The Tutte polynomial and toric Nakajima quiver varieties. Issue 5 (October 2022)

• Record Type: Journal Article
• Authors: Abdelgadir, Tarig; Mellit, Anton; Villegas, Fernando Rodriguez
• Is Part Of: Proceedings. Volume 152:Issue 5(2022)
• Page Start: 1323
• Page End: 1339
• Publication Date: 2022-10
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2021.61
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : 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.