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

Codimension $1$ orbits and semi-invariants for the representations of an oriented graph of type $\mathcal {A}_n$
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Trans. Amer. Math. Soc. 282 (1984), 463-485 Request permission

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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Représentations indécomposables

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Infinite root systems, representations of graphs and invariant theory

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