The Tutte polynomial and toric Nakajima quiver varieties

For a quiver $Q$ with underlying graph $\Gamma$, we take $ {\mathcal {M}}$ an associated toric Nakajima quiver variety. In this article, we give a direct relation between a specialization of the Tutte polynomial of $\Gamma$, the Kac polynomial of $Q$ and the Poincaré polynomial of $ {\mathcal {M}}$....

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2022-10, Vol.152 (5), p.1323-1339
Main Authors: Abdelgadir, Tarig, Mellit, Anton, Villegas, Fernando Rodriguez
Format: Article
Language:English
Subjects:
Decomposition
Graph theory
Polynomials
Trees
Trees (mathematics)
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Summary:For a quiver $Q$ with underlying graph $\Gamma$, we take $ {\mathcal {M}}$ an associated toric Nakajima quiver variety. In this article, we give a direct relation between a specialization of the Tutte polynomial of $\Gamma$, the Kac polynomial of $Q$ and the Poincaré polynomial of $ {\mathcal {M}}$. We do this by giving a cell decomposition of $ {\mathcal {M}}$ indexed by spanning trees of $\Gamma$ 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.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2021.61