Codimension orbits and semi-invariants for the representations of an oriented graph of type

Author:
S. Abeasis

Journal:
Trans. Amer. Math. Soc. **282** (1984), 463-485

MSC:
Primary 14L30; Secondary 14D25, 16A64

MathSciNet review:
732101

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Dynkin diagram with an arbitrary orientation . For a given dimension we consider the corresponding variety of all the representations of on which a group 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 . We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.

**[1]**S. Abeasis,*On the ring of semi-invariants of the representations of an equioriented quiver of type \cal𝐴_{𝑛}*, Boll. Un. Mat. Ital. A (6)**1**(1982), no. 2, 233–240 (English, with Italian summary). MR**663286****[2]**S. Abeasis and A. Del Fra,*Degenerations for the representations of a quiver of type 𝒜_{𝓂}*, J. Algebra**93**(1985), no. 2, 376–412. MR**786760**, 10.1016/0021-8693(85)90166-8**[3]**I. N. Bernšteĭn, I. M. Gel′fand, and V. A. Ponomarev,*Coxeter functors, and Gabriel’s theorem*, Uspehi Mat. Nauk**28**(1973), no. 2(170), 19–33 (Russian). MR**0393065****[4]**Vlastimil Dlab and Claus Michael Ringel,*Indecomposable representations of graphs and algebras*, Mem. Amer. Math. Soc.**6**(1976), no. 173, v+57. MR**0447344****[5]**P. Gabriel,*Représentations indécomposables*, Sèm. Bourbaki, no. 444, 1973/1974, pp. 1-27.**[6]**Dieter Happel,*Relative invariants and subgeneric orbits of quivers of finite and tame type*, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 116–124. MR**654706****[7 V]**V. G. Kac,*Infinite root systems, representations of graphs and invariant theory*, Invent. Math.**56**(1980), no. 1, 57–92. MR**557581**, 10.1007/BF01403155**[8]**-,*Infinite root systems, representations of graphs and invariant theory*. II (to appear).**[9]**M. Sato and T. Kimura,*A classification of irreducible prehomogeneous vector spaces and their relative invariants*, Nagoya Math. J.**65**(1977), 1–155. MR**0430336**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14L30,
14D25,
16A64

Retrieve articles in all journals with MSC: 14L30, 14D25, 16A64

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1984-0732101-8

Keywords:
Dynkin diagrams,
representations,
orbits and semi-invariants

Article copyright:
© Copyright 1984
American Mathematical Society