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

Abeasis, S.

doi:10.1090/S0002-9947-1984-0756033-4

Codimension $1$ orbits and semi-invariants for the representations of an equioriented graph of type $D_{n}$
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by S. Abeasis

Trans. Amer. Math. Soc. 286 (1984), 91-123

DOI: https://doi.org/10.1090/S0002-9947-1984-0756033-4

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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.

References

S. Abeasis, On the ring of semi-invariants of the representations of an equioriented quiver of type ${\cal A}_{n}$, Boll. Un. Mat. Ital. A (6) 1 (1982), no. 2, 233–240 (English, with Italian summary). MR 663286
S. Abeasis, Codimension $1$ orbits and semi-invariants for the representations of an oriented graph of type ${\cal A}_{n}$, Trans. Amer. Math. Soc. 282 (1984), no. 2, 463–485. MR 732101, DOI 10.1090/S0002-9947-1984-0732101-8

Degenerations for the representations of an equioriented quiver of type

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Degenerations for the representations of a quiver of type

S. Abeasis, A. Del Fra, and H. Kraft, The geometry of representations of $A_{m}$, Math. Ann. 256 (1981), no. 3, 401–418. MR 626958, DOI 10.1007/BF01679706
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
Vlastimil Dlab and Claus Michael Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, v+57. MR 447344, DOI 10.1090/memo/0173
Pierre Gabriel, Représentations indécomposables, Séminaire Bourbaki, 26ème année (1973/1974), Lecture Notes in Math., Vol. 431, Springer, Berlin, 1975, pp. Exp. No. 444, pp. 143–169 (French). MR 0485996
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
Dieter Happel, Relative invariants and subgeneric orbits of quivers of finite and tame type, J. Algebra 78 (1982), no. 2, 445–459. MR 680371, DOI 10.1016/0021-8693(82)90092-8
V. G. Kac, Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), no. 1, 57–92. MR 557581, DOI 10.1007/BF01403155
V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), no. 1, 141–162. MR 677715, DOI 10.1016/0021-8693(82)90105-3
M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155. MR 430336, DOI 10.1017/S0027763000017633