Cell decompositions of quiver flag varieties for nilpotent representations of the oriented cycle

TL;DR

This paper establishes an affine cell decomposition for varieties of complete flags in nilpotent quiver representations, generalizing classical results and providing tools for computing their equivariant cohomology.

Contribution

It introduces a new cell decomposition for flag varieties in nilpotent quiver representations, extending known results from type A and Springer fibers.

Findings

Varieties admit an affine cell decomposition parametrized by multi-tableaux.

They possess a torus action and their equivariant cohomology is described via Goresky-Kottwitz-MacPherson theory.

Provides a basis for modules over quiver Hecke algebras related to these representations.

Abstract

Generalizing Schubert cells in type A and a cell decomposition if Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of an oriented cycle admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation and describe the T-equivariant cohomology using Goresky-Kottwitz-MacPherson-theory. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules.