The Tutte polynomial and toric Nakajima quiver varieties

TL;DR

This paper establishes a direct, combinatorial-geometric connection between the Tutte polynomial, Kac polynomial, and Poincaré polynomial of toric Nakajima quiver varieties, using cell decompositions and graph operations.

Contribution

It provides a more hands-on method to relate these polynomials, building on prior theoretical results with explicit geometric and combinatorial constructions.

Findings

Derived a cell decomposition of the quiver variety indexed by spanning trees.

Connected Tutte polynomial specializations to Kac and Poincaré polynomials.

Presented a geometric interpretation of graph deletion and contraction operators.

Abstract

For a quiver $Q$, we take $\mathcal{M}$ an associated toric Nakajima quiver variety and $\Gamma$ the underlying graph. In this article, we give a direct relation between a specialisation of the Tutte polynomial of $\Gamma$, the Kac polynomial of $Q$ and the Poincar\'e polynomial of $\mathcal{M}$. We do this by giving a cell decomposition of $\mathcal{M}$ indexed by spanning trees of $\Gamma$ and `geometrising' the deletion and contraction operators on graphs. These relations have been previously established by Sturmfels-Hausel and (Crawley-Boovey)-Van den Bergh, however the methods here are more hands-on.