Semi-invariants of symmetric quivers of tame type

A symmetric quiver $(Q,\sigma )$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $\sigma$ on $Q_0\sqcup Q_1$\mathrm{.}Theinvolutionallowsustodefineanondegeneratebilinearform$<,>$ on a representation $V$ of $Q$. We shall say that $V$ is orthogonal if $<,>$ is symmetric and symplectic if $<,>$ 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 $(Q,\sigma )$ 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 $c^V$ and, when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $p{f}^V$. To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector.