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On Jacobian Ideals Invariant by a Reducible Action

Authors:	Yung Yu

Institution:	Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan, R.O.C.

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

Keywords:	Invariant polynomial weight irreducible submodule representation completely reducible

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