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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 that 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 that 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 of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*} $$
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Limiting Betti distributions of Hilbert schemes on n points
Abstract Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $$(\mathbb {C}^{2})^{[n]}$$ on $$n$$ points, as $$n\rightarrow +\infty ,$$ is a Gumbel distribution . In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $$((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $$A\geq 2.$$ Furthermore, if $$p_{k}(A;n)$$ denotes the number of partitions of $$n$$ with exactly $$k$$ parts that are multiples of $$A$$ , then we obtain the asymptotic $$ \begin{align*} p_{k}(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^{k}}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, \end{align*} $$ a result which is of independent interest.
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- Award ID(s):
- 2055118
- PAR ID:
- 10404468
- Date Published:
- Journal Name:
- Canadian Mathematical Bulletin
- Volume:
- 66
- Issue:
- 1
- ISSN:
- 0008-4395
- Page Range / eLocation ID:
- 243 to 258
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4153/S0008439522000261
