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Degenerations for the representations of a quiver of type \({\mathcal A}_ m\). (English) Zbl 0598.16030
Let \(Q_ m\) be a quiver of type \(A_ m\), \(d=(d_ 1,...,d_ m)\) an m- tuple of non-negative integers and \(L_ d\) the variety of all representations of \(Q_ m\) of dimension d over a fixed field k. In the paper, to each representation \(A\in L_ d\) one associates a set of non- negative integers \(N^ A=\{N^ A_{uv}|\) \(1\leq u\leq v\leq m\}\) which determines uniquely the orbit \({\mathcal O}_ A\) of A with respect to the natural action of the group \(G=\prod^{m}_{i=1}Gl(d_ i,k)\) on \(L_ d\). The main result asserts that \({\mathcal O}_ B\subset L_ d\) is a degeneration of \({\mathcal O}_ A\subset L_ d\) (that is \({\mathcal O}_ B\) is contained in the closure \(\bar {\mathcal O}_ A\) of \({\mathcal O}_ A)\) if and only if \(N^ B_{uv}\leq N^ A_{uv}\) for every u,v, \(1\leq u\leq v\leq m\). The proof is purely combinatorial. A more elegant treatment of a more general situation the reader can find in a preprint of Ch. Riedtmann [L’Institut Fourier, No.19, Grenoble (1984)].
Reviewer: A.Skowroński

MSC:
16Gxx Representation theory of associative rings and algebras
14L30 Group actions on varieties or schemes (quotients)
14D15 Formal methods and deformations in algebraic geometry
14L24 Geometric invariant theory
16P10 Finite rings and finite-dimensional associative algebras
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