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Transactions of the American Mathematical Society

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Codimension $ 1$ orbits and semi-invariants for the representations of an oriented graph of type $ \mathcal{A}_n$

Author: S. Abeasis
Journal: Trans. Amer. Math. Soc. 282 (1984), 463-485
MSC: Primary 14L30; Secondary 14D25, 16A64
MathSciNet review: 732101
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Abstract: We consider the Dynkin diagram $ \mathcal{A}_n$ with an arbitrary orientation $ \Omega $. For a given dimension $ d = ({d_1}, \ldots ,{d_n})$ we consider the corresponding variety $ {L_d}$ of all the representations of $ (\mathcal{A}_n,\Omega )$ on which a group $ {G_d}$ acts naturally. In this paper we determine the maximal orbit and the codim. $ 1$ orbits of this action, giving explicitly their decomposition in terms of the irreducible representations of $ \mathcal{A}_n$. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.

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Keywords: Dynkin diagrams, representations, orbits and semi-invariants
Article copyright: © Copyright 1984 American Mathematical Society