Semi-invariants of symmetric quivers of finite type

TL;DR

This paper studies semi-invariants of symmetric quivers of finite type, showing they are generated by determinantal semi-invariants and Pfaffians in specific cases, extending classical invariant theory to symmetric quiver representations.

Contribution

It characterizes the rings of semi-invariants for symmetric quivers of finite type, identifying generators as determinantal semi-invariants and Pfaffians, which is a novel extension of invariant theory.

Findings

Semi-invariants are spanned by determinantal semi-invariants $c^V$.

When matrices are skew-symmetric, semi-invariants are generated by Pfaffians $pf^V$.

The results extend classical invariant theory to symmetric quiver representations.

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 $Q_0\sqcup Q_1$. 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 $pf^V$.