Asymptotics of Jack characters

Śniady, Piotr

doi:10.1016/j.jcta.2019.02.020

Cited by 9 publications

(6 citation statements)

References 19 publications

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“…The orbits of this action define an equivalence relation ∼ on X l ; the corresponding equivalence classes have a natural combinatorial interpretation as non-labeled, rooted, bicolored, oriented maps with l edges which is out of scope of the current paper (see [Śni18,Section 1.4…”

Section: 82mentioning

confidence: 99%

Gaussian fluctuations of Jack-deformed random Young diagrams

Dołęga

Śniady

2018

Probab. Theory Relat. Fields

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Section: 82mentioning

confidence: 99%

Gaussian fluctuations of Jack-deformed random Young diagrams

Dołęga

Śniady

2018

Probab. Theory Relat. Fields

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“…However, we emphasize that Dołęga-Śniady establish limit shapes and Gaussian fluctuations for a large class of random partitions in [19] by delicate analysis of asymptotics of Jack characters. For background, see Śniady [58,59]. Relating our methods for Jack measures to the models and techniques in Dołęga-Śniady [19] would be especially exciting in light of the connection between our ribbon paths and the work of Chapuy-Dołęga [14] in §[B].…”

Section: 17mentioning

confidence: 99%

Gaussian Asymptotics of Jack Measures on Partitions from Weighted Enumeration of Ribbon Paths

Moll¹

2020

Preprint

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In this paper we determine two asymptotic results for Jack measures M (v out , v in ), a measure on partitions defined by two specializations v out , v in of Jack polynomials proposed by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming v out = v in , we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to diverging, fixed, and vanishing values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call "ribbon paths", show for arbitrary v out , v in that certain Jack measure joint cumulants are weighted sums of connected ribbon paths on n sites with n − 1 + g pairings, and derive our two results from the contributions of (n, g) = (1, 0) and (2, 0), respectively. Our analysis makes use of Nazarov-Sklyanin's spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack-Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy-Dołęga.

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“…Recently, a generalization of Kerov polynomials involving Jack polynomials has been actively studied in [4,16,21]. In this paper, we work on research in another direction.…”

Section: Kerov Polynomialsmentioning

confidence: 99%

A Spin Analogue of Kerov Polynomials

Matsumoto¹

2018

SIGMA

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Kerov polynomials describe normalized irreducible characters of the symmetric groups in terms of the free cumulants associated with Young diagrams. We suggest wellsuited counterparts of the Kerov polynomials in spin (or projective) representation settings. We show that spin analogues of irreducible characters are polynomials in even free cumulants associated with double diagrams of strict partitions. Moreover, we present a conjecture for the positivity of their coefficients.

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Asymptotics of Jack characters

Cited by 9 publications

References 19 publications

Gaussian fluctuations of Jack-deformed random Young diagrams

Gaussian fluctuations of Jack-deformed random Young diagrams

Gaussian Asymptotics of Jack Measures on Partitions from Weighted Enumeration of Ribbon Paths

A Spin Analogue of Kerov Polynomials

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