Representations of quivers with free module of covariants.
The isomorphism classes of representations with a fixed dimension vector \(\mathbf d\) of a quiver \(Q\) are in a bijective correspondence with the orbits of \(\text{GL}({\mathbf d})\), a product of general linear groups, acting linearly on the representation space \(R(Q,{\mathbf d})\). It is well known that for a Dynkin type quiver \(Q\) the ring \(\text{SI}(Q,{\mathbf d})\) of semi-invariant polynomial functions on \(R(Q,{\mathbf d})\) is a polynomial ring. In the present paper it is proved that if \(Q\) is of Dynkin type \(A_n\), then the ideal in the coordinate ring \(k[R(Q,{\mathbf d})]\) generated by the semi-invariants of positive degree is a complete intersection. As a corollary, when the base field is of characteristic zero, all modules of \(\text{SL}({\mathbf d})\)-covariants in \(k[R(Q,{\mathbf d})]\) are free modules over \(\text{SI}(Q,{\mathbf d})\). An example is presented showing that the corresponding statement does not hold for all Dynkin quivers. A modified statement valid for arbitrary tame quivers was proved by \textit{Ch. Riedtmann} and \textit{G. Zwara} [Comment. Math. Helv. 79, No. 2, 350-361 (2004; Zbl 1063.14052)]. (Also submitted to MR.)
scientific article; zbMATH DE number 2093422
scientific article