On the complement of the dense orbit for a quiver of type $\Aa$

TL;DR

This paper characterizes the irreducible components of the complement of the dense orbit in representations of type A quivers using rank conditions, compares with existing results, and derives a new orbit count formula.

Contribution

It introduces a rank condition-based method to define irreducible components in the complement of the dense orbit for type A quivers, providing new insights and formulas.

Findings

Identifies irreducible components via rank conditions.

Provides a new formula for counting orbits using nilpotent classes.

Compares new results with existing literature by Knight, Zelevinsky, and Ringel.

Abstract

Let $\Aa_t$ be the directed quiver of type $\Aa$ with $t$ vertices. For each dimension vector $d$ there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver $\Aa$ and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit we determine the irreducible components and their codimension. Finally, we consider several particular examples.