Gaussian fluctuations of Jack-deformed random Young diagrams

Dołęga, Maciej; Śniady, Piotr

doi:10.1007/s00440-018-0854-9

Cited by 13 publications

(19 citation statements)

References 31 publications

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“…Theorem 1.9 plays a prominent role in our forthcoming paper [Śni19b] which is devoted to this topic. A convenient tool for proving Gaussianity of fluctuations of random partitions is approximate factorization property for characters [Śni06,DŚ18] which is formulated in the language of certain cumulants. In the case of the linear characters of the symmetric groups this property was known to be true [Śni06].…”

Section: Theorem 13 ([Fś11a]mentioning

confidence: 99%

Linear versus spin: representation theory of the symmetric groups

Matsumoto¹,

Śniady²

2020

Algebraic Combinatorics

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We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition ξ with analogous normalized linear characters evaluated on the double partition D(ξ). We also relate some natural filtration on the usual (linear) Kerov-Olshanski algebra of polynomial functions on the set of Young diagrams with its spin counterpart. Finally, we give a spin counterpart to Stanley formula for the characters of the symmetric groups.2010 Mathematics Subject Classification. 20C25 (Primary); 20C30, 05E05 (Secondary).Key words and phrases. projective representations of the symmetric groups, linear representations of the symmetric groups, asymptotic representation theory, Stanley character formula.

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Section: Theorem 13 ([Fś11a]mentioning

confidence: 99%

Linear versus spin: representation theory of the symmetric groups

Matsumoto¹,

Śniady²

2020

Algebraic Combinatorics

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“…Firstly, since Jack characters are related to Jack polynomials, a better understanding of the former might shed some light on the latter. Secondly, they can be used in order to investigate some natural deformations of classical random Young diagrams [Śni16,DŚ16]. Thirdly, numerical data [Las09b] as well as some partial theoretical results [Las08, Las09a, DFŚ14,Śni15,Śni16] indicate that they might have a rich algebraic-combinatorial or representation-theoretic structure.…”

Section: Dual Combinatorics Of Jack Polynomials Lassalle [Las08 Las09amentioning

confidence: 99%

Bijection Between Oriented Maps and Weighted Non-Oriented Maps

Czyżewska-Jankowska¹,

Śniady

2017

Electron. J. Combin.

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We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions.2010 Mathematics Subject Classification. Primary 05E05; Secondary 20C30, 05C10, 05C30 .

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“…First, we recall that one of the most typical application of cumulant is to show that a certain family of random variables is asymptotically Gaussian. Especially, when one deals with discrete structures, the main technique is to show that cumulants have a certain small cumulant property, which is in the same spirit as our Theorem 1.3; see [4][5][6]26]. It is therefore natural to ask for a probabilistic interpretation of Theorem 1.3.…”

Section: Related Problemsmentioning

confidence: 99%

“…4 We recall that we need to prove that for any positive integer r , and for any partitions λ 1 , . .…”

Section: Strong Factorization Property Of Interpolation Macdonald Polmentioning

confidence: 99%

Strong factorization property of Macdonald polynomials and higher-order Macdonald’s positivity conjecture

Dołęga

2017

J Algebr Comb

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We prove a strong factorization property of interpolation Macdonald polynomials when q tends to 1. As a consequence, we show that Macdonald polynomials have a strong factorization property when q tends to 1, which was posed as an open question in our previous paper with Féray. Furthermore, we introduce multivariate q, t-Kostka numbers and we show that they are polynomials in q, t with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate q, t-Kostka numbers are in fact polynomials in q, t with nonnegative integer coefficients, which generalizes the celebrated Macdonald's positivity conjecture.

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Gaussian fluctuations of Jack-deformed random Young diagrams

Cited by 13 publications

References 31 publications

Linear versus spin: representation theory of the symmetric groups

Linear versus spin: representation theory of the symmetric groups

Bijection Between Oriented Maps and Weighted Non-Oriented Maps

Strong factorization property of Macdonald polynomials and higher-order Macdonald’s positivity conjecture

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