Transactions of the American Mathematical Society

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



Codimension $ 1$ orbits and semi-invariants for the representations of an equioriented graph of type $ D\sb{n}$

Author: S. Abeasis
Journal: Trans. Amer. Math. Soc. 286 (1984), 91-123
MSC: Primary 14L30; Secondary 14D25, 16A64
MathSciNet review: 756033
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Abstract: We consider the Dynkin diagram $ {D_n}$ equioriented and the variety $ \operatorname{Hom}({V_1},{V_3}) \times \Pi_{1 = 2}^n \operatorname{Hom} ({V_i},{V_{i + 1}})$, $ {V_j}$ a vector space over $ K$, on which the group $ G = \prod\nolimits_{i = 1}^n {{\text{GL}}} ({V_i})$ acts. We determine the maximal orbit and the codim. $ 1$ orbits of this action, giving their decomposition in terms of the irreducible representations of $ {D_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, semi-invariants
Article copyright: © Copyright 1984 American Mathematical Society