Semi-invariants of quivers for arbitrary dimension vectors
Let \(Q\) be a quiver. For a dimension vector \(\alpha\) one can define a vector space \(R(Q,\alpha)\) and an algebraic group \(\text{GL}(\alpha)\) such that the orbits correspond to isomorphism classes of \(\alpha\)-dimensional representations of the quiver. \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, No. 2, 585-598 (1990; Zbl 0693.16018)] showed that the ring of invariants for a quiver is generated by traces of closed paths if the characteristic of the base field is 0. \textit{S. Donkin} [Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)] showed that in positive characteristic the ring of invariants is generated by characteristic polynomials of closed paths. For a quiver without oriented cycles, the ring of invariants may be trivial. However, one can also study the ring of semi-invariants. The set of semi-invariant polynomial functions on \(R(Q,\alpha)\) forms a graded ring.NEWLINENEWLINENEWLINEThe first author introduced semi-invariants [in J. Lond. Math. Soc., II. Ser. 43, No. 3, 385-395 (1991; Zbl 0779.16005)]. The main result of this paper is that these invariants always span the ring of semi-invariants if the characteristic of the base field is 0. A similar result for quivers without oriented cycles in arbitrary characteristic was proven by \textit{J. Weyman} and the reviewer [J. Am. Math. Soc. 13, No. 3, 467-479 (2000; Zbl 0993.16011)]. For a quiver without oriented cycles, the Schofield semi-invariants correspond to representations of the quiver, which are in a certain way orthogonal to general representations of dimension \(\alpha\). \textit{M. Domokos} and \textit{A. N. Zubkov} [Transform. Groups 6, No. 1, 9-24 (2000; Zbl 0984.16023)] constructed semi-invariants as determinants and proved that their semi-invariants also generate the ring of semi-invariants.
scientific article; zbMATH DE number 1663813