Distributions on partitions arising from Hilbert schemes and hook lengths

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 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 .

M2-branes and plane partitions

“…where Li p (z) = ∞ k=1 z k /k p for |z| < 1 is the polylogarithm function and the relation [56] We remark that the leading trilogarithm Li 3 (z) in the grand potential (3.38) also appears in the grand potential for the sphere partition function of the ABJM theory [44] and the ADHM theory [4,59,60]. It is crucial for the N 3/2 growth of the free energy.…”

“…Here we set q = e −β and study the hightemperature limit β → 0 of the grand potential. We write the grand potential (3.5) as The asymptotic behavior of (3.34) can be obtained from the following generalized Euler-Maclaurin summation formula [56]:…”

Asymptotics for the twisted eta-product and applications to sign changes in partitions

“…Asymptotics for the coefficients of (ζq; q) −1 ∞ for |ζ| < 1 were also studied in 2017 by Parry [16] in a more general setting, and the case that ζ is any positive real number was studied Wright [21]. Note that the functions (ζq; q) −1 ∞ have also been studied in recent work of Bringmann, Craig, Males, and Ono [6] in the context of distribution of homology of Hilbert schemes and t-hook lengths.…”

M2-branes and plane partitions

“…Note that the function (3.31) can be expanded as The asymptotic behavior of (3.34) can be obtained from the following generalized Euler-Maclaurin summation formula [57]:…”

“…where Li p (z) = ∞ k=1 z k /k p for |z| < 1 is the polylogarithm function and the relation [57] b j=1…”