Classification of gradient space of dimension 8 by a reducible sℓ(2, C) action.

This paper deals with a reducible sℓ(2,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 sℓ(2,C) acts on the formal power series ring via (1.1). Then I( f) = ( ℓ _i1) ⊕ ( ℓ _i2) ⊕... ⊕ ( ℓ _is ) modulo some one dimensional sℓ(2,C) representations where (ℓ _i ) is an irreducible sℓ(2,C) representation of ℓ _i dimension and { ℓ _i1 ℓ _i2,..., ℓ _is } ⊆ { ℓ ₁ , ℓ ₂..., ℓ _r }. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively. [ABSTRACT FROM AUTHOR]