Quiver Grassmannians associated to nilpotent cyclic representations defined by single matrix

TL;DR

This paper investigates the geometry of quiver Grassmannians associated with nilpotent cyclic representations defined by a single matrix, revealing smoothness of certain cells and analyzing their cohomology structure.

Contribution

It provides new insights into the geometry and cohomology of quiver Grassmannians for nilpotent cyclic representations, especially for subrepresentations of dimension (1,...,1).

Findings

Closed Bia{}ynicki-Birula cells are smooth in the studied case.

Describes the multiplicative structure of the cohomology ring.

Identifies the Knutson-Tao basis in the context of equivariant cohomology.

Abstract

In the present paper we study the geometry of the closed Bia{\l}ynicki-Birula cells of the quiver Grassmannians associated to a nilpotent representation of a cyclic quiver defined by a single matrix. For the special case, where we choose subrepresentations of dimension $\mathbf{1}=(1,\dots,1)$, the main result of this paper is that the closed Bia{\l}ynicki-Birula cells are smooth. We also discuss the multiplicative structure of the cohomology ring of such spaces. Namely, we describe the so-called Knutson-Tao basis in context to the basis of equivariant cohomology that is dual to fundamental classes in equivariant homology.