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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On Jacobian Ideals Invariant by a Reducible $s\ell (2, \mathbf {C})$ Action


Author: Yung Yu
Journal: Trans. Amer. Math. Soc. 348 (1996), 2759-2791
MSC (1991): Primary 17B10, 17B20; Secondary 14B05
MathSciNet review: 1355078
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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 $(0.1)$. 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.


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Additional Information

Yung Yu
Affiliation: Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan, R.O.C.
Email: yungyu@mail.ncku.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01633-9
PII: S 0002-9947(96)01633-9
Keywords: Invariant polynomial, weight, irreducible submodule, representation, completely reducible
Received by editor(s): April 28, 1995
Additional Notes: Research partially supported by N.S.C.
Article copyright: © Copyright 1996 American Mathematical Society