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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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