Semi-invariants of symmetric quivers of finite type

Abstract

Let $(Q,\sigma )$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\bigsqcup Q_1$. 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 $c^V$ and, in the case when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $p{f}^V$.

References in corpus (4)

• Cluster Complexes via Semi-Invariants
• Symmetric quivers, invariant theory, and saturation theorems for the classical groups
• Semi-invariants of Symmetric Quivers
• Semi-invariants of symmetric quivers of tame type

Cited by in corpus (1)

• Semi-invariants of symmetric quivers of tame type