Semi-invariants of symmetric quivers of finite type

Let (Q, σ) be a symmetric quiver, where Q = (Q0,Q1) is a finite quiver without oriented cycles and σ is a contravariant involution on Q0Q1. The involution allows us to define a nondegenerate bilinear form <, > on a representation V of Q. We shall call the representation orthogonal if <, > is symmetric, and symplectic if <, > is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type cVand, in the case when the matrix defining cVis skew-symmetric, by the Pfaffians pfV. In particular we give an explicit finite set of generators. © 2012 Springer Science+Business Media B.V.