we
use the notation
${\hat{d}_{i_{1}j_{1}}}\wedge{\hat{d}_{i_{2}j_{2}}}\wedge\cdots\wedge{\hat{d}_{i_{k}j_{k}}}$
to denote the (10-k)-tuple where $d_{i_{1}j_{1}},\cdots,d_{i_{k}j_{k}}$ have
been omitted in Table 4.

To summarize Proposition 4.3 and Proposition 4.4, we have proved the following
statement in this section:

Theorem 4.6 Any singular vector for $E(5,10)$-module $M$ is of the form:

$\xi^{0,n}=e^{-\frac{1}{2}P}\xi_{0,n},\ \ n\in\overline{1,4}$

where the leading term $\xi_{0,n}$ satisfies the equation:

$[(E_{53}+\frac{1}{2}(x_{5}\partial_{x_{3}})_{0}^{\prime})\partial_{y_{12}}+(E_{52}+\frac{1}{2}(x_{5}\partial_{x_{2}})_{0}^{\prime})(-1)^{1+|13|}\partial_{y_{13}}+(E_{51}+\frac{1}{2}(x_{5}\partial_{x_{1}})_{0}^{\prime})(-1)^{|23|}\partial_{y_{23}}].\xi_{0,n}=0.$
$None$

Moreover, $\xi_{0,n}$ is the maximal vector lying in one of the following
$sl_{5}$-tensor modules:

$\xi_{0,1}\in V(\omega_{2})\otimes V(\lambda),\xi_{0,2}\in
V(\omega_{1}+\omega_{3})\otimes V(\lambda),\xi_{0,3}\in
V(2\omega_{1}+\omega_{4})\otimes V(\lambda),\xi_{0,4}\in V(3\omega_{1})\otimes
V(\lambda).$ $None$

##  5 Singular vectors degree by degree

In this section, we work out all the singular vectors in Theorem 4.6
explicitly degree by degree. Before turning to the calculation, we introduce
some formula which we are going to use in the remainder of this section.

Recall that we could endow any vector space $\Lambda^{m}V$ an $sl_{5}$\-
module structure with the action
$(x_{i}\partial_{x_{i}})_{0}^{{}^{\prime}}-(x_{i+1}\partial_{x_{i+1}})_{0}^{{}^{\prime}}\
(i\in\overline{1,4}),\ (x_{i}\partial_{x_{j}})_{0}^{\prime}\ (i\neq j)$, which
is isomorphic to the tensor product module $\wedge^{m}V(\omega_{2})\otimes
V(\lambda)$ in Section 4.1. Now we define the following differential operator
on the $sl_{5}$\- module $\Lambda^{m}V$:

$\tilde{c}=\frac{1}{10}[\sum\limits_{i=1}^{4}((x_{i}\partial_{x_{i}})_{0}^{{}^{\prime}}-(x_{i+1}\partial_{x_{i+1}})_{0}^{{}^{\prime}})h_{i}^{*}+\sum\limits_{1\leq
i\neq j\leq 5}(x_{i}\partial_{x_{j}})_{0}^{{}^{\prime}}E_{ji}],$ $None$
$T_{i,jkl}=[E_{ij}+\frac{1}{2}(x_{i}\partial_{x_{j}})_{0}^{\prime}](-1)^{|kl|}\partial_{y_{kl}}+[E_{ik}+\frac{1}{2}(x_{i}\partial_{x_{k}})_{0}^{\prime}](-1)^{1+|jl|}\partial_{y_{jl}}+[E_{il}+\frac{1}{2}(x_{i}\partial_{x_{l}})_{0}^{\prime}](-1)^{|jk|}\partial_{y_{jk}}.$
$None$

Lemma 5.1 Assume
$Q_{ij}^{0}\in\mbox{Span}_{\mathbb{F}}\\{y_{ij}\partial_{y_{kl}}\ |\ 1\leq
i<j\leq 5,\ 1\leq k<l\leq 5\\}$. Then

$\sum\limits_{1\leq i<j\leq
5}Q_{ij}^{0}.(-1)^{|ij|}\partial_{y_{ij}}.\tilde{c}|_{\Lambda^{m}V}=\sum\limits_{1\leq
i<j\leq 5}Q_{ij}^{1}(-1)^{|ij|}\partial_{y_{ij}}|_{\wedge^{m}V},$

where

$Q_{ij}^{1}=Q_{ij}^{0}(\tilde{c}+\sum\limits_{k=1}^{4}\frac{s_{k}^{ij}}{10}h_{k}^{*})+\frac{1}{10}\sum\limits_{m\neq
i,j}(Q_{im}^{0}E_{jm}-Q_{jm}^{0}E_{im}),$
$s_{k}^{ij}\stackrel{{\scriptstyle\mbox{by}(4.16)}}{{=}}t_{k}(\underline{n})-t_{k}(\underline{n}-\epsilon_{ij})$
$None$

for any $k\in\overline{1,4},\underline{n}\in T^{10},(ij)\in S^{\prime}$.

Proof Indeed, the formula (5.3) follows from:

$[(-1)^{|ij|}\partial_{y_{ij}},\tilde{c}]|_{\wedge^{m}V}=\frac{1}{10}(\sum\limits_{k\neq
i,j}(-1)^{|ik|}\partial_{y_{ik}}E_{kj}-\sum\limits_{k\neq
i,j}(-1)^{|jk|}\partial_{y_{jk}}E_{ki}+\sum\limits_{k=1}^{4}s_{k}^{ij}(-1)^{|ij|}\partial_{y_{ij}}h_{k}^{*})|_{\wedge^{m}V}.$
$None$

$\Box$

Lemma 5.2 We could define the following intertwining operators between the
$sl_{5}$-module $V(\omega_{1}+\omega_{2})\otimes\Lambda^{m}V$ and
$\Lambda^{m-1}V$ by:

$T^{m}:V(\omega_{1}+\omega_{2})\otimes\Lambda^{m}V\rightarrow\Lambda^{m-1}V;v_{30,1}^{\omega_{1}+\omega_{2}}\otimes\xi\mapsto
T_{5,123}(\xi),$ $None$

where $v_{30,1}^{\omega_{1}+\omega_{2}}$ is the lowest weight vector for
$V(\omega_{1}+\omega_{2})$ (cf. Table 9) and $\xi$ is any maximal vector in
$sl_{5}$-module $\Lambda^{m}V$.

Proof Since the $sl_{5}$-module $V(\omega_{1}+\omega_{2})\otimes\Lambda^{m}V$
is generated by such vectors of $v_{30,1}^{\omega_{1}+\omega_{2}}\otimes\xi$
by part (4) of Lemma 3.3, the assertion follows from the following formula:

$[(x_{i+1}\partial_{x_{i}})_{0}^{\prime},T_{5,123}]|_{\wedge^{m}V}=0,\
i\in\overline{1,4},$
$[(x_{s}\partial_{x_{t}})_{0}^{\prime},T_{i,jkl}]|_{\wedge^{m}V}=\delta_{t,i}T_{s,jkl}-\delta_{s,j}T_{i,tkl}-\delta_{s,k}T_{i,jtl}-\delta_{s,l}T_{i,jkt}.$
$None$

$\Box$

###  5.1 Singular vectors of degree one

Theorem 5.3 All the possible degree one singular vectors are listed in the
following:

$d_{12}v_{\lambda},\ \mbox{where}\ \lambda=(m,n,0,0),\ m,n\in\mathbb{N};$
$\prod\limits_{\mbox{wt}(d_{15})<\sigma\leq\omega_{2}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{15})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{15}v_{\lambda},\
\mbox{where}\ \lambda=(m,0,0,n),\ m\in\mathbb{N},\ 1\leq n\in\mathbb{N};$
$\prod\limits_{\mbox{wt}(d_{45})<\sigma\leq\omega_{2}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{45})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{45}v_{\lambda},\
\mbox{where}\ \lambda=(0,0,m,n),\ 1\leq m\in\mathbb{N},\ n\in\mathbb{N}.$

Proof The leading term of any singular vector of degree one can be written as

$\xi_{0,1}=\sum\limits_{1\leq i<j\leq 5}d_{ij}v_{ij},\ v_{ij}\in V(\lambda),$
$None$

which should satisfy :

$T_{5,123}.\xi_{0,1}=[\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}].\xi_{0,1}=0,$
$None$

i.e.

$E_{53}v_{12}-E_{52}v_{13}+E_{51}v_{23}=0.$ $None$

Note that $(x_{i}\partial_{x_{j}})_{0}.\xi_{0,1}=0$ $(1\leq i<j\leq 5)$ imply
that

$v_{13}=-E_{23}v_{12},\ v_{23}=E_{13}v_{12}=-E_{12}v_{13},\
v_{14}=-E_{34}v_{13},\ v_{15}=-E_{25}v_{12}=-E_{35}v_{13},\ $
$v_{25}=E_{15}v_{12}=-E_{35}v_{23},\
v_{35}=E_{15}v_{13}=-E_{23}v_{25}=E_{25}v_{23},\ v_{45}=-E_{34}v_{35}.$ $None$

Obviously, $v_{12}\neq 0$.

Case 1. wt($\xi_{0,1})=$ wt$(d_{12}v_{\lambda})$.

In this case, $v_{13}=v_{23}=0$, $v_{12}=v_{\lambda}$. And (5.9) implies that
$E_{5,3}.v_{12}=E_{5,3}.v_{\lambda}=0$. That is to say,
$\lambda=(m,n,0,0),(m,n)\in\mathbb{N}^{2}$.

Case 2.
wt($\xi_{0,1})\in\\{\mbox{wt}(d_{13}v_{\lambda}),\mbox{wt}(d_{14}v_{\lambda}),\mbox{wt}(d_{15}v_{\lambda})\\}$.

In these three cases, we have $v_{23}=0,v_{13}\neq 0$.

Case 2.1
$wt(\xi_{0,1})\in\\{\mbox{wt}(d_{13}v_{\lambda}),\mbox{wt}(d_{14}v_{\lambda})\\}$

In these two cases, $v_{15}=0,v_{13}\neq 0$,
$(h_{2}+h_{3}+h_{4}).v_{13}=(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}.$
Hence,

$\displaystyle
0=E_{25}(E_{53}.v_{12}-E_{52}.v_{13})=(E_{23}+E_{53}E_{25})v_{12}-(h_{2}+h_{3}+h_{4}+E_{52}E_{25})v_{13}$
$\displaystyle=$
$\displaystyle-(1+h_{2}+h_{3}+h_{4})v_{13}-E_{53}v_{15}=-(1+h_{2}+h_{3}+h_{4})v_{13}=-(1+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}\hskip
28.45274pt(5.11)$

provides a contradiction.

Case 2.2 $wt(\xi_{0,1})=\mbox{wt}(d_{15}v_{\lambda})$

In this case, $v_{15}=v_{\lambda}$ and
$\mbox{wt}(v_{13})=\lambda-\alpha_{3}-\alpha_{4}$. And

$0=E_{25}(E_{53}.v_{12}-E_{52}.v_{13})=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}-E_{53}v_{15},$
$0=E_{35}E_{25}(E_{53}.v_{12}-E_{52}.v_{13})=-E_{35}(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13}-E_{35}E_{53}v_{15}=(\lambda_{2}+\lambda_{3}+\lambda_{4}-h_{3}-h_{4})v_{15}=\lambda_{2}v_{15}$
$None$

forces $\lambda_{2}=0$.

Case 2.2.1. $\lambda_{3}=0,\ \lambda_{4}>0$.

Suppose

$T_{5,123}.\tilde{c}^{k}=\sum\limits_{1\leq i<j\leq
5}Q_{ij}^{k-1}(-1)^{|ij|}\partial_{y_{ij}}.$ $None$

By (5.3),
$Q_{15}^{1}v_{\lambda}=(Q_{12}^{0}E_{52}+Q_{13}^{0}E_{53})v_{\lambda}=0$. Then

$T_{5,123}.\xi_{0,1}=[\partial_{y_{12}}E_{53}+(-1)^{1+|13|}\partial_{y_{13}}E_{52}+(-1)^{|23|}\partial_{y_{23}}E_{51}].\tilde{c}.d_{15}v_{\lambda}=Q_{15}^{1}.v_{\lambda}=0.$
$None$

Therefore, $\lambda=(m,0,0,n),m\in\mathbb{N},n>0$.

Case 2.2.2. $\lambda_{3}>0,\ \lambda_{4}>0$.

Note that $E_{54}v_{\lambda}\neq 0$. Since
$T_{5,123}.\tilde{c}.d_{15}v_{\lambda}=Q_{15}^{1}.v_{\lambda}=0$, the equation
(3.8) implies

$T_{5,123}.\xi_{0,1}=T_{5,123}.\tilde{c}^{2}.d_{15}v_{\lambda}=Q_{15}^{2}.v_{\lambda}=(E_{53}E_{42}E_{54}-E_{52}E_{43}E_{54})v_{\lambda}=0;$
$None$

which yields

$0=E_{34}E_{25}(E_{53}E_{42}E_{54}-E_{52}E_{43}E_{54})v_{\lambda}=-\lambda_{3}(1+\lambda_{3}+\lambda_{4})E_{54}v_{\lambda}.$
$None$

A contradiction arises.

Case 3.
wt($\xi_{0,1})\in\\{\mbox{wt}(d_{23}v_{\lambda}),\mbox{wt}(d_{24}v_{\lambda}),\mbox{wt}(d_{34}v_{\lambda}),\mbox{wt}(d_{25}v_{\lambda}),\mbox{wt}(d_{35}v_{\lambda}),\mbox{wt}(d_{45}v_{\lambda})\\}$.

In these cases, $v_{13}\neq 0,v_{23}\neq 0$. Set

$Q=E_{53}.v_{12}-E_{52}.v_{13}+E_{51}.v_{23},\ E_{15}Q=Q_{1},\
E_{25}Q_{1}=Q_{2},E_{35}Q_{1}=Q_{2}^{\prime}\ .$ $None$

Then

$0=Q_{1}=E_{13}v_{12}+E_{53}E_{15}v_{12}-E_{12}v_{13}-E_{52}E_{15}v_{13}+(h_{1}+h_{2}+h_{3}+h_{4})v_{23}.$
$None$

Case 3.1.
$wt(\xi_{0,1})\in\\{\mbox{wt}(d_{23}v_{\lambda}),\mbox{wt}(d_{24}v_{\lambda}),\mbox{wt}(d_{34}v_{\lambda})\\}$.

In these three cases, we have $v_{23}\neq 0$, $v_{25}=v_{35}=0$. So

$0=Q_{1}=(2+|\mbox{wt}(v_{23})|)v_{23}$ $None$

induces a contradiction.

Case 3.2. $wt(\xi_{0,1})=\mbox{wt}(d_{25}v_{\lambda})$.

In this case, $v_{35}=0$ and
$\mbox{wt}(v_{23})=\lambda-\alpha_{3}-\alpha_{4}$. So

$0=Q_{1}=(2+|\mbox{wt}(v_{23})|)v_{23}+E_{53}v_{25},0=Q_{2}^{\prime}=-(|\lambda|+2-h_{3}-h_{4})v_{25}=-(\lambda_{1}+\lambda_{2}+1)v_{25}$
$None$

force $\lambda_{1}+\lambda_{2}+1=0$. A contradiction arises.

Case 3.3.
$wt(\xi_{0,1})\in\\{\mbox{wt}(d_{35}v_{\lambda}),\mbox{wt}(d_{45}v_{\lambda})\\}$.

In these two cases, $v_{35}\neq 0$. And the equations

$0=Q_{1}=(2+|\mbox{wt}(v_{23})|)v_{23}+E_{53}v_{25}-E_{52}v_{35},0=Q_{2}=(1+|\mbox{wt}(v_{23})|-h_{2}-h_{3}-h_{4})v_{35}=\lambda_{1}v_{35}$
$None$

imply $\lambda_{1}=0$. Recall the intertwining operator defined in Lemma 5.2.
In these two cases, $T_{5,123}(\xi_{0,1})=0$ is equivalent to
$T^{1}|_{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,1}))}=0$. Assume
$v_{(\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,1}))}^{\lambda}$ is any
maximal vector of weight $\lambda$ appearing in the tensor decomposition
$V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,1}))$. Then
$T^{1}|_{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,1}))}=0$ iff
$T^{1}(v_{(\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,1}))}^{\lambda})=0$.

Case 3.3.1
$wt(\xi_{0,1})=\mbox{wt}(d_{35}v_{\lambda})=(0,\lambda_{2}-1,\lambda_{3}+1,\lambda_{4}-1)$.

Indeed,
$T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2}-1,\lambda_{3}+1,\lambda_{4}-1)}^{\lambda})$
in this case could be written as:

$\displaystyle
T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2}-1,\lambda_{3}+1,\lambda_{4}-1)}^{\lambda})$
$\displaystyle=$
$\displaystyle(T_{4,345}+T_{2,235})\xi_{0,1}+\frac{1}{2}(T_{1,135}-T_{2,235})\xi_{0,1}-\frac{1}{1+\lambda_{3}}(T_{3,345}-T_{2,245}).(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}$
$\displaystyle-$
$\displaystyle\frac{1}{2+2\lambda_{3}}(T_{2,245}-T_{1,145}).(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}-\frac{3}{1+\lambda_{2}+\lambda_{3}}T_{2,345}.x_{4}\partial_{x_{3}}.x_{3}\partial_{x_{2}}\xi_{0,1}$
$\displaystyle+$
$\displaystyle\frac{6+3\lambda_{3}}{(1+\lambda_{2}+\lambda_{3})(1+\lambda_{3})}T_{2,345}.(x_{3}\partial_{x_{2}})_{0}.(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}$
$\displaystyle-$
$\displaystyle\frac{6+3\lambda_{3}}{(1+\lambda_{2}+\lambda_{3})(1+\lambda_{3})}T_{1,345}.(x_{2}\partial_{x_{1}})_{0}.(x_{3}\partial_{x_{2}})_{0}.(x_{4}\partial_{x_{3}})_{0}.\xi_{0,1}$
$\displaystyle+$
$\displaystyle\frac{3}{1+\lambda_{2}+\lambda_{3}}T_{1,345}.(x_{4}\partial_{x_{3}})_{0}.(x_{2}\partial_{x_{1}})_{0}.(x_{3}\partial_{x_{2}})_{0}.\xi_{0,1}=\frac{(2+\lambda_{3})(\lambda_{2}+\lambda_{3}+7))}{(1+\lambda_{3})(\lambda_{2}+\lambda_{3}+1))}v_{35}\neq
0.\hskip 34.14322pt(5.22)$

Case 3.3.2
$wt(\xi_{0,1})=\mbox{wt}(d_{45}v_{\lambda})=(0,\lambda_{2},\lambda_{3}-1,\lambda_{4})$.

Suppose $\lambda_{2}\neq 0$. Then

$\displaystyle
T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2},\lambda_{3}-1,\lambda_{4})}^{\lambda})=\frac{-2\lambda_{2}}{3}(T_{3,345}-T_{2,245})\xi_{0,1}-\frac{\lambda_{2}}{3}(T_{2,245}-T_{1,145})\xi_{0,1}$
$\displaystyle+$ $\displaystyle
T_{2,345}.(x_{3}\partial_{x_{2}})_{0}.\xi_{0,1}-T_{1,345}.(x_{2}\partial_{x_{1}})_{0}.(x_{3}\partial_{x_{2}})_{0}.\xi_{0,1}=\frac{2\lambda_{2}(\lambda_{2}+3)}{3}v_{45}\neq
0\hskip 79.6678pt(5.23)$

induces a contradiction. Assume $\lambda_{2}=0$. Then it is easily checked
that

$T^{1}(v_{(1,1,0,0)\otimes(0,\lambda_{2},\lambda_{3}-1,\lambda_{4})}^{\lambda})=[2(T_{3,345}-T_{2,245})+(T_{2,245}-T_{1,145})]\xi_{0,1}=0.$
$None$

Thus $\lambda=(0,0,m,n)$. The proof is complete by Lemma 3.3. $\Box$

###  5.2 Singular vectors of degree two

Theorem 5.4 All the possible degree two singular vectors are listed in the
following:

$\prod\limits_{\mbox{wt}(d_{12}d_{15})<\sigma\leq\omega_{1}+\omega_{3}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{12}d_{15})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{12}d_{15}v_{\lambda},\
\mbox{where}\ \ \lambda=(m,0,0,1),\ m\in\mathbb{N}.$

Proof The leading term of any singular vector of degree two could be written
as:

$\xi_{0,2}=\sum\limits_{j\in\overline{1,35},k\in\overline{1,\mbox{mult}(\overrightarrow{w}_{j}^{\omega_{1}+\omega_{3}})}}v_{j,k}^{\omega_{1}+\omega_{3}}v_{j,k}^{\lambda},\
v_{j,k}^{\lambda}\in V(\lambda),$ $None$

which should satisfy $T_{5,123}.\xi_{0,2}=0$. Assume

$T_{5,123}.\xi_{0,2}=\sum\limits_{1\leq i<j\leq 5}d_{ij}t_{ij},t_{ij}\in
V(\lambda).$ $None$

Then we could derive the following equations:

$t_{12}=E_{52}v_{1,1}^{\lambda}-E_{51}v_{2,1}^{\lambda}=0,t_{15}=E_{53}v_{5,1}^{\lambda}-E_{52}v_{10,1}^{\lambda}+E_{51}(v_{11,2}^{\lambda}+v_{11,3}^{\lambda})+v_{1,1}^{\lambda}=0,$
$t_{13}=E_{53}v_{1,1}^{\lambda}-E_{51}v_{6,1}^{\lambda}=0,t_{25}=E_{53}v_{9,1}^{\lambda}+E_{52}(v_{11,1}^{\lambda}+v_{11,3}^{\lambda})+E_{51}v_{19,1}^{\lambda}+v_{2,1}^{\lambda}=0,$
$t_{23}=E_{53}v_{2,1}^{\lambda}-E_{52}v_{6,1}^{\lambda}=0,t_{35}=E_{53}(v_{11,1}^{\lambda}+v_{11,2}^{\lambda})-E_{52}v_{17,1}^{\lambda}+E_{51}v_{22,1}^{\lambda}+v_{6,1}^{\lambda}=0.$
$None$

It follows from $v_{1,1}^{\lambda}\neq 0$ that one of $v_{5,1}^{\lambda},\
v_{10,1}^{\lambda},\ v_{11,2}^{\lambda}+v_{11,3}^{\lambda}$ should be nonzero.
Hence, the information of the weights in Table 5 implies that
$\mbox{wt}(\xi_{0,2})$ should be restricted to the following cases:

$\displaystyle\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{5}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{9}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{10}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{15}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},$
$\displaystyle\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}(i\in\overline{24,35})\\}\hskip
73.97733pt(5.30)$

Case 1
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{5}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{10}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{15}^{\omega_{1}+\omega_{3}}\\}$

In these three cases, $v_{2,1}^{\lambda}=v_{6,1}^{\lambda}=0$.

Case 1.1
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{5}^{\omega_{1}+\omega_{3}}$

We have $wt(v_{1,1}^{\lambda})=\lambda-\alpha_{3}-\alpha_{4}$ and
$\lambda_{4}>0$. So
$0=E_{25}.t_{12}=E_{25}.E_{52}v_{1,1}^{\lambda}=(h_{2}+h_{3}+h_{4})v_{1,1}^{\lambda}=(\lambda_{2}+\lambda_{3}+\lambda_{4}-1)v_{1,1}^{\lambda}$
yields $(\lambda_{2},\lambda_{3},\lambda_{4})=(0,0,1).$ Then
$\chi_{\overrightarrow{w}_{1}^{\omega_{1}+\omega_{3}}}(\tilde{c})=\frac{3\lambda_{1}+2}{25}$
by Lemma 3.5. And

$T_{5,123}.\xi_{0,2}=T_{5,123}.(\tilde{c}-\chi_{\overrightarrow{w}_{1}^{\omega_{1}+\omega_{3}}}(\tilde{c})).d_{12}d_{15}.v_{\lambda}=T_{5,123}.\tilde{c}.d_{12}d_{15}.v_{\lambda}-\chi_{\overrightarrow{w}_{1}^{\omega_{1}+\omega_{3}}}(\tilde{c})d_{15}E_{53}.v_{\lambda}=0,$
$None$

since

$T_{5,123}.\tilde{c}.d_{12}d_{15}.v_{\lambda}\stackrel{{\scriptstyle\mbox{by}\
(5.13)}}{{=}}Q_{12}^{1}d_{15}v_{\lambda}-Q_{15}^{1}d_{12}v_{\lambda};$

where

$Q_{12}^{1}\stackrel{{\scriptstyle\mbox{by}(5.3)}}{{=}}Q_{12}^{0}(\tilde{c}+\frac{h_{2}^{*}}{10})+\frac{1}{10}Q_{13}^{0}E_{23}-\frac{1}{10}Q_{23}^{0}E_{13},\
Q_{15}^{1}\stackrel{{\scriptstyle\mbox{by}(5.3)}}{{=}}\frac{1}{10}(Q_{12}^{0}E_{52}+Q_{13}^{0}E_{53}),$
$Q_{12}^{0}=E_{53}+\frac{(x_{5}\partial_{x_{3}})_{0}^{\prime}}{2},\
Q_{13}^{0}=-E_{52}-\frac{(x_{5}\partial_{x_{2}})_{0}^{\prime}}{2},Q_{23}^{0}=E_{51}+\frac{(x_{5}\partial_{x_{1}})_{0}^{\prime}}{2},$
$\tilde{c}.d_{15}v_{\lambda}\stackrel{{\scriptstyle\mbox{by}(5.1)}}{{=}}\frac{1}{10}[d_{15}(h_{1}^{*}-h_{4}^{*}).v_{\lambda}+\sum\limits_{i=2}^{4}d_{1i}E_{5i}v_{\lambda}]=\frac{1}{10}[\frac{3\lambda_{1}-3}{5}d_{15}v_{\lambda}+\sum\limits_{i=2}^{4}d_{1i}E_{5i}v_{\lambda}].$
$None$

That is, $\lambda=(m,0,0,1).$

Case 1.2
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{10}^{\omega_{1}+\omega_{3}}$

We have
$\mbox{wt}(v_{1,1}^{\lambda})=\lambda-\alpha_{2}-\alpha_{3}-\alpha_{4}$ and
$\lambda_{2}>0,\lambda_{4}>0$, $E_{25}.v_{1,1}^{\lambda}=-v_{10,1}^{\lambda}$.
So

$0=E_{25}t_{12}=(h_{2}+h_{3}+h_{4})v_{1,1}^{\lambda}+E_{52}.E_{25}.v_{1,1}^{\lambda}=(\lambda_{2}+\lambda_{3}+\lambda_{4}-2)v_{1,1}^{\lambda}-E_{52}.v_{10,1}^{\lambda},$
$None$
$0=E_{25}^{2}t_{12}=E_{25}.[(\lambda_{2}+\lambda_{3}+\lambda_{4}-2)v_{1,1}^{\lambda}-E_{52}.v_{10,1}^{\lambda}]=-2(\lambda_{2}+\lambda_{3}+\lambda_{4}-1)v_{10,1}^{\lambda}$
$None$

yields a contradiction.

Case 1.3
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{15}^{\omega_{1}+\omega_{3}}$

We have $\mbox{wt}(v_{10,1}^{\lambda})=\lambda-\alpha_{3}$ and
$\lambda_{3}>0$, $E_{23}.v_{1,5}^{\lambda}=-v_{10,1}^{\lambda}$. Then

$E_{53}v_{5,1}^{\lambda}-E_{52}v_{10,1}^{\lambda}+v_{1,1}^{\lambda}=0,$ $None$
$0=E_{25}(E_{53}v_{5,1}^{\lambda}-E_{52}v_{10,1}^{\lambda}+v_{1,1}^{\lambda})=E_{23}v_{5,1}^{\lambda}-(h_{2}+h_{3}+h_{4})v_{10,1}^{\lambda}+E_{25}v_{1,1}^{\lambda}$
$None$

imply $(\lambda_{2},\lambda_{3},\lambda_{4})=(0,0,0).$ A contradiction arises.

Case 2
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{9}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}(i\in\overline{24,35})\\}$

Case 2.1
$wt(\xi_{0,2})=\lambda+\overrightarrow{w}_{9}^{\omega_{1}+\omega_{3}}$

We have $v_{2,1}^{\lambda}\neq 0$, since
$E_{35}v_{2,1}^{\lambda}=-v_{9,1}^{\lambda}$. Note that $v_{6,1}^{\lambda}=0$,
$E_{15}v_{1,1}^{\lambda}=E_{15}v_{2,1}^{\lambda}=0$,
$E_{12}v_{1,1}^{\lambda}=-v_{2,1}^{\lambda}$,
$\mbox{wt}(v_{2,1}^{\lambda})=\lambda-\alpha_{3}-\alpha_{4}$ and
$\lambda_{1}>0,\lambda_{4}>0$. Then

$0=E_{15}.t_{12}=E_{15}.(E_{52}v_{1,1}^{\lambda}-E_{51}v_{2,1}^{\lambda})=(E_{12}+E_{52}E_{15})v_{1,1}^{\lambda}-(\sum\limits_{i=1}^{4}h_{i}+E_{51}E_{15})v_{2,1}^{\lambda}$
$None$

implies $|\lambda|=0$. A contradiction arises.

Case 2.2
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}(i\in\overline{24,35})\\}$

Case 2.2.1
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{11}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{17}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{19}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{21}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{24}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{26}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{27}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{29}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{30}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{33}^{\omega_{1}+\omega_{3}}.\\}$

In these cases, $v_{25,1}^{\lambda}=v_{28,1}^{\lambda}=0$. Assume
$v_{11,2}^{\lambda}+v_{11,3}^{\lambda}\neq 0$. Then

$0=E_{15}t_{15}\stackrel{{\scriptstyle\mbox{by}(5.29)}}{{=}}(2+\sum\limits_{i=1}^{4}h_{i})(v_{11,2}^{\lambda}+v_{11,3}^{\lambda})=0$
$None$

yields a contradiction. Hence, $v_{11,2}^{\lambda}+v_{11,3}^{\lambda}=0$.
Furthermore, either the assertion $v_{10,1}^{\lambda}\neq 0,\
(h_{2}+h_{3}+h_{4})v_{10,1}^{\lambda}=0$ or the assertion
$v_{10,1}^{\lambda}=0,\ v_{5,1}^{\lambda}\neq 0,\
(h_{3}+h_{4}-1)v_{5,1}^{\lambda}=0$ holds. By detailed check case by case,
only the cases
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{24}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{26}^{\omega_{1}+\omega_{3}}\\}$
satisfy this assertion.

For the case
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{16}^{\omega_{1}+\omega_{3}}$,
we get $v_{10,1}^{\lambda}=0,\ v_{5,1}^{\lambda}\neq 0,\
(\lambda_{3},\lambda_{4})=(1,0)$. And we could write
$\xi_{0,2}=y_{12}\xi_{0,2}^{\prime}$, where
$\xi_{0,2}^{\prime}=\sum\limits_{1\leq i<j\leq 5}d_{ij}v_{ij}$. Hence,
$T_{5,123}.\xi_{0,2}=T_{5,123}.y_{12}\xi_{0,2}^{\prime}=([T_{5,123},y_{12}]+y_{12}T_{5,123}).\xi_{0,2}^{\prime}=(x_{5}\partial_{x_{3}}+\frac{1}{2}(-1)^{|34,45|}y_{45}\partial_{34}).\xi_{0,2}^{\prime}=\frac{1}{2}(-1)^{|34,45|}y_{45}\partial_{34}.\xi_{0,2}^{\prime}\neq
0$.

For the case
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{24}^{\omega_{1}+\omega_{3}}$,
we get $v_{19,1}^{\lambda}\neq 0$. The equation $E_{15}t_{25}=0$ implies that

$0=E_{15}v_{2,1}^{\lambda}+(E_{13}+E_{53}E_{15})v_{9,1}^{\lambda}+(E_{12}+E_{52}E_{15})(v_{11,1}^{\lambda}+v_{11,3}^{\lambda})+\sum\limits_{i=1}^{4}h_{i}v_{19,1}^{\lambda}=(3+|\lambda|)v_{19,1}^{\lambda}.$
$None$

For the case
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{26}^{\omega_{1}+\omega_{3}}$,
one of $v_{19,1}^{\lambda}$ and $v_{22,1}^{\lambda}$ should be nonzero,
otherwise $v_{26,i}^{\lambda}=0$. Then, the equation $E_{15}t_{35}=0$ implies
that

$0=E_{15}v_{6,1}^{\lambda}+(E_{13}+E_{53}E_{15})(v_{11,1}^{\lambda}+v_{11,2}^{\lambda})-(E_{12}+E_{52}E_{15})v_{17,1}^{\lambda})+\sum\limits_{i=1}^{4}h_{i}.v_{22,1}^{\lambda}=(1+|\lambda|)v_{22,1}^{\lambda}.$
$None$

Case 2.2.2
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{25}^{\omega_{1}+\omega_{3}}$

In this case, $v_{25,1}^{\lambda}\neq 0$, $v_{28,1}^{\lambda}=0$ and
$wt(\xi_{0,2})=\lambda+\mbox{wt}(d_{15}d_{25})$. Then

$0=E_{15}t_{15}=(2+\sum\limits_{i=1}^{4}h_{i})(v_{11,2}^{\lambda}-v_{11,3}^{\lambda})-E_{53}v_{25,1}^{\lambda},$
$E_{35}.E_{15}t_{15}=(2+|\mbox{wt}(\overrightarrow{w}_{25}^{\omega_{1}+\omega_{3}})|-h_{3}-h_{4})v_{25,1}^{\lambda}=0.$
$None$

So $\lambda_{1}+\lambda_{2}+1=0$. A contradiction arises.

Case 2.2.3
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{28}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{31}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{34}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{35}^{\omega_{1}+\omega_{3}},\\}$

In these cases, $v_{25,1}^{\lambda}\neq 0$ and $v_{28,1}^{\lambda}\neq 0$.

Case 2.2.3.1
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{28}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{31}^{\omega_{1}+\omega_{3}}\\}$

The equations

$0=E_{15}t_{12}=(E_{12}+E_{52}E_{15})v_{1,1}^{\lambda}-(\sum\limits_{i=1}^{4}h_{i}+E_{51}E_{15})v_{2,1}^{\lambda}=-(1+|\mbox{wt}(\overrightarrow{w}_{2}^{\omega_{1}+\omega_{3}})|)v_{1,1}^{\lambda}+E_{52}(v_{11,2}^{\lambda}+v_{11,3}^{\lambda}),$
$None$
$E_{25}^{2}E_{15}t_{12}=2(\lambda_{2}+\lambda_{3}+\lambda_{4}-1)v_{28,1}^{\lambda}=0$
$None$

induce a contradiction.

Case 2.2.3.2
$\mbox{wt}(\xi_{0,2})\in\\{\lambda+\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{34}^{\omega_{1}+\omega_{3}},\lambda+\overrightarrow{w}_{35}^{\omega_{1}+\omega_{3}}\\}$

First, we have $v_{32,1}^{\lambda}\neq 0$ in these cases, since
$E_{34}v_{32,1}^{\lambda}=-v_{34,1}^{\lambda},\
E_{24}v_{32,1}^{\lambda}=v_{35,1}^{\lambda}$. Then

$\displaystyle
0=E_{15}t_{35}=E_{15}v_{6,1}^{\lambda}+(E_{13}+E_{53}E_{15})(v_{11,1}^{\lambda}+v_{11,2}^{\lambda})-(E_{12}+E_{52}E_{15})v_{17,1}^{\lambda}$
$\displaystyle+(\sum\limits_{i=1}^{4}h_{i})v_{22,1}^{\lambda}=(1+|\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})|)v_{22,1}^{\lambda}+E_{53}v_{32,1}^{\lambda},$
$E_{35}E_{15}t_{35}=(-1-|\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})|+h_{3}+h_{4})v_{32,1}^{\lambda}=0.$
$None$

If
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}}$,
then $\mbox{wt}(\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}})=\lambda$ and
$\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{3}-\alpha_{4}$.
If
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{34}^{\omega_{1}+\omega_{3}}$,
then
$\mbox{wt}(\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{3}$
and
$\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})=\lambda-2\alpha_{3}-\alpha_{4}$.
If
$\mbox{wt}(\xi_{0,2})=\lambda+\overrightarrow{w}_{35}^{\omega_{1}+\omega_{3}}$,
then
$\mbox{wt}(\overrightarrow{w}_{32}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{2}-\alpha_{3}$
and
$\mbox{wt}(\overrightarrow{w}_{22}^{\omega_{1}+\omega_{3}})=\lambda-\alpha_{2}-2\alpha_{3}-\alpha_{4}$.
Thus, (5.44) yields $\lambda_{1}+\lambda_{2}=0$ or
$\lambda_{1}+\lambda_{2}+1=0$. A contradiction arises. $\Box$

###  5.3 Singular vectors of degree three

Theorem 5.5 All the possible degree three singular vectors are listed in the
following:

$e^{-\frac{P}{2}}.\prod\limits_{\mbox{wt}(d_{15}d_{25}d_{45})<\sigma\leq
2\omega_{1}+\omega_{4}}\frac{\tilde{c}-\chi_{\sigma+\lambda}(\tilde{c})}{\chi_{\mbox{wt}(d_{15}d_{25}d_{45})+\lambda}(\tilde{c})-\chi_{\sigma+\lambda}(\tilde{c})}.d_{15}d_{25}d_{45}v_{\lambda},$
$\ \mbox{where}\ \ \lambda=(0,0,m,n),\ 1\leq m\in\mathbb{N},\ 2\leq
n\in\mathbb{N}.$

Proof The leading term of any singular vector of degree three could be written
as:

$\xi_{0,3}=\sum\limits_{j\in\overline{1,55},k\in\overline{1,\mbox{mult}(\overrightarrow{w}_{j}^{2\omega_{1}+\omega_{4}})}}v_{j,k}^{2\omega_{1}+\omega_{4}}v_{j,k}^{\lambda},\
v_{j,k}^{\lambda}\in V(\lambda),$ $None$

which should satisfy $T_{5,123}.\xi_{0,3}=0$ and
$(x_{i}\partial_{x_{j}})_{0}.\xi_{0,3}=0$ ( $1\leq i<j\leq 5$). Assume

$T_{5,123}.\xi_{0,3}=\sum\limits_{i_{1}<j_{1},i_{2}<j_{2},(i_{1},j_{1})\neq(i_{2},j_{2})}d_{i_{1}j_{1}}d_{i_{2}j_{2}}t_{i_{1}j_{1},i_{2}j_{2}},\
t_{i_{1}j_{1},i_{2}j_{2}}\in V(\lambda).$ $None$

Since $v_{1,1}^{\lambda}\neq 0$, the equation

$0=t_{14,15}=-v_{1,1}^{\lambda}+E_{53}v_{7,1}^{\lambda}-E_{52}v_{13,1}^{\lambda}+E_{51}(v_{18,1}^{\lambda}-v_{18,4}^{\lambda})$
$None$

implies that one of the terms $v_{7,1}^{\lambda}$, $v_{13,1}^{\lambda}$ and
$v_{18,1}^{\lambda}-v_{18,4}^{\lambda}$ should be nonzero. Hence, the
information of the weights of Table 6-7 induces that $wt(\xi_{0,3})$ could be
restricted to the following cases:

$\displaystyle
wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{7}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{12}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{13}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{17}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{18}^{2\omega_{1}+\omega_{4}},$
$\displaystyle\lambda+\overrightarrow{w}_{21}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{22}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{23}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{24}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{i}^{2\omega_{1}+\omega_{4}}(i\in\overline{26,55})\\}.\hskip
51.21504pt(5.48)$

Case 1 $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{7}^{2\omega_{1}+\omega_{4}}$

In this case, $v_{6,1}^{\lambda}=0$, $v_{3,1}^{\lambda}\neq 0$,
$wt(v_{3,1}^{\lambda})=\lambda-\alpha_{3}$,
$E_{34}v_{3,1}^{\lambda}=-v_{7,1}^{\lambda}$. Then

$E_{25}t_{12,15}=E_{25}(E_{52}v_{3,1}^{\lambda}-E_{51}v_{6,1}^{\lambda})=(h_{2}+h_{3}+h_{4})v_{3,1}^{\lambda}=(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{3,1}^{\lambda}=0$
$None$

contradicts $\lambda_{3}>0$.

Case 2
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{12}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{17}^{2\omega_{1}+\omega_{4}}\\}$

Note that
$E_{34}v_{6,1}^{\lambda}=-v_{12,1}^{\lambda},E_{12}v_{12,1}^{\lambda}=-v_{17,1}^{\lambda},E_{12}v_{3,1}^{\lambda}=-2v_{6,1}^{\lambda}.$
Hence, $v_{3,1}^{\lambda}\neq 0$ and $v_{6,1}^{\lambda},\neq 0$. Since
$wt(v_{6,1}^{\lambda})=\lambda-\alpha_{3}$ for
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{12}^{2\omega_{1}+\omega_{4}}$;
$wt(v_{6,1}^{\lambda})=\lambda-\alpha_{1}-\alpha_{3}$ for
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{17}^{2\omega_{1}+\omega_{4}}$. The
equation
$E_{15}t_{12,15}=-(2+|\mbox{wt}v_{6,1}^{\lambda}|)v_{6,1}^{\lambda}=0$ induces
that $|\lambda|<0$ in both cases.

Case 3
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{13}^{2\omega_{1}+\omega_{4}}\vspace{0.1cm}$

We have $E_{25}v_{1,1}^{\lambda}=-v_{13,1}^{\lambda}$,
$E_{23}v_{7,1}^{\lambda}=-v_{13,1}^{\lambda}$. Then

$0=E_{25}t_{14,15}\stackrel{{\scriptstyle\mbox{by}(5.47)}}{{=}}-E_{25}v_{1,1}^{\lambda}+E_{23}v_{7,1}^{\lambda}-\sum\limits_{i=2}^{4}h_{i}v_{13,1}^{\lambda}=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{13,1}^{\lambda}=0,$
$None$

contradicts $\lambda_{2}>0$.

Case 4
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{22}^{2\omega_{1}+\omega_{4}}$

We have $v_{11,1}^{\lambda}\neq 0$, since
$E_{24}v_{11,1}^{\lambda}=v_{22,1}^{\lambda}$. Then

$0=E_{15}t_{13,15}=E_{15}(E_{53}v_{3,1}^{\lambda}-E_{51}v_{11,1}^{\lambda})=-(2+|\lambda|)v_{11,1}^{\lambda}$
$None$

yields a contradiction.

Case 5
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{24}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{28}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{30}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{35}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{36}^{2\omega_{1}+\omega_{4}}\\}$

For
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{24}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{30}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{36}^{2\omega_{1}+\omega_{4}}\\}$,
we have $v_{28,1}^{\lambda}=0$. Then

$0=E_{25}t_{15,25}=E_{25}(E_{52}v_{24,1}^{\lambda}-E_{51}v_{28,1}^{\lambda})=(h_{2}+h_{3}+h_{4})v_{24,1}^{\lambda}$
$None$

yields $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$ or $1$. For
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{28}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{35}^{2\omega_{1}+\omega_{4}}\\},$
the equation

$0=E_{15}t_{15,25}=-(h_{1}+h_{2}+h_{3}+h_{4})v_{28,1}^{\lambda}+(E_{12}+E_{52}E_{15})v_{24,1}^{\lambda}$
$None$

yields $1+|\lambda|=0$.

Case 6
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{23}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{29}^{2\omega_{1}+\omega_{4}}\\}$

We have
$E_{35}v_{10,1}^{\lambda}=v_{23,1}^{\lambda},E_{25}v_{10,1}^{\lambda}=-v_{29,1}^{\lambda},E_{23}v_{23,1}^{\lambda}=-v_{29,1}^{\lambda}$.
Consider the equation

$t_{14,45}=-v_{10,1}^{\lambda}+E_{53}v_{23,1}^{\lambda}-E_{52}v_{29,1}^{\lambda}-E_{51}(v_{32,1}^{\lambda}-v_{32,4}^{\lambda}).$
$None$

For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{23}^{2\omega_{1}+\omega_{4}}$,
$E_{35}t_{14,45}=(\lambda_{3}+\lambda_{4}-1)v_{23,1}^{\lambda}=0$ induces
$\lambda_{3}+\lambda_{4}=1$, which contradicts $\lambda_{3}>1$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{29}^{2\omega_{1}+\omega_{4}}$, we
have
$E_{25}t_{14,45}=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{29,1}^{\lambda}=0$
induces $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$, which contradicts
$\lambda_{2}>0,\lambda_{3}>0$.

Case 7
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{27}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{39}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{42}^{2\omega_{1}+\omega_{4}}\\}$

Note that in these cases, the equations are derived:
$E_{35}v_{15,1}^{\lambda}=v_{27,1}^{\lambda},E_{15}v_{15,1}^{\lambda}=v_{39,1}^{\lambda},E_{13}v_{27,1}^{\lambda}=v_{39,1}^{\lambda},E_{12}v_{32,4}^{\lambda}=-v_{39,1}^{\lambda},E_{12}v_{32,1}^{\lambda}=0=E_{12}v_{32,3}^{\lambda},E_{25}v_{15,1}^{\lambda}=-\sum\limits_{i=1}^{4}v_{32,i}^{\lambda},E_{23}v_{27,1}^{\lambda}=-v_{32,1}^{\lambda}-2v_{32,3}^{\lambda}-v_{32,4}^{\lambda},E_{34}v_{32,1}^{\lambda}=-v_{42,1}^{\lambda},E_{34}v_{32,2}^{\lambda}=v_{42,1}^{\lambda},E_{34}v_{32,3}^{\lambda}=0=E_{34}v_{32,4}^{\lambda}.$
Consider the equation

$t_{24,45}=-v_{15,1}^{\lambda}+E_{53}v_{27,1}^{\lambda}-E_{52}(v_{32,1}^{\lambda}+v_{32,3}^{\lambda}+v_{32,4}^{\lambda})+E_{51}v_{39,1}^{\lambda}.$
$None$

For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{27}^{2\omega_{1}+\omega_{4}}$,
$E_{35}t_{24,45}=(\lambda_{3}+\lambda_{4}-1)v_{27,1}^{\lambda}=0$ induces
$\lambda_{3}+\lambda_{4}=1$, which contradicts $\lambda_{3}>1$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{39}^{2\omega_{1}+\omega_{4}}$,
$E_{15}t_{24,45}=(1+|\lambda|)v_{39,1}^{\lambda}=0$ induces $1+|\lambda|=0$.
For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{42}^{2\omega_{1}+\omega_{4}}$,
Then
$E_{15}t_{24,45}=v_{32,2}^{\lambda}-v_{32,3}^{\lambda}-(\lambda_{2}+\lambda_{3}+\lambda_{4})(v_{32,1}^{\lambda}+v_{32,3}^{\lambda}+v_{32,4}^{\lambda})=0$,
and
$E_{34}E_{15}t_{24,45}=(1+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{42,1}^{\lambda}=0$
induces $1+\lambda_{2}+\lambda_{3}+\lambda_{4}=0$.

Case 8
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{31}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{33}^{2\omega_{1}+\omega_{4}}\\}$

Observe these equations are derived:
$E_{15}v_{4,1}^{\lambda}=v_{31,1}^{\lambda},E_{13}v_{17,1}^{\lambda}=v_{31,1}^{\lambda},E_{12}v_{21,1}^{\lambda}=-v_{31,1}^{\lambda},E_{12}v_{21,2}^{\lambda}=v_{31,1}^{\lambda},E_{12}v_{21,3}^{\lambda}=0,E_{12}v_{21,4}^{\lambda}=0.$
Consider the equation

$t_{24,25}=-v_{4,1}^{\lambda}+E_{53}v_{17,1}^{\lambda}-E_{52}(2v_{21,1}^{\lambda}+v_{21,2}^{\lambda}+v_{21,3}^{\lambda})+E_{51}v_{31,1}^{\lambda}.$
$None$

It follows from $E_{15}t_{24,25}=0$ that $(1+|\lambda|)v_{31,1}^{\lambda}=0$.

Case 9
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{37}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{38}^{2\omega_{1}+\omega_{4}}\\}$

Note
$E_{15}v_{14,1}^{\lambda}=v_{38,1}^{\lambda},E_{13}v_{26,2}^{\lambda}=v_{38,1}^{\lambda},E_{12}v_{37,1}^{\lambda}=-v_{38,1}^{\lambda},E_{25}v_{14,1}^{\lambda}=-v_{37,1}^{\lambda},E_{23}v_{26,2}^{\lambda}=-v_{37,1}^{\lambda}.$
Consider the equation

$t_{34,35}=-v_{14,1}^{\lambda}+E_{53}v_{26,2}^{\lambda}-E_{52}v_{37,1}^{\lambda}+E_{51}v_{38,1}^{\lambda}.$
$None$

For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{37}^{2\omega_{1}+\omega_{4}}$,
$E_{25}t_{34,35}=-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{37,1}^{\lambda}=0$
implies $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$, which which contradicts
$\lambda_{2}>2$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{38}^{2\omega_{1}+\omega_{4}}$,
$E_{15}t_{34,35}=(1+|\lambda|)v_{38,1}^{\lambda}=0$.

Case 10
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{40}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{44}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{48}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{51}^{2\omega_{1}+\omega_{4}}\\}$

Note
$E_{25}v_{19,1}^{\lambda}=-v_{40,1}^{\lambda},E_{23}v_{32,3}^{\lambda}=-v_{40,1}^{\lambda},E_{15}v_{19,1}^{\lambda}=v_{44,1}^{\lambda},E_{13}v_{32,3}^{\lambda}=v_{44,1}^{\lambda},E_{12}v_{40,1}^{\lambda}=-v_{44,1}^{\lambda}.$
Consider the equation

$t_{34,45}=-v_{19,1}^{\lambda}+E_{53}v_{32,3}^{\lambda}-E_{52}v_{40,1}^{\lambda}+E_{51}v_{44,1}^{\lambda},$
$None$

For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{40}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{48}^{2\omega_{1}+\omega_{4}},$
the equation $E_{25}t_{34,45}=0$ implies
$-(\lambda_{2}+\lambda_{3}+\lambda_{4})v_{40,1}^{\lambda}=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{44}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{51}^{2\omega_{1}+\omega_{4}},$
the equation $E_{15}t_{34,45}=0$ implies $(1+|\lambda|)v_{44,1}^{\lambda}=0$.

Case 11
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{45}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{52}^{2\omega_{1}+\omega_{4}}\\}$

Note
$E_{15}v_{21,3}^{\lambda}=-v_{45,1}^{\lambda},E_{15}v_{21,4}^{\lambda}=v_{45,1}^{\lambda},E_{13}v_{35,1}^{\lambda}=v_{45,1}^{\lambda},E_{12}v_{41,1}^{\lambda}=-v_{45,1}^{\lambda},E_{12}v_{41,i}^{\lambda}=0\
(i\in\overline{2,4}),E_{15}v_{18,3}^{\lambda}=v_{41,1}^{\lambda}-v_{41,4}^{\lambda},E_{15}v_{18,4}^{\lambda}=-v_{41,1}^{\lambda}-v_{41,3}^{\lambda},E_{13}v_{30,1}^{\lambda}=v_{41,1}^{\lambda}-v_{41,2}^{\lambda}-v_{41,3}^{\lambda},E_{12}v_{36,1}^{\lambda}=-2v_{41,1}^{\lambda}-v_{41,2}^{\lambda},E_{15}v_{30,1}^{\lambda}=E_{35}v_{41,3}^{\lambda}=-E_{35}v_{41,4}^{\lambda}=-v_{52,1}^{\lambda}.$
Consider the equation

$t_{25,45}=-v_{21,3}^{\lambda}-v_{21,4}^{\lambda}+E_{53}v_{35,1}^{\lambda}-E_{52}(v_{41,1}^{\lambda}+v_{41,2}^{\lambda}+v_{41,3}^{\lambda})+E_{51}v_{45,1}^{\lambda}=0.$
$None$

For $wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{45}^{2\omega_{1}+\omega_{4}},$
the equation $E_{15}t_{25,45}=0$ implies $(2+|\lambda|)v_{45,1}^{\lambda}=0$.
Now suppose
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{52}^{2\omega_{1}+\omega_{4}}.$ The
equation $t_{15,45}=0$ implies that

$E_{15}t_{15,45}=0=(|\lambda|+2)v_{41,1}^{\lambda}+(1-|\lambda|)v_{41,3}^{\lambda}+v_{41,4}^{\lambda}-E_{53}v_{52,1}^{\lambda},$
$0=E_{35}E_{15}t_{15,45}=(\lambda_{1}+\lambda_{2})v_{52,1}^{\lambda},$ $None$

i.e. $\lambda_{1}=\lambda_{2}=0$. Then
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{52}^{2\omega_{1}+\omega_{4}}=(0,1,\lambda_{3}-1,\lambda_{4}-2).$
Observe that

$\Pi((\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3}))\bigcap\Pi((\omega_{1}+\omega_{3})\otimes\lambda)=\\{\mbox{wt}(\xi_{0,3})+(0,0,1,0),\mbox{wt}(\xi_{0,3})+(1,-1,0,1)\\}.$
$None$
$\Pi((\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,3})+(0,0,1,0)))\bigcap\\{\lambda\\}=\emptyset,\
\Pi((\omega_{1}+\omega_{2})\otimes(\mbox{wt}(\xi_{0,3})+(1,-1,0,1)))\bigcap\\{\lambda\\}=\emptyset.$
$None$

Then (5.62) implies that

$T^{2}|_{V(\omega_{1}+\omega_{2})\otimes
V(\mbox{wt}(\xi_{0,3})+(0,0,1,0))}=0,T^{2}|_{V(\omega_{1}+\omega_{2})\otimes
V(\mbox{wt}(\xi_{0,3})+(1,-1,0,1))}=0.$ $None$

Note that $T_{5,123}\xi_{0,3}=0$ iff $T^{3}|_{V(\omega_{1}+\omega_{2})\otimes
V(\mbox{wt}(\xi_{0,3}))}=0$ by Lemma 5.2. Assume
$v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(0,1,\lambda_{3},\lambda_{4}-2)}$
(resp.
$v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(1,0,\lambda_{3}-1,\lambda_{4}-1)}$
) is any maximal vector of weight $(0,1,\lambda_{3},\lambda_{4}-2)$ (resp.
$(1,0,\lambda_{3}-1,\lambda_{4}-1)$ ) appearing in the tensor decomposition
$V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,3}))$. Since
$T^{3}{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,3}))}\subseteq
V(\omega_{1}+\omega_{3})\otimes V(\lambda)$,
$T^{3}|_{V(\omega_{1}+\omega_{2})\otimes V(\mbox{wt}(\xi_{0,3}))}=0$ iff
$T^{3}.(v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(0,1,\lambda_{3},\lambda_{4}-2)})=0,T^{3}(v_{(\omega_{1}+\omega_{2})\otimes\mbox{wt}(\xi_{0,3})}^{(1,0,\lambda_{3}-1,\lambda_{4}-1)})=0$.
Otherwise, we could get a singular vector of degree two with weights
$(0,1,\lambda_{3},\lambda_{4}-2)$ and $(1,0,\lambda_{3}-1,\lambda_{4}-1)$
respectively by (5.63), which contradicts proposition 5.3. Hence, we get
$\lambda=(0,0,m,n)$.

Case 12
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{43}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{47}^{2\omega_{1}+\omega_{4}}\\}$

Consider the equation

$t_{15,45}=-v_{18,3}^{\lambda}-v_{18,4}^{\lambda}+E_{53}v_{30,1}^{\lambda}-E_{52}v_{36,1}^{\lambda}+E_{51}v_{41,1}^{\lambda}-E_{51}v_{41,3}^{\lambda},$
$None$

Note
$E_{15}v_{18,3}^{\lambda}=v_{41,1}^{\lambda}-v_{41,4}^{\lambda},E_{15}v_{18,4}^{\lambda}=-v_{41,1}^{\lambda}-v_{41,3}^{\lambda},E_{13}v_{30,1}^{\lambda}=v_{41,1}^{\lambda}-v_{41,2}^{\lambda}-v_{41,3}^{\lambda},E_{12}v_{36,1}^{\lambda}=-2v_{41,1}^{\lambda}-v_{41,2}^{\lambda}.$
Then $E_{15}t_{15,45}=0$ induces that
$3v_{41,1}^{\lambda}+v_{41,4}^{\lambda}+\sum\limits_{i=1}^{4}h_{i}(v_{41,1}^{\lambda}-v_{41,3}^{\lambda})=0$.
Since
$E_{45}v_{41,4}^{\lambda}=-v_{43,1}^{\lambda},E_{45}v_{41,i}^{\lambda}=0(i\in\overline{1,3})$
and
$E_{34}v_{41,3}^{\lambda}=-v_{47,1}^{\lambda},E_{34}v_{41,i}^{\lambda}=0(i\in\\{1,2,4\\})$.
Hence, $E_{45}E_{15}t_{15,45}=0$ and $E_{34}E_{15}t_{15,45}=0$ imply
$v_{41,3}^{\lambda}=0$ and $|\lambda|v_{47,1}^{\lambda}=0$ in these two cases,
respectively.

Case 13
$wt(\xi_{0,3})\in\\{\lambda+\overrightarrow{w}_{46}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{49}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{50}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{53}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{54}^{2\omega_{1}+\omega_{4}},\lambda+\overrightarrow{w}_{55}^{2\omega_{1}+\omega_{4}}\\}$

Observe
$E_{25}v_{26,3}^{\lambda}=2v_{46,1}^{\lambda},E_{25}v_{26,4}^{\lambda}=-v_{46,1}^{\lambda},E_{23}v_{41,2}^{\lambda}=-v_{46,1}^{\lambda},E_{15}v_{26,3}^{\lambda}=-2v_{49,1}^{\lambda},E_{15}v_{26,4}^{\lambda}=v_{49,1}^{\lambda},E_{13}v_{41,2}^{\lambda}=v_{49,1}^{\lambda},E_{12}v_{46,1}^{\lambda}=-v_{49,1}^{\lambda},E_{25}v_{41,2}^{\lambda}=-v_{54,1}^{\lambda},E_{35}v_{46,1}^{\lambda}=-v_{54,1}^{\lambda}.$
Consider the equation

$t_{35,45}=-v_{26,3}^{\lambda}-v_{26,4}^{\lambda}+E_{53}v_{41,2}^{\lambda}-E_{52}v_{46,1}^{\lambda}+E_{51}v_{49,1}^{\lambda},$
$None$

For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{46}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{50}^{2\omega_{1}+\omega_{4}},$
the equation $E_{25}t_{35,45}=0$ implies
$2+\lambda_{2}+\lambda_{3}+\lambda_{4}=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{49}^{2\omega_{1}+\omega_{4}},\overrightarrow{w}_{53}^{2\omega_{1}+\omega_{4}},$
the equation $E_{15}t_{35,45}=0$ implies $3+|\lambda|=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{54}^{2\omega_{1}+\omega_{4}},$ the
equation $E_{35}E_{25}t_{35,45}=0$ implies
$(1+\lambda_{2})v_{54,1}^{\lambda}=0$. For
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{55}^{2\omega_{1}+\omega_{4}},$ the
equation $E_{35}E_{15}t_{35,45}=0$ implies $\lambda_{1}+\lambda_{2}+2=0$.

Case 14
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{21}^{2\omega_{1}+\omega_{4}}$

We have $v_{21,3}^{\lambda}+2v_{21,4}^{\lambda}+E_{45}v_{15,1}^{\lambda}=0$.
And the equations $t_{25,45}=0$ and $t_{24,45}=0$ induce that
$v_{21,3}^{\lambda}+v_{21,4}^{\lambda}=0$, $v_{15,1}^{\lambda}=0$. Thus,
$v_{21,3}^{\lambda}=v_{21,4}^{\lambda}=0$. Then

$0=E_{25}.t_{24,25}=E_{25}.[-v_{4,1}^{\lambda}+E_{53}v_{17,1}^{\lambda}-E_{52}(2v_{21,1}^{\lambda}+v_{21,2}^{\lambda})]=0,$
$None$
$0=E_{15}.t_{14,25}=E_{15}[-v_{2,1}^{\lambda}+E_{53}v_{12,1}^{\lambda}-E_{52}(v_{18,1}^{\lambda}+v_{18,2}^{\lambda}-v_{18,3}^{\lambda})+E_{51}v_{21,1}^{\lambda}]=0$
$None$

yield

$2(1+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{21,1}^{\lambda}+(2+\lambda_{2}+\lambda_{3}+\lambda_{4})v_{21,2}^{\lambda}=0,\
(2+|\lambda|)v_{21,1}^{\lambda}=0$ $None$

i.e. $v_{21,i}^{\lambda}=0$ for $i\in\overline{1,4}$. A contradiction arises.

Case 15
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{26}^{2\omega_{1}+\omega_{4}}$

Note that

$t_{24,35}=-v_{8,1}^{\lambda}+E_{53}\sum\limits_{i=1}^{4}v_{21,i}^{\lambda}-E_{52}(v_{26,2}^{\lambda}+\sum\limits_{i=1}^{4}v_{26,i}^{\lambda}).$
$None$

We have
$v_{26,2}^{\lambda}+v_{26,3}^{\lambda}+2v_{26,4}^{\lambda}+E_{45}v_{19,1}^{\lambda}=0$,
$v_{26,2}^{\lambda}+v_{26,3}^{\lambda}+E_{23}v_{21,3}^{\lambda}=0$ and
$v_{26,4}^{\lambda}+E_{23}v_{21,4}^{\lambda}=0$. And the equations
$t_{35,45}=0$ , $t_{34,45}=0$ and $t_{25,45}=0$ induce that
$v_{26,3}^{\lambda}+v_{26,4}^{\lambda}=0$, $v_{19,1}^{\lambda}=0$ and
$v_{26,2}^{\lambda}+v_{26,3}^{\lambda}+v_{26,4}^{\lambda}=0$. Thus,
$v_{26,i}^{\lambda}=0$ for $i\in\overline{2,4}$. The equation $t_{24,35}=0$
induce that

$E_{25}.[-v_{8,1}^{\lambda}+E_{53}(v_{21,1}^{\lambda}+v_{21,2}^{\lambda})-E_{52}v_{26,1}^{\lambda}]=0,$
$None$

which means $\lambda_{2}+\lambda_{3}+\lambda_{4}=0$. A contradiction arises.

Case 16
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{32}^{2\omega_{1}+\omega_{4}}$

Note that
$v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+E_{14}v_{18,3}^{\lambda}=0,v_{32,1}^{\lambda}+v_{32,4}^{\lambda}+E_{14}v_{18,4}^{\lambda}=0,v_{32,1}^{\lambda}-2v_{32,3}^{\lambda}-v_{32,4}^{\lambda}+E_{24}v_{21,3}^{\lambda}=0,v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+E_{24}v_{21,4}^{\lambda}=0,2v_{32,1}^{\lambda}+E_{34}v_{26,3}^{\lambda}=0,v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+E_{34}v_{26,4}^{\lambda}=0.$
And the equations follow:

$t_{25,45}=v_{21,3}^{\lambda}+v_{21,4}^{\lambda}=0,t_{35,45}=v_{26,3}^{\lambda}+v_{26,4}^{\lambda}=0,$
$t_{34,45}=-v_{19,1}^{\lambda}+E_{53}v_{32,3}^{\lambda}=0,t_{15,45}=-v_{18,3}^{\lambda}-v_{18,4}^{\lambda}+E_{53}v_{30,1}^{\lambda}=0.$
$None$

Thus,

$-2v_{32,1}^{\lambda}+2v_{32,3}^{\lambda}-v_{32,2}^{\lambda}+v_{32,4}^{\lambda}=0,\
3v_{32,1}^{\lambda}+v_{32,2}^{\lambda}=0,v_{32,2}^{\lambda}+(\lambda_{3}+\lambda_{4}-1)v_{32,3}^{\lambda}=0,2v_{32,1}^{\lambda}+v_{32,2}^{\lambda}+v_{32,4}^{\lambda}=0.$
$None$

yield the contradiction:$2+\lambda_{3}+\lambda_{4}=0$.

Case 17
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{41}^{2\omega_{1}+\omega_{4}}$

Observe that
$-v_{41,1}^{\lambda}-v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+v_{41,4}^{\lambda}+E_{24}v_{28,1}^{\lambda}=0,v_{41,2}^{\lambda}+2v_{41,3}^{\lambda}+v_{41,4}^{\lambda}+E_{34}v_{34,1}^{\lambda}=0,-v_{41,1}^{\lambda}-v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+E_{25}v_{21,3}^{\lambda}=0,v_{41,1}^{\lambda}+v_{41,2}^{\lambda}+v_{41,4}^{\lambda}+E_{25}v_{21,4}^{\lambda}=0,v_{41,1}^{\lambda}+2v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+E_{23}v_{35,1}^{\lambda}=0,2v_{41,3}^{\lambda}+E_{35}v_{26,3}^{\lambda}=0,-v_{41,2}^{\lambda}-v_{41,3}^{\lambda}+v_{41,4}^{\lambda}+E_{35}v_{26,4}^{\lambda}=0.$
Then

$0=E_{25}t_{25,45}=E_{25}[-v_{21,3}^{\lambda}-v_{21,4}^{\lambda}+E_{53}v_{35,1}^{\lambda}-E_{52}(v_{41,1}^{\lambda}+v_{41,2}^{\lambda}+v_{41,3}^{\lambda})]=0,$
$None$
$0=E_{35}t_{35,45}=E_{35}[-v_{26,3}^{\lambda}-v_{26,4}^{\lambda}+E_{53}v_{41,2}^{\lambda}]=0.$
$None$

which mean

$v_{41,4}^{\lambda}-v_{41,1}^{\lambda}-2v_{41,2}^{\lambda}=(\sum\limits_{i=2}^{4}\lambda_{i})(\sum\limits_{i=1}^{3}v_{41,i}^{\lambda}),\
v_{41,3}^{\lambda}-v_{41,2}^{\lambda}+v_{41,4}^{\lambda}+(\lambda_{3}+\lambda_{4})v_{41,2}^{\lambda}).$
$None$

We claim that one of $v_{28,1}^{\lambda}$ and $v_{34,1}^{\lambda}$ should be
nonzero. Otherwise, the equation implies:
$-v_{41,1}^{\lambda}-v_{41,2}^{\lambda}+v_{41,3}^{\lambda}+v_{41,4}^{\lambda}=0$
and $v_{41,2}^{\lambda}+2v_{41,3}^{\lambda}+v_{41,4}^{\lambda}=0$, which
provide a contradiction. Now,
$0=E_{15}t_{15,35}=E_{15}(E_{53}v_{24,1}^{\lambda}-E_{51}v_{34,1}^{\lambda})=E_{13}v_{24,1}^{\lambda}-\sum\limits_{i=1}^{4}h_{i}v_{34,1}^{\lambda}=0$
and
$0=E_{15}t_{15,25}=E_{15}(E_{52}v_{24,1}^{\lambda}-E_{51}v_{28,1}^{\lambda})=E_{12}v_{24,1}^{\lambda}-\sum\limits_{i=1}^{4}h_{i}v_{34,1}^{\lambda}=0$.
Thus $1+|\lambda|=0$. A contradiction arises.

Case 18
$wt(\xi_{0,3})=\lambda+\overrightarrow{w}_{18}^{2\omega_{1}+\omega_{4}}$

Note that
$E_{15}v_{1,1}^{\lambda}=v_{18,1}^{\lambda}-2v_{18,3}^{\lambda},E_{25}v_{2,1}^{\lambda}=-v_{18,1}^{\lambda}-v_{18,2}^{\lambda}-v_{18,3}^{\lambda},E_{13}v_{7,1}^{\lambda}=v_{18,1}^{\lambda}-2v_{18,2}^{\lambda}-2v_{18,4}^{\lambda},E_{13}v_{3,1}^{\lambda}=-2v_{11,1}^{\lambda},E_{23}v_{12,1}^{\lambda}=-v_{18,1}^{\lambda}-2v_{18,2}^{\lambda}-v_{18,4}^{\lambda},E_{34}v_{11,1}^{\lambda}=-v_{18,2}^{\lambda}-v_{18,3}^{\lambda}-2v_{18,4}^{\lambda}.$
We have $v_{11,1}^{\lambda}=0$. Otherwise,
$0=E_{15}t_{13,15}=-(2+|\lambda|)v_{11,1}^{\lambda}=0$. Hence,

$0=-v_{18,2}^{\lambda}-v_{18,3}^{\lambda}-2v_{18,4}^{\lambda}=E_{34}v_{11,1}^{\lambda}.$
$None$

Moreover,

$0=t_{15,45}=v_{18,3}^{\lambda}+v_{18,4}^{\lambda},0=E_{15}t_{14,15}=3v_{18,1}^{\lambda}+2v_{18,3}^{\lambda}-2v_{18,4}^{\lambda}+|\lambda|(v_{18,1}^{\lambda}-v_{18,4}^{\lambda}).$
$None$

Observe that

$t_{14,25}=-v_{2,1}^{\lambda}+E_{53}v_{12,1}^{\lambda}-E_{52}(v_{18,1}^{\lambda}+v_{18,2}^{\lambda}-v_{18,3}^{\lambda})+E_{51}(v_{21,1}^{\lambda}-v_{21,3}^{\lambda}-v_{21,4}^{\lambda})=0.$
$None$

The equation $E_{25}t_{14,25}=0$ induces

$-v_{18,2}^{\lambda}+v_{18,3}^{\lambda}-v_{18,4}^{\lambda}-(\lambda_{2}+\lambda_{3}+\lambda_{4})(v_{18,1}^{\lambda}+v_{18,2}^{\lambda}-v_{18,3}^{\lambda})=0.$
$None$

All these equations yield the contradiction:

$\frac{3+|\lambda|}{4+|\lambda|}=-(\lambda_{2}+\lambda_{3}+\lambda_{4}).$
$None$

$\Box$

###  5.4 Singular vectors of degree four

Theorem 5.6 All the possible degree four singular vectors are listed in the
following:

$d_{12}d_{13}d_{14}d_{15}v_{\lambda},\ \lambda=(m,0,0,0),\ m\in\mathbb{N}.$

Proof The leading term of any singular vector of degree four could be written
as:

$\xi_{0,4}=\sum\limits_{j\in\overline{1,35}}v_{j}^{3\omega_{1}}v_{j}^{\lambda},$
$None$

which should satisfy $T_{5,123}.\xi_{0,4}=0$. Since

$[T_{5,123}-(E_{53}\partial_{y_{12}}+E_{52}(-1)^{1+|13|}\partial_{y_{13}}+E_{51}(-1)^{|23|}\partial_{y_{23}}),(x_{i}\partial_{x_{j}})_{0}^{\prime}]=0$
$None$

for $1\leq j<i\leq 5$, we have

$[T_{5,123}-(E_{53}\partial_{y_{12}}+E_{52}(-1)^{1+|13|}\partial_{y_{13}}+E_{51}(-1)^{|23|}\partial_{y_{23}})].|_{V(3\omega_{1})\otimes
V(\lambda)}=0.$ $None$

Hence,

$T_{5,123}.\xi_{0,4}=[E_{53}\partial_{y_{12}}+E_{52}(-1)^{1+|13|}\partial_{y_{13}}+E_{51}(-1)^{|23|}\partial_{y_{23}}].\xi_{0,4}=0.$
$None$

Case 1
$wt(\xi_{0,4})=\overrightarrow{w}_{1}^{3\omega_{1}}=\mbox{wt}(d_{12}d_{13}d_{14}d_{15}v_{\lambda})$

The vector $d_{12}d_{13}d_{14}d_{15}v_{\lambda}$ is singular iff
$\lambda=(m,0,0,0)$.

Case 2
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{2}^{3\omega_{1}},\lambda+\overrightarrow{w}_{3}^{3\omega_{1}},\lambda+\overrightarrow{w}_{5}^{3\omega_{1}},\lambda+\overrightarrow{w}_{6}^{3\omega_{1}},\lambda+\overrightarrow{w}_{8}^{3\omega_{1}},\lambda+\overrightarrow{w}_{10}^{3\omega_{1}},\lambda+\overrightarrow{w}_{12}^{3\omega_{1}},\lambda+\overrightarrow{w}_{13}^{3\omega_{1}},\lambda+\overrightarrow{w}_{15}^{3\omega_{1}},\lambda+\overrightarrow{w}_{16}^{3\omega_{1}},\lambda+\overrightarrow{w}_{17}^{3\omega_{1}},\lambda+\overrightarrow{w}_{22}^{3\omega_{1}},\lambda+\overrightarrow{w}_{23}^{3\omega_{1}},\lambda+\overrightarrow{w}_{27}^{3\omega_{1}},\lambda+\overrightarrow{w}_{30}^{3\omega_{1}}\\}$

In these cases, the equations are derived:
$E_{51}v_{i}^{\lambda}-E_{52}v_{j}^{\lambda}=0,\
E_{12}v_{j}^{\lambda}=-s_{ij}v_{i}^{\lambda},\
\mbox{wt}(v_{i}^{\lambda})=\lambda,(i,j,s_{ij})\in\\{(2,1,3),(3,2,2),(5,3,1),(8,6,1),(10,7,2),(12,9,1),(13,10,1),(15,11,2),(16,14,1),(17,15,1)\\}.$
Then, the equation
$E_{15}.(E_{51}v_{i}^{\lambda}-E_{52}v_{j}^{\lambda})=(|\lambda|+s_{ij})v_{i}^{\lambda}=0$
yields a contradiction.

Case 3
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{4}^{3\omega_{1}},\lambda+\overrightarrow{w}_{9}^{3\omega_{1}},\lambda+\overrightarrow{w}_{14}^{3\omega_{1}},\lambda+\overrightarrow{w}_{19}^{3\omega_{1}},\lambda+\overrightarrow{w}_{20}^{3\omega_{1}},\lambda+\overrightarrow{w}_{21}^{3\omega_{1}},\lambda+\overrightarrow{w}_{25}^{3\omega_{1}},\lambda+\overrightarrow{w}_{26}^{3\omega_{1}},\lambda+\overrightarrow{w}_{29}^{3\omega_{1}},\lambda+\overrightarrow{w}_{33}^{3\omega_{1}}\\}$

In these cases, we derive that:
$E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda}=0,\
E_{13}v_{j}^{\lambda}=-t_{ij}v_{i}^{\lambda},\
\mbox{wt}(v_{i}^{\lambda})=\lambda,(i,j,t_{ij})\in\\{(4,1,3),(9,4,2),(14,7,2),(19,11,2),(20,9,1),(21,14,1),(25,18,1),(26,19,1),(29,24,1),(33,28,1)\\}.$Then,
$E_{15}.(E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda})=(|\lambda|+t_{ij})v_{i}^{\lambda}=0$
yields a contradiction.

Case 4
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{7}^{3\omega_{1}},\lambda+\overrightarrow{w}_{11}^{3\omega_{1}},\lambda+\overrightarrow{w}_{18}^{3\omega_{1}},\lambda+\overrightarrow{w}_{24}^{3\omega_{1}},\lambda+\overrightarrow{w}_{28}^{3\omega_{1}}\\}$

In these cases, we have $\lambda_{3}+\lambda_{4}>0$. And
$E_{52}v_{\lambda}=E_{53}v_{\lambda}=0$, i.e. $\lambda=(m,0,0,0)$. A
contradiction arises.

Case 5
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{31}^{3\omega_{1}},\lambda+\overrightarrow{w}_{32}^{3\omega_{1}},\lambda+\overrightarrow{w}_{34}^{3\omega_{1}},\lambda+\overrightarrow{w}_{35}^{3\omega_{1}}\\}$

Case 5.1
$wt(\xi_{0,4})\in\\{\lambda+\overrightarrow{w}_{31}^{3\omega_{1}},\lambda+\overrightarrow{w}_{32}^{3\omega_{1}},\lambda+\overrightarrow{w}_{34}^{3\omega_{1}}\\}$

In these cases, the following equations are derived:
$E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda}=0,\
E_{13}v_{j}^{\lambda}=-q_{ij}v_{i}^{\lambda},\
\mbox{wt}(v_{i}^{\lambda})=\lambda-\alpha_{3},\mbox{wt}(v_{j}^{\lambda})=\lambda-\alpha_{1}-\alpha_{2}-\alpha_{3},(i,j,t_{ij})\in\\{(25,18,1),(30,24,1),(34,28,1).$
Then,
$E_{15}.(E_{51}v_{i}^{\lambda}-E_{53}v_{j}^{\lambda})=(|\lambda|+q_{ij})v_{i}^{\lambda}=0$
yields a contradiction.

Case 5.2 $wt(\xi_{0,4})=\lambda+\overrightarrow{w}_{35}^{3\omega_{1}}$

Note that $E_{51}v_{33}^{\lambda}-E_{53}v_{28}^{\lambda}=0,\
E_{51}v_{30}^{\lambda}-E_{52}v_{28}^{\lambda}=0,\
\mbox{wt}(v_{33}^{\lambda})=\lambda-\alpha_{3}-\alpha_{4},\
\mbox{wt}(v_{30}^{\lambda})=\lambda-\alpha_{2}-\alpha_{3}-2\alpha_{4},\
\mbox{wt}(v_{28}^{\lambda})=\lambda-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4},E_{13}v_{28}^{\lambda}=-v_{33}^{\lambda},E_{15}v_{28}^{\lambda}=-v_{35}^{\lambda},E_{35}v_{33}^{\lambda}=-v_{35}^{\lambda},E_{12}v_{28}^{\lambda}=-v_{30}^{\lambda},E_{25}v_{30}^{\lambda}=-v_{35}^{\lambda}.$
Then

$E_{35}E_{15}(E_{51}v_{33}^{\lambda}-E_{53}v_{28}^{\lambda})=0=E_{35}(|\lambda|v_{33}^{\lambda}+E_{53}v_{35}^{\lambda})=-(\lambda_{1}+\lambda_{2})v_{35}^{\lambda}$
$None$

induces $\lambda_{1}=\lambda_{2}=0$. And

$E_{35}E_{15}(E_{51}v_{30}^{\lambda}-E_{52}v_{28}^{\lambda})=0=E_{35}[(|\lambda|-1)v_{30}^{\lambda}+E_{52}v_{35}^{\lambda}]=(1-\lambda_{1})v_{35}^{\lambda}$
$None$

induces $\lambda_{1}=1$. A contradiction arises. $\Box$

Acknowledgement: Part of this work was completed while the author was visiting
the Massachusetts Institute of Technology in 2011. Se is very grateful to all
of the faculty and the staff members of the institution for their hospitality
and support during her visit. The author would like to thank Professor Victor
Kac for his introducing this interesting conjecture that motivates this work.
The author would also like to thank Professor Dihua Jiang for his
encouragements. Finally, the author acknowledges the financial support by CSC.

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Table 1: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(\omega_{2})$ $i$ |  $\overrightarrow{w}_{i}^{\omega_{2}}$ | $v_{i}^{\omega_{2}}$ | $i$ |  $\overrightarrow{w}_{i}^{\omega_{2}}$ | $v_{i}^{\omega_{2}}$
---|---|---|---|---|---
1 |  $(0,1,0,0)=\omega_{2}$ | $d_{12}$ | 6 |  $(0,-1,0,1)=\omega_{2}-\alpha_{1}-2\alpha_{2}-\alpha_{3}$ | $d_{34}$
2 |  $(1,-1,1,0)=\omega_{2}-\alpha_{2}$ | $d_{13}$ | 7 |  $(1,0,0,-1)=\omega_{2}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{15}$
3 |  $(-1,0,1,0)=\omega_{2}-\alpha_{1}-\alpha_{2}$ | $d_{23}$ | 8 |  $(-1,1,0,-1)=\omega_{2}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{25}$
4 |  $(1,0,-1,1)=\omega_{2}-\alpha_{2}-\alpha_{3}$ | $d_{14}$ | 9 |  $(0,-1,1,-1)=\omega_{2}-\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ | $d_{35}$
5 |  $(-1,1,-1,1)=\omega_{2}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ | $d_{24}$ | 10 |  $(0,0,-1,0)=\omega_{2}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ | $d_{45}$
Table 2: Decomposition for wedge module $\Lambda^{k}W$ ($k\in\overline{1,10}$) $\Lambda^{k}W$ |  irreducible components for $\Lambda^{k}W$ |  maximal vector for the irreducible components
---|---|---
$\wedge^{1}W$ |  $V(\omega_{2})$ |  $d_{12}$
$\wedge^{2}W$ |  $V(\omega_{1}+\omega_{3})$ |  ${d_{12}}\wedge{d_{13}}$
$\wedge^{3}W$ |  $V(2\omega_{3})\oplus V(2\omega_{1}+\omega_{4})$ |  ${d_{12}}\wedge{d_{13}}\wedge{d_{23}},{d_{12}}\wedge{d_{13}}\wedge{d_{14}}$
$\wedge^{4}W$ |  $V(3\omega_{1})\oplus V(\omega_{1}+\omega_{3}+\omega_{4})$ |  ${d_{12}}\wedge{d_{13}}\wedge{d_{14}}\wedge{d_{15}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}$
$\wedge^{5}W$ |  $V(2\omega_{1}+\omega_{3})\oplus V(\omega_{2}+2\omega_{4})$ |  ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{15}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}$
$\wedge^{6}W$ |  $V(3\omega_{4})\oplus V(\omega_{1}+\omega_{2}+\omega_{4})$ |  ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{15}}$
$\wedge^{7}W$ |  $V(\omega_{1}+2\omega_{4})\oplus V(2\omega_{2})$ |  ${d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}},{d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{15}}\wedge{d_{25}}$
$\wedge^{8}W$ |  $V(\omega_{2}+\omega_{4})$ |  ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}}\wedge{d_{25}}$
$\wedge^{9}W$ |  $V(\omega_{3})$ |  ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}}\wedge{d_{25}}\wedge{d_{35}}$
$\wedge^{10}W$ |  $V(0)$ |  ${d_{12}}\wedge{d_{13}}\wedge{d_{23}}\wedge{d_{14}}\wedge{d_{24}}\wedge{d_{34}}\wedge{d_{15}}\wedge{d_{25}}\wedge{d_{35}}\wedge{d_{45}}$
Table 3: Tensor decomposition for $V(k\omega_{4})\otimes V(\mu)$ $\mu$ |  highest weight in the decomposition $V(k\omega_{4})\otimes V(\mu)$
---|---
$\omega_{2}$ |  $\omega_{(0,1,0,k)},\omega_{(1,0,0,k-1)}$
$\omega_{1}+\omega_{3}$ |  $\omega_{(1,0,1,k)},\omega_{(0,0,1,k-1)},\omega_{(1,1,0,k-1)},\omega_{(0,1,0,k-2)}$
$2\omega_{3}$ |  $\omega_{(0,0,2,k)},\omega_{(0,1,1,k-1)},\omega_{(0,2,0,k-2)}$
$2\omega_{1}+\omega_{4}$ |  $\omega_{(2,0,0,k+1)},\omega_{(1,0,0,k)},\omega_{(0,0,0,k-1)},\omega_{(2,0,1,k-1)},\omega_{(1,0,1,k-2)},\omega_{(0,0,1,k-3)}$
$\omega_{1}+\omega_{3}+\omega_{4}$ |  $\omega_{(1,0,1,k+1)},\omega_{(0,0,1,k)},\omega_{(1,1,0,k)},\omega_{(0,1,0,k-1)},\omega_{(1,0,2,k-1)},\omega_{(0,0,2,k-2)},\omega_{(1,1,1,k-2)},\omega_{(0,1,1,k-3)}$
$3\omega_{1}$ |  $\omega_{(3,0,0,k)},\omega_{(2,0,0,k-1)},\omega_{(1,0,0,k-2)},\omega_{(0,0,0,k-3)}$
$\omega_{2}+2\omega_{4}$ |  $\omega_{(0,1,0,k+2)},\omega_{(1,0,0,k+1)},\omega_{(0,1,1,k)},\omega_{(1,0,1,k-1)},\omega_{(0,1,2,k-2)},\omega_{(1,0,2,k-3)}$
$2\omega_{1}+\omega_{3}$ |  $\omega_{(2,0,1,k)},\omega_{(1,0,1,k-1)},\omega_{(0,0,1,k-2)},\omega_{(2,1,0,k-1)},\omega_{(1,1,0,k-2)},\omega_{(0,1,0,k-3)}$
$3\omega_{4}$ |  $\omega_{(0,0,0,k+3)},\omega_{(0,0,1,k+1)},\omega_{(0,0,2,k-1)},\omega_{(0,0,3,k-3)}$
$\omega_{1}+\omega_{2}+\omega_{4}$ |  $\omega_{(1,1,0,k+1)},\omega_{(0,1,0,k)},\omega_{(2,0,0,k)},\omega_{(1,0,0,k-1)},\omega_{(1,1,1,k-1)},\omega_{(0,1,1,k-2)},\omega_{(2,0,1,k-2)},\omega_{(1,0,1,k-3)}$
$\omega_{1}+2\omega_{4}$ |  $\omega_{(1,0,0,k+2)},\omega_{(0,0,0,k+1)},\omega_{(1,0,1,k)},\omega_{(0,0,1,k-1)},\omega_{(1,0,2,k-2)},\omega_{(0,0,2,k-3)}$
$2\omega_{2}$ |  $\omega_{(0,2,0,k)},\omega_{(1,1,0,k-1)},\omega_{(2,0,0,k-2)}$
$\omega_{2}+\omega_{4}$ |  $\omega_{(0,1,0,k+1)},\omega_{(1,0,0,k)},\omega_{(0,1,1,k-1)},\omega_{(1,0,1,k-2)},$
$\omega_{3}$ |  $\omega_{(0,0,1,k)},\omega_{(0,1,0,k-1)}$
Table 4: The leading term $l_{\mu}$ $\mu$ |  $l_{\mu}$
---|---
$\omega_{2}$ |  $d_{12},d_{15}$
$\omega_{1}+\omega_{3}$ |  $d_{12}\wedge d_{13},2d_{12}\wedge d_{35}+d_{23}\wedge d_{15}-d_{13}\wedge d_{25},d_{12}\wedge d_{15},d_{15}\wedge d_{25}$
$2\omega_{3}$ |  $d_{12}\wedge d_{13}\wedge d_{23},d_{12}\wedge d_{13}\wedge d_{25}-d_{12}\wedge d_{23}\wedge d_{15},d_{12}\wedge d_{15}\wedge d_{25}$
$2\omega_{1}+\omega_{4}$ |  $d_{12}\wedge d_{13}\wedge d_{14},d_{15}\wedge d_{25}\wedge d_{35},2d_{12}\wedge d_{13}\wedge d_{45}-2d_{12}\wedge d_{14}\wedge d_{35}+2d_{13}\wedge d_{14}\wedge d_{25}-d_{13}\wedge d_{24}\wedge d_{15}-d_{23}\wedge d_{14}\wedge d_{15}+d_{12}\wedge d_{34}\wedge d_{15},d_{12}\wedge d_{35}\wedge d_{45}-d_{13}\wedge d_{25}\wedge d_{45}+d_{23}\wedge d_{15}\wedge d_{45}+d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}+d_{34}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{15},d_{12}\wedge d_{15}\wedge d_{35}-d_{13}\wedge d_{15}\wedge d_{25}$
$\omega_{1}+\omega_{3}+\omega_{4}$ |  $d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14},d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15},2d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{45}-d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{35}-d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{15}-d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{25}+d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{15},d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15},d_{12}\wedge d_{13}\wedge d_{25}\wedge d_{45}-d_{12}\wedge d_{23}\wedge d_{15}\wedge d_{45}-d_{12}\wedge d_{14}\wedge d_{25}\wedge d_{35}+d_{12}\wedge d_{24}\wedge d_{15}\wedge d_{35}-d_{12}\wedge d_{34}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{25}\wedge d_{35}-d_{12}\wedge d_{23}\wedge d_{15}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{15}\wedge d_{25}\wedge d_{35}$
$3\omega_{1}$ |  $d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{15},d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{45}-d_{12}\wedge d_{14}\wedge d_{15}\wedge d_{35}+d_{13}\wedge d_{14}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{15}\wedge d_{35}\wedge d_{45}-d_{13}\wedge d_{15}\wedge d_{25}\wedge d_{45}+d_{14}\wedge d_{15}\wedge d_{25}\wedge d_{35},d_{15}\wedge d_{25}\wedge d_{35}\wedge d_{45}$
$\omega_{2}+2\omega_{4}$ |  $d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{24},2d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{45}+2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{35}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{34}\wedge d_{15}-d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{24}\wedge d_{15},d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{25}-d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{15},3d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{35}-3d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{25}+4d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{45}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}\wedge d_{35}-d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{25}\wedge d_{35}$
$2\omega_{1}+\omega_{3}$ |  $d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15},4d_{12}\wedge d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{45}-2d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}\wedge d_{35}-d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{35}+3d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{35}-3d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{15}\wedge d_{25},2d_{12}\wedge d_{13}\wedge d_{25}\wedge d_{35}\wedge d_{45}-2d_{12}\wedge d_{23}\wedge d_{15}\wedge d_{35}\wedge d_{45}+d_{12}\wedge d_{34}\wedge d_{15}\wedge d_{25}\wedge d_{35}+2d_{13}\wedge d_{23}\wedge d_{15}\wedge d_{25}\wedge d_{45}-d_{13}\wedge d_{24}\wedge d_{15}\wedge d_{25}\wedge d_{35}+d_{23}\wedge d_{14}\wedge d_{15}\wedge d_{25}\wedge d_{35},d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{15}\wedge d_{25},d_{12}\wedge d_{15}\wedge d_{25}\wedge d_{35}\wedge d_{45}$
$3\omega_{4}$ |  ${\hat{d}_{15}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{24}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}}+{\hat{d}_{14}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{45}}-{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},$
$\omega_{1}+\omega_{2}+\omega_{4}$ |  ${\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{12}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}}-{\hat{d}_{13}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}+{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{45}}+{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{35}}-{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}},{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}}-{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}},{\hat{d}_{12}}\wedge{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}}+{\hat{d}_{12}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}-{\hat{d}_{13}}\wedge{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{45}}-{\hat{d}_{13}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}+{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}}-{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}+2{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{13}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}}+2{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}-{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}}$
$\omega_{1}+2\omega_{4}$ |  ${\hat{d}_{25}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{34}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{25}}-{\hat{d}_{24}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{35}}+{\hat{d}_{14}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{35}}-{\hat{d}_{23}}\wedge{\hat{d}_{15}}\wedge{\hat{d}_{45}}+{\hat{d}_{13}}\wedge{\hat{d}_{25}}\wedge{\hat{d}_{45}}-{\hat{d}_{12}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{24}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},2{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{15}}-2{\hat{d}_{14}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{25}}-{\hat{d}_{23}}\wedge{\hat{d}_{14}}\wedge{\hat{d}_{45}}+{\hat{d}_{13}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{45}}-{\hat{d}_{12}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}}+2{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{35}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{14}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}},$
$2\omega_{2}$ |  ${\hat{d}_{34}}\wedge{\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{35}}+{\hat{d}_{23}}\wedge{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{24}}\wedge{\hat{d}_{34}},$
$\omega_{2}+\omega_{4}$ |  ${\hat{d}_{35}}\wedge{\hat{d}_{45}},{\hat{d}_{23}}\wedge{\hat{d}_{45}}+{\hat{d}_{24}}\wedge{\hat{d}_{35}}-{\hat{d}_{34}}\wedge{\hat{d}_{25}},{\hat{d}_{34}}\wedge{\hat{d}_{45}},{\hat{d}_{24}}\wedge{\hat{d}_{34}}$
$\omega_{3}$ |  ${\hat{d}_{45}},{\hat{d}_{34}}$
Table 5: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(\omega_{1}+\omega_{3})$ $(i,j)$ |  $\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}$ |  $v_{i,j}^{\omega_{1}+\omega_{3}}$ | $(i,j)$ |  $\overrightarrow{w}_{i}^{\omega_{1}+\omega_{3}}$ |  $v_{i,j}^{\omega_{1}+\omega_{3}}$
---|---|---|---|---|---
(1,1) |  $(1,0,1,0)=\omega_{1}+\omega_{3}$ |  $d_{12}\wedge d_{13}$ | (18,1) |  $(-1,-1,1,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}$ |  $d_{23}\wedge d_{34}$
(2,1) |  $(-1,1,1,0)=\omega_{1}+\omega_{3}-\alpha_{1}$ |  $d_{12}\wedge d_{23}$ | (19,1) |  $(-2,1,1,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{25}$
(3,1) |  $(1,1,-1,1)=\omega_{1}+\omega_{3}-\alpha_{3}$ |  $d_{12}\wedge d_{14}$ | (20,1) |  $(1,-1,-1,2)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ |  $d_{14}\wedge d_{34}$
(4,1) |  $(-1,2,-1,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{3}$ |  $d_{12}\wedge d_{24}$ | (21,1) |  $(1,-1,0,0)==\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{35}-d_{13}\wedge d_{45}$
(5,1) |  $(1,1,0,-1)=\omega_{1}+\omega_{3}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{15}$ | (21,2) |  $(1,-1,0,0)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{35}-d_{34}\wedge d_{15}$
(6,1) |  $(0,-1,2,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}$ |  $d_{13}\wedge d_{23}$ | (21,3) |  $(1,-1,0,0)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{34}\wedge d_{15}+d_{13}\wedge d_{45}$
(7,1) |  $(2,-1,0,1)=\omega_{1}+\omega_{3}-\alpha_{2}-\alpha_{3}$ |  $d_{13}\wedge d_{14}$ | (22,1) |  $(-1,-1,2,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{35}$
(8,1) |  $(0,0,0,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $d_{12}\wedge d_{34}-d_{13}\wedge d_{24}$ | $(23,1)$ |  $(-1,0,-1,2)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ |  $d_{24}\wedge d_{34}$
(8,2) |  $(0,0,0,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $d_{12}\wedge d_{34}+d_{23}\wedge d_{14}$ | (24,1) |  $(-2,2,-1,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{24}\wedge d_{25}$
(8,3) |  $(0,0,0,1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $d_{13}\wedge d_{24}-d_{23}\wedge d_{14}$ | (25,1) |  $(0,1,0,-2)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{15}\wedge d_{25}$
(9,1) |  $(-1,2,0,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{25}$ | (26,1) |  $(-1,0,0,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{34}\wedge d_{25}-d_{24}\wedge d_{35}$
(10,1) |  $(2,-1,1,-1)=\omega_{1}+\omega_{3}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{15}$ | (26,2) |  $(-1,0,0,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{45}-d_{24}\wedge d_{35}$
(11,1) |  $(0,0,1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{35}-d_{13}\wedge d_{25}$ | (26,3) |  $(-1,0,0,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{34}\wedge d_{25}+d_{23}\wedge d_{45}$
(11,2) |  $(0,0,1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{35}+d_{23}\wedge d_{15}$ | (27,1) |  $(1,0,-2,1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-3\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{45}$
(11,3) |  $(0,0,1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{15}-d_{13}\wedge d_{25}$ | (28,1) |  $(1,-1,1,-2)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{15}\wedge d_{35}$
(12,1) |  $(1,-2,1,1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-\alpha_{3}$ |  $d_{13}\wedge d_{34}$ | (29,1) |  $(-1,1,-2,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-\alpha_{4}$ |  $d_{24}\wedge d_{45}$
(13,1) |  $(-2,1,0,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $d_{23}\wedge d_{24}$ | (30,1) |  $(0,-2,1,0)=\omega_{1}+\omega_{3}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{34}\wedge d_{35}$
(14,1) |  $(0,1,-2,2)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}$ |  $d_{14}\wedge d_{24}$ | (31,1) |  $(1,0,-1,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ |  $d_{15}\wedge d_{45}$
(15,1) |  $(2,0,-1,0)=\omega_{1}+\omega_{3}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{15}$ | (32,1) |  $(-1,0,1,-2)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{25}\wedge d_{35}$
(16,1) |  $(0,1,-1,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{45}-d_{14}\wedge d_{25}$ | (33,1) |  $(0,-1,-1,1)=\omega_{1}+\omega_{3}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$ |  $d_{34}\wedge d_{45}$
(16,2) |  $(0,1,-1,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{45}+d_{24}\wedge d_{15}$ | (34,1) |  $(-1,1,-1,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ |  $d_{25}\wedge d_{45}$
(16,3) |  $(0,1,-1,0)=\omega_{1}+\omega_{3}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{25}-d_{24}\wedge d_{15}$ | (35,1) |  $(0,-1,0,-1)=\omega_{1}+\omega_{3}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ |  $d_{35}\wedge d_{45}$
(17,1) |  $(1,-2,2,-1)=\omega_{1}+\omega_{3}-\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{35}$ |  |  |
Table 6: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(2\omega_{1}+\omega_{4})$ $(i,j)$ | $\overrightarrow{w}_{i}^{2\omega_{1}+\omega_{4}}$ |  $v_{i,j}^{2\omega_{1}+\omega_{4}}$
---|---|---
(1,1) | $(2,0,0,1)=2\omega_{1}+\omega_{4}$ |  $d_{12}\wedge d_{13}\wedge d_{14}$
(2,1) | $(0,1,0,1)=2\omega_{1}+\omega_{4}-\alpha_{1}$ |  $d_{12}\wedge d_{23}\wedge d_{14}+d_{12}\wedge d_{13}\wedge d_{24}$
(3,1) | $(2,0,1,-1)=2\omega_{1}+\omega_{4}-\alpha_{4}$ |  $d_{12}\wedge d_{13}\wedge d_{15}$
(4,1) | $(-2,2,0,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}$ |  $d_{12}\wedge d_{23}\wedge d_{24}$
(5,1) | $(1,-1,1,1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}$ |  $d_{13}\wedge d_{23}\wedge d_{14}+d_{12}\wedge d_{13}\wedge d_{34}$
(6,1) | $(0,1,1,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{4}$ |  $d_{12}\wedge d_{23}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{25}$
(7,1) | $(2,1,-1,0)=2\omega_{1}+\omega_{4}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{14}\wedge d_{15}$
(8,1) | $(-1,0,1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}$ |  $d_{13}\wedge d_{23}\wedge d_{24}+d_{12}\wedge d_{23}\wedge d_{34}$
(9,1) | $(-2,2,1,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{4}$ |  $d_{12}\wedge d_{23}\wedge d_{25}$
(10,1) | $(1,0,-1,2)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $-d_{13}\wedge d_{14}\wedge d_{24}+d_{12}\wedge d_{14}\wedge d_{34}$
(11,1) | $(1,-1,2,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{4}$ |  $d_{13}\wedge d_{23}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{35}$
(12,1) | $(0,2,-1,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{25}$
(13,1) | $(3,-1,0,0)=2\omega_{1}+\omega_{4}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{14}\wedge d_{15}$
(14,1) | $(0,-2,2,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}$ |  $d_{13}\wedge d_{23}\wedge d_{34}$
(15,1) | $(-1,1,-1,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $d_{12}\wedge d_{24}\wedge d_{34}-d_{23}\wedge d_{14}\wedge d_{24}$
(16,1) | $(-1,0,2,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{4}$ |  $d_{12}\wedge d_{23}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{25}$
(17,1) | $(-2,3,-1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{24}\wedge d_{25}$
(18,1) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{24}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{25}+d_{23}\wedge d_{14}\wedge d_{15},$
(18,2) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{24}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{34}\wedge d_{15},$
(18,3) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{24}\wedge d_{15}-d_{13}\wedge d_{14}\wedge d_{25}-d_{12}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{35}$
(18,4) | $(1,0,0,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $-d_{23}\wedge d_{14}\wedge d_{15}+d_{13}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{45}$
(19,1) | $(0,-1,0,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}$ |  $d_{13}\wedge d_{24}\wedge d_{34}-d_{23}\wedge d_{14}\wedge d_{34}$
(20,1) | $(0,-2,3,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{4}$ |  $d_{13}\wedge d_{23}\wedge d_{35}$
(21,1) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{24}\wedge d_{35}+2d_{13}\wedge d_{24}\wedge d_{25}+d_{23}\wedge d_{14}\wedge d_{25}+d_{23}\wedge d_{24}\wedge d_{15}$
(21,2) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{34}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{25}$
(21,3) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $-d_{23}\wedge d_{14}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{23}\wedge d_{45}$
(21,4) | $(-1,1,0,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{24}\wedge d_{35}-d_{12}\wedge d_{34}\wedge d_{25}+d_{23}\wedge d_{24}\wedge d_{15}-d_{23}\wedge d_{14}\wedge d_{25}$
(22,1) | $(2,-2,1,0)=2\omega_{1}+\omega_{4}-\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{34}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{35}$
(23,1) | $(1,1,-2,1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{14}\wedge d_{45}+d_{14}\wedge d_{24}\wedge d_{15}$
(24,1) | $(1,0,1,-2)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-\alpha_{3}-2\alpha_{4}$ |  $d_{12}\wedge d_{15}\wedge d_{35}-d_{13}\wedge d_{15}\wedge d_{25}$
(25,1) | $(0,0,-2,3)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}$ |  $d_{14}\wedge d_{24}\wedge d_{34}$
(26,1) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{24}\wedge d_{35}+d_{13}\wedge d_{34}\wedge d_{25}+d_{23}\wedge d_{14}\wedge d_{35}+d_{23}\wedge d_{34}\wedge d_{15}$
(26,2) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $-d_{23}\wedge d_{14}\wedge d_{35}+d_{13}\wedge d_{34}\wedge d_{25}+2d_{13}\wedge d_{24}\wedge d_{35}+d_{12}\wedge d_{34}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{45}$
(26,3) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{24}\wedge d_{35}+d_{13}\wedge d_{23}\wedge d_{45}-d_{23}\wedge d_{14}\wedge d_{35}$
(26,4) | $(0,-1,1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{34}\wedge d_{15}-d_{23}\wedge d_{14}\wedge d_{35}-d_{13}\wedge d_{34}\wedge d_{25}+d_{13}\wedge d_{24}\wedge d_{35}$
(27,1) | $(-1,2,-2,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{24}\wedge d_{45}$
(28,1) | $(-1,1,1,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-\alpha_{3}-2\alpha_{4}$ |  $d_{12}\wedge d_{25}\wedge d_{35}-d_{23}\wedge d_{15}\wedge d_{25}$
(29,1) | $(2,-1,-1,1)=2\omega_{1}+\omega_{4}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{34}\wedge d_{15}+d_{13}\wedge d_{14}\wedge d_{45}$
(30,1) | $(1,1,-1,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{12}\wedge d_{15}\wedge d_{45}-d_{14}\wedge d_{15}\wedge d_{25}$
Table 7: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(2\omega_{1}+\omega_{4})$ $(i,j)$ | $\overrightarrow{w}_{i}^{2\omega_{1}+\omega_{4}}$ |  $v_{i,j}^{2\omega_{1}+\omega_{4}}$
---|---|---
(31,1) | $(-3,2,0,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{24}\wedge d_{25}$
(32,1) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{24}\wedge d_{45}+2d_{14}\wedge d_{24}\wedge d_{35}-d_{14}\wedge d_{34}\wedge d_{25}-d_{23}\wedge d_{14}\wedge d_{45}+d_{24}\wedge d_{34}\wedge d_{15}$
(32,2) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{24}\wedge d_{35}-d_{14}\wedge d_{34}\wedge d_{25}+d_{24}\wedge d_{34}\wedge d_{15}$
(32,3) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{12}\wedge d_{34}\wedge d_{45}+d_{13}\wedge d_{24}\wedge d_{45}+d_{14}\wedge d_{34}\wedge d_{25}+d_{14}\wedge d_{24}\wedge d_{35}$
(32,4) | $(0,0,-1,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{24}\wedge d_{45}+d_{23}\wedge d_{14}\wedge d_{45}+d_{14}\wedge d_{34}\wedge d_{25}+d_{24}\wedge d_{34}\wedge d_{15}$
(33,1) | $(-2,0,1,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-2\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{24}\wedge d_{35}+d_{23}\wedge d_{34}\wedge d_{25}$
(34,1) | $(0,-1,2,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-\alpha_{3}-2\alpha_{4}$ |  $d_{13}\wedge d_{25}\wedge d_{35}-d_{23}\wedge d_{15}\wedge d_{35}$
(35,1) | $(-1,2,-1,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{12}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{15}\wedge d_{25}$
(36,1) | $(2,-1,0,-1)=2\omega_{1}+\omega_{4}-\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{13}\wedge d_{15}\wedge d_{45}-d_{14}\wedge d_{15}\wedge d_{35}$
(37,1) | $(1,-3,2,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{34}\wedge d_{35}$
(38,1) | $(-1,-2,2,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{34}\wedge d_{35}$
(39,1) | $(-2,1,-1,1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-2\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{24}\wedge d_{45}+d_{24}\wedge d_{34}\wedge d_{25}$
(40,1) | $(1,-2,0,1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{34}\wedge d_{45}+d_{14}\wedge d_{34}\wedge d_{35}$
(41,1) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{13}\wedge d_{25}\wedge d_{45}+d_{23}\wedge d_{15}\wedge d_{45}-d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}$
(41,2) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{12}\wedge d_{35}\wedge d_{45}+d_{13}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{15}\wedge d_{35}-d_{34}\wedge d_{15}\wedge d_{25}$
(41,3) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{13}\wedge d_{25}\wedge d_{45}-d_{23}\wedge d_{15}\wedge d_{45}+d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}$
(41,4) | $(0,0,0,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{14}\wedge d_{25}\wedge d_{35}-d_{24}\wedge d_{15}\wedge d_{35}+d_{34}\wedge d_{15}\wedge d_{25}$
(42,1) | $(0,1,-3,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{24}\wedge d_{45}$
(43,1) | $(0,0,1,-3)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-2\alpha_{3}-3\alpha_{4}$ |  $d_{15}\wedge d_{25}\wedge d_{35}$
(44,1) | $(-1,-1,0,1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-2\alpha_{3}-\alpha_{4}$ |  $d_{23}\wedge d_{34}\wedge d_{45}+d_{24}\wedge d_{34}\wedge d_{35}$
(45,1) | $(-2,1,0,-1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-2\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{23}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{25}\wedge d_{35}$
(46,1) | $(1,-2,1,-1)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{13}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{15}\wedge d_{35}$
(47,1) | $(0,1,-2,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ |  $d_{14}\wedge d_{25}\wedge d_{45}-d_{24}\wedge d_{15}\wedge d_{45}$
(48,1) | $(1,-1,-2,2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$ |  $d_{14}\wedge d_{34}\wedge d_{45}$
(49,1) | $(-1,-1,1,-1)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-2\alpha_{3}-2\alpha_{4}$ |  $d_{23}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{25}\wedge d_{35}$
(50,1) | $(1,-1,-1,0)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ |  $d_{14}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{15}\wedge d_{45}$
(51,1) | $(-1,0,-2,2)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-\alpha_{4}$ |  $d_{24}\wedge d_{34}\wedge d_{45}$
(52,1) | $(0,1,-1,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-2\alpha_{2}-3\alpha_{3}-3\alpha_{4}$ |  $d_{15}\wedge d_{25}\wedge d_{45}$
(53,1) | $(-1,0,-1,0)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-2\alpha_{4}$ |  $d_{24}\wedge d_{35}\wedge d_{45}-d_{34}\wedge d_{25}\wedge d_{45}$
(54,1) | $(1,-1,0,-2)=2\omega_{1}+\omega_{4}-2\alpha_{1}-3\alpha_{2}-3\alpha_{3}-3\alpha_{4}$ |  $d_{15}\wedge d_{35}\wedge d_{45}$
(55,1) | $(-1,0,0,-2)=2\omega_{1}+\omega_{4}-3\alpha_{1}-3\alpha_{2}-3\alpha_{3}-3\alpha_{4}$ |  $d_{25}\wedge d_{35}\wedge d_{45}$
Table 8: Weights and weight vectors for $\mbox{sl}_{5}$ module $V(3\omega_{1})$ $i$ |  $\overrightarrow{w}_{i}^{3\omega_{1}}$ |  $v_{i}^{3\omega_{1}}$
---|---|---
1 |  $(3,0,0,0)=3\omega_{1}$ |  $d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{15}$
2 |  $(1,1,0,0)=3\omega_{1}-\alpha_{1}$ |  $d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{25}$
3 |  $(-1,2,0,0)=3\omega_{1}-2\alpha_{1}$ |  $d_{12}\wedge d_{23}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{25}$
4 |  $(2,-1,1,0)=3\omega_{1}-\alpha_{1}-\alpha_{2}$ |  $d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{35}$
5 |  $(-3,3,0,0)=3\omega_{1}-3\alpha_{1}$ |  $d_{12}\wedge d_{23}\wedge d_{24}\wedge d_{25}$
6 |  $(0,0,1,0)=3\omega_{1}-2\alpha_{1}-\alpha_{2}$ |  $d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{15}+d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{35}$
7 |  $(2,0,-1,1)=3\omega_{1}-\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $-d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{15}+d_{12}\wedge d_{13}\wedge d_{14}\wedge d_{45}+d_{12}\wedge d_{14}\wedge d_{34}\wedge d_{15}$
8 |  $(-2,1,1,0)=3\omega_{1}-3\alpha_{1}-\alpha_{2}$ |  $d_{13}\wedge d_{23}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{34}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{24}\wedge d_{35}$
9 |  $(1,-2,2,0)=3\omega_{1}-2\alpha_{1}-2\alpha_{2}$ |  $d_{13}\wedge d_{23}\wedge d_{34}\wedge d_{15}+d_{13}\wedge d_{23}\wedge d_{14}\wedge d_{35}+d_{12}\wedge d_{13}\wedge d_{34}\wedge d_{35}$
10 |  $(0,1,-1,1)=3\omega_{1}-2\alpha_{1}-\alpha_{2}-\alpha_{3}$ |  $-d_{23}\wedge d_{14}\wedge d_{24}\wedge d_{15}-d_{13}\wedge d_{14}\wedge d_{24}\wedge d_{25}+d_{12}\wedge d_{23}\wedge d_{14}\wedge d_{45}+d_{12}\wedge d_{13}\wedge d_{24}\wedge d_{45}+d_{12}\wedge d_{24}\wedge d_{34}\wedge d_{15}+d_{12}\wedge d_{14}\wedge d_{34}\wedge d_{25}$
11 |  $(2,0,0,-1)=3\omega_{1}-\alpha_{1}-\alpha_{2}-\alpha_{3}-\alpha_{4}$ |  $d_{13}\wedge d_{14}\wedge d_{15}\wedge d_{25}+d_{12}\wedge d_{13}\wedge d_{15}\wedge d_{45}-d_{12}\wedge d_{14}\wedge d_{15}\wedge d_{35}$