Tautological bases of CoHA modules

TL;DR

This paper constructs canonical bases for CoHA modules associated with quivers, using tautological vector bundles over non-commutative Hilbert schemes, generalizing classical geometric results.

Contribution

It introduces canonical bases for CoHA modules built from tautological Chern classes, extending classical geometric bases to non-commutative Hilbert schemes.

Findings

Construction of canonical bases for CoHA modules.

Generalization of classical bases from Grassmannians and flag varieties.

Analysis of cell decompositions of non-commutative Hilbert schemes.

Abstract

Given a quiver, we consider its cohomological Hall algebra (CoHA) as well as CoHA modules built of cohomology groups of non-commutative Hilbert schemes. We investigate cell decompositions of non-commutative Hilbert schemes and the corresponding (non-canonical) bases of CoHA modules. We construct canonical bases of CoHA modules, which consist of products of Chern classes of tautological vector bundles over non-commutative Hilbert schemes. This result generalizes classical results for Grassmannians and (partial) flag varieties.