Universität zu Köln

Bringmann, Kathrin, Craig, William, Males, Joshua and Ono, Ken (2022). Distributions on partitions arising from Hilbert schemes and hook lengths. Forum Math. Sigma, 10. CAMBRIDGE: CAMBRIDGE UNIV PRESS. ISSN 2050-5094

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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 C-2. For the Hilbert schemes, we prove that homology is equidistributed as n -> infinity. For t-hooks, we prove distributions that are often not equidistributed. The cases where t is an element of {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 F-1 (xi; q) := Pi(infinity)(n=1) (1- xi q(n)), F-2 (xi; q) := Pi(infinity)(n=1) (1- (xi q)(n)) and F-3 (xi; q) := Pi(infinity)(n=1) (1- xi(-1)(xi q)(n)).

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Bringmann, Kathrin UNSPECIFIED UNSPECIFIED UNSPECIFIED Craig, William UNSPECIFIED UNSPECIFIED UNSPECIFIED Males, Joshua UNSPECIFIED UNSPECIFIED UNSPECIFIED Ono, Ken UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-674602
DOI:	10.1017/fms.2022.45
Journal or Publication Title:	Forum Math. Sigma
Volume:	10
Date:	2022
Publisher:	CAMBRIDGE UNIV PRESS
Place of Publication:	CAMBRIDGE
ISSN:	2050-5094
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language FORMULA; PROOF Multiple languages Mathematics, Applied; Mathematics Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/67460