Topology of Z_3 equivariant Hilbert schemes

Deborah Castro; Dustin Ross

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arXiv · 1810.05750

Topology of Z_3 equivariant Hilbert schemes

Deborah Castro·Dustin Ross

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Abstract

Motivated by work of Gusein-Zade, Luengo, and Melle-Hern\'andez, we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincar\'e polynomials of Z_3-equivariant Hilbert schemes of points in the plane, where Z_3 acts diagonally. This generating series has a conjectural product formula, a proof of which has remained elusive over the last ten years. We introduce a new combinatorial correspondence between partitions of n and {1,2}-compositions of n, which behaves well with respect to the statistic in question. As an application, we use this correspondence to compute the highest Betti numbers of the Z_3 equivariant Hilbert schemes.

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Deborah Castro, Dustin Ross. 2018-10-12. Topology of Z_3 equivariant Hilbert schemes. https://arxiv.org/abs/1810.05750

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