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arXiv · 1910.01633

The Tutte polynomial and toric Nakajima quiver varieties

Tarig Abdelgadir·Anton Mellit·Fernando Rodriguez-Villegas
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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.

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Tarig Abdelgadir, Anton Mellit, Fernando Rodriguez-Villegas. 2019-10-03. The Tutte polynomial and toric Nakajima quiver varieties. https://arxiv.org/abs/1910.01633

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