Bounded Littlewood identities

Rains, Eric M.; Warnaar, S. Ole

doi:10.1090/memo/1317

Cited by 10 publications

(11 citation statements)

References 117 publications

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“…The node in row k and column j has hook length ℎ(𝑘, 𝑗) := (𝜆 𝑘 − 𝑘) + (𝜆 𝑗 − 𝑗) + 1, where 𝜆 𝑗 is the number of nodes in column j. These numbers play many significant roles in combinatorics, number theory and representation theory (for example, see [17,26]). We investigate those hook lengths that are multiples of a fixed positive integer t, the so-called t-hooks.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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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%

“…The node in row k and column j has hook length , where is the number of nodes in column j . These numbers play many significant roles in combinatorics, number theory and representation theory (for example, see [17, 26]).…”

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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“…Our main result is a generalization of the refined Littlewood identity (17) to the case of the spin Hall-Littlewood functions.…”

Section: S Gavrilovamentioning

confidence: 92%

“…For a comprehensive study of Littlewood identities for Hall-Littlewood polynomials, we refer the reader to [14,Ch. III] and to [18,17] for recent developments concerning boxed Littlewood formulae for Macdonald polynomials.…”

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confidence: 99%

Refined Littlewood identity for spin Hall–Littlewood symmetric rational functions

Гаврилова¹

2023

Algebraic Combinatorics

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Fully inhomogeneous spin Hall-Littlewood symmetric rational functions F λ are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials and can be viewed as partition functions in sl(2) higher spin six vertex models.We obtain a refined Littlewood identity expressing a weighted sum of F λ 's over all signatures λ with even multiplicities as a certain Pfaffian. This Pfaffian can be derived as a partition function of the six vertex model in a triangle with suitably decorated domain wall boundary conditions. The proof is based on the Yang-Baxter equation.

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“…In this section, we briefly introduce plethystic notation and substitutions. For more details, see [13,18,23,25].…”

Section: Plethystic Notationmentioning

confidence: 99%