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

Abstract

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) = (li1) ⊕ (li2) ⊕... ⊕ (lis) modulo some one dimensional sl(2,C) representations where (li) is an irreducible sl(2,C) representation of li dimension and {li1li2,...,lis} ⊆ {l1, l2...,lr}. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively.