From partitions to Hodge numbers of Hilbert schemes of surfaces

Gillman, Nate; Gonzalez, Xavier; Ono, Ken; Rolen, Larry; Schoenbauer, Matthew

doi:10.1098/rsta.2018.0435

Cited by 7 publications

(7 citation statements)

References 18 publications

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“…This is captured by the pleasing formula of Göttsche [10, 11] These q -infinite products often essentially specialise to modular forms, which then leads to asymptotics and distribution results via a standard application of the Circle Method. Indeed, the fourth author and his collaborators carried this out in [9]. Here we consider a prominent situation involving partitions, where modular forms do not arise, a fact that complicates the computation of asymptotics and distributions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distributions on partitions arising from Hilbert schemes and hook lengths

Bringmann

Craig

Males

et al. 2022

Forum of Mathematics, Sigma

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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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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distributions on partitions arising from Hilbert schemes and hook lengths

Bringmann

Craig

Males

et al. 2022

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

“…These q-infinite products often essentially specialize to modular forms, which then leads to asymptotics and distribution results via a standard application of the Circle Method. Indeed, the fourth author and his collaborators carried this out in [9]. Here we consider a prominent situation involving partitions, where modular forms do not arise, a fact which complicates the computation of asymptotics and distributions.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distributions on partitions arising from Hilbert schemes and hook lengths

Bringmann¹,

Craig²,

Males³

et al. 2021

Preprint

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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 C 2 . For the Hilbert schemes, we prove that homology is equidistributed as n → ∞. For t-hooks, we prove distributions which are often not equidistributed. The cases where t ∈ {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(1 − (ξq) n ) and F3(ξ; q) := ∞ n=1 1 − ξ −1 (ξq) n .

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“…In [2], the authors introduce techniques in order to compute the bivariate asymptotic behaviour of coefficients for a Jacobi form in order to answer Dyson's conjecture on the bivariate asymptotic behaviour of the partition crank. This method is used in numerous other papers -for example, in relation to the rank of a partition [8], ranks and cranks of cubic partitions [13], and certain genera of Hilbert schemes [15] (a result that has recently been extended to a complete classification with exact formulae using the Hardy-Ramanujan circle method [9]), along with many other partition-related statistics. Using Wright's circle method [22,23] and following the same approach as [2] we show the following theorem.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%