Three formulas for CSM classes of open quiver loci

TL;DR

This paper introduces three formulas for computing the Chern-Schwartz-MacPherson classes of open quiver loci in type A quiver representations, refining quiver polynomial data with geometric and combinatorial methods.

Contribution

It provides one geometric and two combinatorial formulas for equivariant CSM classes of open quiver loci, including new pipe dream variants and streamlined quiver polynomial formulas.

Findings

Three formulas for CSM classes of open quiver loci.

Chained generic pipe dreams as a new combinatorial tool.

Streamlined formulas for quiver polynomials with fewer terms.

Abstract

In the space of equioriented type $A$ quiver representations, we define subvarieties called "open quiver loci" by placing strict rank conditions on the maps within representations. The closures of these subvarieties are the quiver loci, whose equivariant cohomology classes are the quiver polynomials of Buch and Fulton. We present one geometric formula and two combinatorial formulas that compute equivariant Chern-Schwartz-MacPherson (CSM) classes of open quiver loci; these classes refine the data of the quiver polynomials. The second combinatorial formula is in terms of "chained generic pipe dreams," which modify the pipe dreams of Bergeron and Billey to more strongly resemble the lacing diagrams of Abeasis and Del Fra. We also present two new formulas for quiver polynomials; these are streamlined versions of known formulas due to Knutson, Miller, and Shimozono, in the sense that they contain fewer terms.