On the zero set of semi-invariants for [formula omitted]-quivers

Abstract

Let Q be a quiver of type D n , d a dimension vector for Q, and T a representative of the open orbit of the variety rep ( Q , d ) of d-dimensional representations of Q, under the product Gl ( d ) of the general linear groups at all vertices of Q. Let T = T 1 λ 1 ⊕ ⋯ ⊕ T r λ r be a decomposition of T into pairwise non-isomorphic indecomposable representations T i with multiplicities λ i . We show that it depends on the multiplicity of at most one such direct summand whether or not the set of common zeros of all non-constant semi-invariants for rep ( Q , d ) , with respect to the action of Gl ( d ) , is a set theoretical complete intersection.