This paper deals with a reducible
s
ℓ
(
2
,
C
)
s\ell (2, \mathbf {C})
action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that
s
ℓ
(
2
,
C
)
s\ell (2, \mathbf {C})
acts on the formal power series ring via
(
0.1
)
(0.1)
. Then
I
(
f
)
=
(
ℓ
i
1
)
⊕
(
ℓ
i
2
)
⊕
⋯
⊕
(
ℓ
i
s
)
I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}})
modulo some one dimensional
s
ℓ
(
2
,
C
)
s\ell (2, \mathbf {C})
representations where
(
ℓ
i
)
(\ell _{i})
is an irreducible
s
ℓ
(
2
,
C
)
s\ell (2, \mathbf {C})
representation of dimension
ℓ
i
\ell _{i}
or empty set and
{
ℓ
i
1
,
ℓ
i
2
,
…
,
ℓ
i
s
}
⊆
{
ℓ
1
,
ℓ
2
,
…
,
ℓ
r
}
\{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\}
. Unlike classical invariant theory which deals only with irreducible action and 1–dimensional representations, we treat the reducible action and higher dimensional representations succesively.