Distributions on partitions arising from Hilbert schemes and hook lengths

TL;DR

This paper studies the distribution of integer partitions based on invariants from algebraic combinatorics and geometry, proving equidistribution results for Hilbert scheme Betti numbers and analyzing distributions related to hook lengths.

Contribution

It extends the Nekrasov-Okounkov hook product formula to distribution analysis and proves homology equidistribution for Hilbert schemes as n grows large.

Findings

Homology of Hilbert schemes is equidistributed as n approaches infinity.

Distribution of t-hooks often not equidistributed, especially for t=2,3.

Derived asymptotics of infinite products near roots of unity.

Abstract

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of $n$ points on $\mathbb{C}^2.$ For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty.$ For $t$-hooks, we prove distributions which are often not equidistributed. The cases where $t\in \{2, 3\}$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results which are of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\text {and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). $$