Semi-Invariants of Symmetric Quivers of Tame Type

Abstract

A symmetric quiver (Q, σ) is a finite quiver without oriented cycles Q = (Q ₀, Q ₁) equipped with a contravariant involution σ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $\langle -,-\rangle_V$ on a representation V of Q. We shall say that V is orthogonal if $\langle -,-\rangle_V$ is symmetric and symplectic if $\langle -,-\rangle_V$ 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, σ) 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 the matrix defining c ^V 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.