Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

TL;DR

This paper classifies exceptional $ ext{Delta}$-filtered modules for certain algebras and demonstrates the existence of open orbits in the nilpotent radical of seaweed Lie algebras, with specific counterexamples.

Contribution

It establishes the uniqueness of exceptional $ ext{Delta}$-filtered modules for fixed dimension vectors and confirms the existence of open orbits in seaweed Lie algebras in type A.

Findings

Unique exceptional $ ext{Delta}$-filtered modules for each dimension vector.

Existence of open adjoint orbits in seaweed Lie algebras of type A.

Counterexample of no open orbit in type E8 seaweed Lie algebra.

Abstract

In this paper, we study the set of $\Delta$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $\Delta$-dimension vector, there is a unique (up to isomorphism) exceptional $\Delta$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in $\mathrm{gl}_{n}(\field)$, thus answering positively in this $\mathrm{gl}_{n}(\field)$ case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie algebra in a simple Lie algebra of type $E_{8}$ not admitting an open orbit in its nilpotent radical is given.