Invariant theory of algebra representations (Q2759659)

This paper gives an overview of some results regarding invariants and semi-invariants of quiver representations.NEWLINENEWLINENEWLINEIn the first part the author studies invariants of quiver representations and semisimple representations of quivers. \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, No. 2, 585-598 (1990; Zbl 0693.16018)] showed in characteric 0 that the traces of closed paths will generate the ring of invariants. In positive characteristic, \textit{S. Donkin} [Comment. Math. Helv. 69, No. 1, 137-141 (1994; Zbl 0816.16015)] showed that the characteristic polynomials of closed paths generate the invariant ring. The author studies the geometry of the categorical quotient (using for example Luna's slice theorem). For a given dimension vector, there is an easy criterion whether it is simple, i.e., whether there exists a simple representation of that dimension. The author shows that the categorical quotient of the representation space of any dimension vector is essentially a product of categorical quotients of representation spaces of simple dimension vectors.NEWLINENEWLINENEWLINEIf the quiver has no oriented cycles, the ring of invariants will always be trivial. However, there still may be an interesting ring of semi-invariants. The author first studies rings of semi-invariants for bipartite quivers and then shows how the semi-invariants for arbitrary quivers can be found by reducing the problem to bipartite quivers. See also \textit{M. Domokos} and \textit{A. N. Zubkov} [Transform. Groups 6, No. 1, 9-24 (2000; Zbl 0984.16023)]. Another approach to semi-invariants can be found in papers by \textit{A. Schofield} and \textit{M. Van den Bergh} [Indag. Math., New Ser. 12, No. 1, 125-138 (2001; see the following review Zbl 1004.16013)], and by \textit{J. Weyman} and the reviewer [J. Am. Math. Soc. 13, No. 3, 467-479 (2000; Zbl 0993.16011)]. Both teams showed that the semi-invariants introduced by Schofield generate the ring of semi-invariants.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].