Quiver quotient varieties and complete intersections. (Q1780994)

Denote by \(S_\alpha(Q)\) the algebraic quotient of the space of representations of the quiver \(Q\) with dimension vector \(\alpha\) under the action of the base change group \(\text{GL}(\alpha)\) (working over the field of complex numbers); this variety parameterizes the semisimple representations. The author proves for symmetric quiver settings (each arrow has a pair going in the opposite direction) without loops that \(S_\alpha(Q)\) is a complete intersection if and only if \((Q,\alpha)\) can be reduced by a successive application of three simple reduction steps to a member of an explicitly given list of quiver settings. The reduction steps were introduced by the author [\textit{R. Bocklandt}, J. Algebra 253, No. 2, 296-313 (2002; Zbl 1041.16010)], where he classified coregular quiver settings in a similar vein. The proof involves an analysis of concrete rings of polynomial invariants. (Also submitted to MR.)