Quiver-graded Richardson Orbits

TL;DR

This paper generalizes Richardson orbits to a quiver-graded setting using a new algebraic framework, linking orbit existence to rigid modules and providing explicit examples and calculations.

Contribution

It introduces the nilpotent quiver algebra and connects Richardson orbit existence to rigid $ extit{ extbf{ extDelta}}$-filtered modules, extending classical Lie theory concepts.

Findings

Existence of Richardson orbits is characterized by rigid $ extit{ extbf{ extDelta}}$-filtered modules.

Explicit calculations of Richardson orbits in specific examples.

Identification of cases where no Richardson orbit exists.

Abstract

In Lie theory, a dense orbit in the unipotent radical of a parabolic group under the adjoint action is called a Richardson orbit. We define a quiver-graded version of Richardson orbits generalising the classical definition in the case of the general linear group. In our setting a product of parabolic subgroups of general linear groups acts on a closed subvariety of the representation space of a quiver. Such dense orbits do not exist in general. We define a quasi-hereditary algebra called the nilpotent quiver algebra whose isomorphism classes of $\Delta$-filtered modules correspond to orbits in our generalised setting. We translate the existence of a Richardson orbit into the existence of a rigid $\Delta$-filtered module of a given dimension vector. We study an idempotent recollement of this algebra whose associated intermediate extension functor can be used to produce Richardson orbits in some situations. This can be explicitly calculated in examples. We also give examples where no Richardson orbit exists.