an:03964104
Zbl 0598.16030
Abeasis, S.; Del Fra, A.
Degenerations for the representations of a quiver of type \({\mathcal A}_ m\)
EN
J. Algebra 93, 376-412 (1985).
00150038
1985
j
16Gxx 14L30 14D15 14L24 16P10
quiver of type \(A_ m\); representations; orbit; natural action; degeneration
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 \textit{Ch. Riedtmann} [L'Institut Fourier, No.19, Grenoble (1984)].
A.Skowro??ski