A positivity conjecture for Jack polynomials

Lassalle, Michel

doi:10.4310/mrl.2008.v15.n4.a6

Cited by 30 publications

(44 citation statements)

References 10 publications

(16 reference statements)

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“…Lassalle obtained a similar identity. Specifically, (3.3) in [14] implies that, for partitions ν and μ such that |ν| ≥ |μ| + (μ) and for any m ≥ 0, (1) , n) = δ μ, (1) .…”

Section: Proposition 813mentioning

confidence: 98%

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

Matsumoto

2011

Ramanujan J

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We study symmetric polynomials whose variables are odd-numbered Jucys-Murphy elements. They define elements of the Hecke algebra associated to the Gelfand pair of the symmetric group with the hyperoctahedral group. We evaluate their expansions in zonal spherical functions and in double coset sums. These evaluations are related to integrals of polynomial functions over orthogonal groups. Furthermore, we give their extension based on Jack polynomials.

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“…Lassalle obtained a similar identity. Specifically, (3.3) in [14] implies that, for partitions ν and μ such that |ν| ≥ |μ| + (μ) and for any m ≥ 0, (1) , n) = δ μ, (1) .…”

Section: Proposition 813mentioning

confidence: 98%

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

Matsumoto

2011

Ramanujan J

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“…While Jack characters have been studied for a long time, the idea, due to Lassalle, to look at them as a function of λ as above is quite recent [Las08,Las09]. Among other things, he proved that, as in the case α = 1, the functions Ch (α) µ span linearly a subalgebra of functions on all Young diagrams, which has a nice characterization: we present these results in Section 2, see in particular Proposition 2.9.…”

Section: Remarkmentioning

confidence: 99%

“…The last characterization is the definition of what is usually called an α-shifted symmetric function [OO97,Las08]. It would be equivalent to ask in the definition of α-polynomial functions that…”

Section: α-Polynomial Functionsmentioning

confidence: 99%

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Dołęga

Féray

2016

Duke Math. J.

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In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian β-ensemble, already suggested by some work of Matsumoto.Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it.2010 Mathematics Subject Classification. 60C05, 05E05 (secondary:60B20).

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“…It was suggested by Lassalle that a combinatorial description of these objects might exist. This combinatorial setup was indicated by some polynomiality and positivity conjectures that he stated in a series of papers [Las08,Las09]. Although these conjectures are not fully resolved, it was proven by us together withŚniady [DFS14] that in some special cases bipartite maps together with some statistics that "measures their non-orientability" give the desired combinatorial setup.…”

Section: Related Problemsmentioning

confidence: 99%

Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

Dołęga¹

2017

Electron. J. Combin.

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Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; 1, 1 + β) with an additional parameter β that may be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has the property that for β ∈ {0, 1}, it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefficients of ψ are polynomials in β with positive integer coefficients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of β is an integer-valued statistic that in some sense "measures the non-orientability" of the corresponding bipartite map.We show that except for two special values of β = 0, 1 for which the combinatorial interpretation of the coefficients of ψ is known, there exists a third special value β = −1 for which the coefficients of ψ indexed by two partitions µ, ν, and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black/white vertex degrees given by µ/ν, respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of ψ. As a consequence, we introduce a collection of integer-valued statistics of maps (η) such that the top-degree of the multivariate generating series of rooted bipartite maps with only one face (called unicellular) with respect to η gives the top-degree of the appropriate coefficients of ψ. Finally, we show that the b-conjecture holds true for all rooted, unicellular bipartite maps of genus at most 2.

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A positivity conjecture for Jack polynomials

Abstract: We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley's ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for partitions having a rectangular shape.

Cited by 30 publications

References 10 publications

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

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