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

Dołęga, Maciej

doi:10.1007/s10801-017-0750-x

Cited by 6 publications

(10 citation statements)

References 26 publications

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“…Remarkably, extensive computer simulations suggest that Macdonald cumulants are, in fact, Schur-positive, which we conjectured in our recent paper [Doł17a]:…”

Section: Introductionmentioning

confidence: 58%

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Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions

Dołęga

2017

Preprint

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We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund, Haiman and Loehr for Macdonald polynomials. We provide several applications of our formula. Firstly, it allows us to give a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover, we prove that Macdonald cumulants are q, t-positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials indexed by hooks. Our combinatorial formula relates Macdonald cumulants to the generating function of G-parking functions, or equivalently to a certain specialization of the Tutte polynomials.

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“…Remarkably, extensive computer simulations suggest that Macdonald cumulants are, in fact, Schur-positive, which we conjectured in our recent paper [Doł17a]:…”

Section: Introductionmentioning

confidence: 58%

“…Macdonald positivity ex-conjecture has seen many generalizations in different directions up to these days. One example of such a generalization called higher-order Macdonald positivity conjecture was presented in our recent work [Doł17a] and will be the main subject of this paper. 1.2.…”

Section: Introductionmentioning

confidence: 95%

Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions

Dołęga

2017

Preprint

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“…In this case the cumulants measure the discrepancy between these two multiplication structures on A. This situation arises naturally in many branches of algebraic combinatorics, for example in the case of Macdonald cumulants [Doł17a,Doł17b] and cumulants of Jack characters [DF17,Śn16].…”

Section: Frameworkmentioning

confidence: 99%

Algebras with two multiplications and their cumulants

Burchardt

2019

J Algebr Comb

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Cumulants are a notion that comes from the classical probability theory, they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov-Shiraev's formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters.

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“…After submission of the current paper, the first author has found a proof of Conjecture 1.5; see [Doł16a].…”

Section: Note Added In Revisionmentioning

confidence: 99%

Cumulants of Jack symmetric functions and the $b$-conjecture

Dołęga

Féray

2017

Trans. Amer. Math. Soc.

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ABSTRACT. Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They made the following conjecture: the coefficients of ψ(x, y, z; t, 1 + β) in the power-sum basis are polynomials in β with nonnegative integer coefficients (by construction, these coefficients are rational functions in β).We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. A key step of the proof is a strong factorization property of Jack polynomials when the Jack-deformation parameter α tends to 0, that may be of independent interest.

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

Cited by 6 publications

References 26 publications

Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions

Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions

Algebras with two multiplications and their cumulants

Cumulants of Jack symmetric functions and the $b$-conjecture

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