Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type 𝒜_{𝓃}

Abeasis, S.

doi:10.1090/S0002-9947-1984-0732101-8

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Codimension 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. 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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