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Semi-invariants of quivers. (English) Zbl 0779.16005

This interesting article on invariant theory for representations of quivers seems to make valuable progress in the geometrical approach to representation theory.
Let \(Q\) be a quiver with no oriented cycles, \(k\) an algebraically closed field and \(\alpha\) a dimension vector for \(Q\). Let \(R(\alpha)\) denote the space of representations of \(Q\) with \(\underline{\text{dim}} \alpha\) and let \(\text{Gl}(\alpha)\) be the obvious product of full linear groups whose orbits over \(R(\alpha)\) define the isoclasses of representations of \(Q\) which lie on \(R(\alpha)\). Following V. G. Kac [see J. Algebra 78, 141-162 (1982; Zbl 0497.17007)] \(\alpha\) is a real Schur root of \(Q\) if \(R(\alpha)\) has an open orbit whose belonging representation, denoted by \(G(\alpha)\), is indecomposable. Given two dimension vectors \(\alpha\), \(\beta\) that are orthogonal with respect to the usual bilinear form associated to \(Q\), a polynomial on \(p\), \(q\), \(P_{\alpha,\beta}(p,q)\) (where \(p\) is in \(R(\alpha)\) and \(q\) is in \(R(\beta)\)) is introduced. After some study is carried on, the author is able to do as follows. Given \(\alpha\) such that \(Gl(\alpha)\) has an open orbit, certain representations \(S_ i\) are naturally introduced and the main result asserts that the polynomials \(P_{\beta_ i,\alpha}(S_ i,q)\) (where \(\beta_ i\) is \(\underline{\text{dim}}(S_ i)\)) are algebraically independent, are semi-invariants and generate all semi-invariants.

MSC:

16G20 Representations of quivers and partially ordered sets
14L30 Group actions on varieties or schemes (quotients)
15A72 Vector and tensor algebra, theory of invariants

Citations:

Zbl 0497.17007
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