Symmetric degenerations are not in general induced by type A degenerations

TL;DR

This paper investigates the relationship between symmetric degenerations and type A degenerations in quiver representations, providing explicit examples where orbit closure relations differ across classical Lie algebra types.

Contribution

It offers the first explicit example showing symmetric degenerations are not always induced by type A degenerations, especially in type D.

Findings

Orbit closure relations are induced in types B and C.

Orbit closure relations are not induced in type D.

Explicit example of a finite type quiver illustrating these differences.

Abstract

We consider a symmetric quiver with relations. Its (symmetric) representations of a fixed symmetric dimension vector are encoded in the (symmetric) representation varieties. The orbits by a (symmetric) base change group action are the isomorphism classes of (symmetric) representations. The symmetric orbits are induced by simply restricting the nonsymmetric orbits. However, when it comes to orbit closure relations, it is so far an open question under which assumptions they are induced. In connection with Borel orbits of 2-nilpotent matrices of classical Lie algebras, we describe an explicit example of a quiver of finite representation type for which orbit closure relations are induced in types B and C, but not in type D.