The Tutte polynomial and toric Nakajima quiver varieties

For a quiver Q with underlying graph Γ, we take M an associated toric Nakajima quiver variety. In this article, we give a direct relation between a specialization of the Tutte polynomial of Γ, the Kac polynomial of Q and the Poincaré polynomial of M. We do this by giving a cell decomposition of M indexed by spanning trees of Γ and ‘geometrizing’ the deletion and contraction operators on graphs. These relations have been previously established in Hausel–Sturmfels [6] and Crawley-Boevey–Van den Bergh [3], however the methods here are more hands-on.