DT-invariants of quivers and the Steinberg character of GL_n

TL;DR

This paper links DT-invariants of double quivers to the Steinberg character, providing new formulas and connections to quiver varieties and previous work on Weyl group representations.

Contribution

It offers a simple description of DT-invariants via the Steinberg character and relates them to Poincaré polynomials of singular quiver varieties.

Findings

DT-invariants correspond to Steinberg character multiplicities

Indivisible dimension vectors yield Poincaré polynomial expressions

Connections established with prior work on Weyl group representations

Abstract

In this paper we give a simple description of DT-invariants of double quivers without potential as the multiplicity of the Steinberg character in some representation associated with the quiver. When the dimension vector is indivisible we use this description to express these DT-invariants as the Poincar\'e polynomial of some singular quiver varieties. Finally we explain the connections with previous work of Hausel-Letellier-Villegas where DT-invariants are expressed as the graded multiplicities of the trivial representation of some Weyl group in the cohomology of some non-singular quiver varieties attached to an extended quiver.