Semi-invariants of symmetric quivers of tame type

TL;DR

This paper studies semi-invariants of symmetric quivers of tame type, showing they are generated by determinantal semi-invariants and Pfaffians, with a detailed description of their generic decompositions.

Contribution

It characterizes the semi-invariants for symmetric quivers of tame type, extending classical results to orthogonal and symplectic representations.

Findings

Semi-invariants are spanned by determinantal semi-invariants.

Pfaffians generate semi-invariants when matrices are skew-symmetric.

Provides a description of symplectic and orthogonal generic decompositions.

Abstract

A symmetric quiver $(Q,\sigma)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $\sigma$ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ 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 $pf^V$. To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector.