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Degenerations for the representations of an equioriented quiver of type $$D_ m$$. (English) Zbl 0537.16025
Given a field k and the oriented Dynkin diagram $\begin{matrix} && 1'\\ D_{m+1}:&& \downarrow \\ & 1 & \to & 2 & \to & 3 & \to &... & \to & m+1 \end{matrix}$ any K-linear representation $V= \begin{pmatrix} && V' \\ && \downarrow f \\ V_ 1 & \overset {f_ 1} {} & V_ 2 & \overset {f_ 2} {} & V_ 3 {&} &... & \overset {f_ m} {} & V_{m+1}\end{pmatrix}$ of dimension $d= \begin{pmatrix} d' \\ d_ 1,d_ 2,...,d_{m+1} \end{pmatrix}$ with $$d_ j=\dim_ KV_ j$$ can be considered as an element of the algebraic variety $L_ d=Hom_ K(V',V_ 2)\times \prod^{m}_{i=1}Hom_ K(V_ i,V_{i+1}).$ The algebraic group $$G_ d=\prod^{m+1}_{i=1}Gl(V_ i)$$ acts naturally on $$L_ d$$, the number of orbits of this action is finite and each orbit $$O_ V$$ corresponds to the isomorphism class of the representation V. The paper under review is devoted to the following degeneration problem: Given an orbit $$O_ V$$ in $$L_ d$$ characterize all orbits $$0_ W$$ in $$L_ d$$ such that $$O_ W$$ is contained in the Zariski closure $$\bar O_ V$$ of $$O_ V$$ i.e. $$O_ W$$ is a degeneration of $$O_ V$$. A solution is given in terms of an ordering $$\leq$$ between rank parameter sets $$N^ V$$, $$N^ W$$ (of integers) associated to representations V,W. It is proved that $$O_ W$$ is contained in $${\bar O}_ V$$ if and only if $$N^ W\leq N^ V$$. It is also shown that if $$O_ W\subseteq \bar O_ V$$ and $$O_ W$$ is open in $${\bar O}_ V-O_ V$$ then there exists a subrepresentation U of V such that $$W\cong U\oplus(V/U).$$
Reviewer: D.Simson

##### MSC:
 16Gxx Representation theory of associative rings and algebras 14L30 Group actions on varieties or schemes (quotients) 20G05 Representation theory for linear algebraic groups
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##### References:
 [1] \scS. Abeasis, Codimension 1 orbits and semi-invariants for the representations of an oriented graph of type An, Trans. Amer. Math. Soc., in press. [2] \scS. Abeasis, Codimension 1 orbits and semi-invariants for the representations of an equioriented graph of type Dm, in press. [3] Abeasis, S; Del Fra, A, Degenerations for the representations of an equioriented quiver of type Am, Boll. un. mat. ital. suppl. algebra geometria, 157-171, (1980) · Zbl 0449.16024 [4] \scS. Abeasis and A. Del Fra, Degenerations for the representations of a quiver of type Am, J. Algebra, in press. · Zbl 0598.16030 [5] Artin, M, On Azumaya algebras and finite dimensional representation rings, J. algebra, 11, 532-563, (1969) · Zbl 0222.16007 [6] Bernstein, I.N; Gelfand, I.M; Ponomarev, V.A, Coxeter functors and Gabriel’s theorem, Uspechi math. nauk, Russian math. surveys, 28, 17-32, (1973), transl. · Zbl 0279.08001 [7] Dlab, V; Ringel, G.M, Indecomposable representations of graphs and algebras, Mem. amer. math. soc., 6, 173, 1-57, (1976) · Zbl 0332.16015 [8] Gabriel, P, Unzerlegbare darstellungen I, Manuscripta math., 6, 71-103, (1972) · Zbl 0232.08001 [9] Gabriel, P, Representations indecomposables, Sem. bourboki, 444, 1-27, (1973/1974) [10] Kac, V, Infinite root system, representations of graphs and invariant theory, Invent. math., 56, 57-92, (1980) · Zbl 0427.17001 [11] Kempf, G, Instability in invariant theory, Ann. of math., 108, (1978) · Zbl 0406.14031
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