Semi-Invariants of symmetric quivers of tame type

A symmetric quiver (Q, s) is a finite quiver without oriented cycles Q = (Q0, Q1) equipped with a contravariant involution s on Q0 u Q1. The involution allows us to define a nondegenerate bilinear form -, - V on a representation V of Q. We shall say that V is orthogonal if (-, -) V is symmetric and symplectic if (-, -)V is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if (Q, s) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type cV and, when the matrix defining cV is skew-symmetric, by the Pfaffians pf V. To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector. © Springer Science+Business Media B.V. 2011.