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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

16Gxx Representation theory of associative rings and algebras
14L30 Group actions on varieties or schemes (quotients)
20G05 Representation theory for linear algebraic groups
Full Text: DOI
[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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