Motivated by work of Gusein-Zade, Luengo, and Melle-Hernández, we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincaré polynomials of $\mathbb{Z}_3$-equivariant Hilbert schemes of points in the plane, where $\mathbb{Z}_3$ 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 $\mathbb{Z}_3$-equivariant Hilbert schemes.
@article{10_37236_8290,
author = {Deborah Castro and Dustin Ross},
title = {Topology of {\(\mathbb{Z}_3\)-equivariant} {Hilbert} schemes},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {1},
doi = {10.37236/8290},
zbl = {1409.14010},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8290/}
}
TY - JOUR
AU - Deborah Castro
AU - Dustin Ross
TI - Topology of \(\mathbb{Z}_3\)-equivariant Hilbert schemes
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8290/
DO - 10.37236/8290
ID - 10_37236_8290
ER -
%0 Journal Article
%A Deborah Castro
%A Dustin Ross
%T Topology of \(\mathbb{Z}_3\)-equivariant Hilbert schemes
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8290/
%R 10.37236/8290
%F 10_37236_8290
Deborah Castro; Dustin Ross. Topology of \(\mathbb{Z}_3\)-equivariant Hilbert schemes. The electronic journal of combinatorics, Tome 26 (2019) no. 1. doi: 10.37236/8290
