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Structure constants of Jack characters

Adam Burchardt

Abstract

In 1996 Goulden and Jackson introduced a family of coefficients $(c_{\mu, \nu}^{\lambda})$ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $J^{(\alpha )}_\pi$. Goulden and Jackson suggested that there is a combinatorics of matchings hidden behind the coefficients $c_{\mu,\nu}^{\lambda}$. This \emph{Matchings-Jack Conjecture} remains open. Jack characters are a generalization of the characters of the symmet\-ric groups, they provide a kind of dual information about the Jack polynomials. We investigate the structure constants $g_{\mu,\nu}^{\lambda}$ for Jack characters. They are a generalization of the connection coefficients for the symmetric groups. We give formulas for the top-degree part of $g_{\mu,\nu}^{\lambda}$ and $c_{\mu,\nu}^{\lambda}$. We present those results in context of Matchings-Jack Conjecture of Goulden and Jackson. 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 expresses a given nested product with respect to those two multiplications as a sum of products of the cumulants. This formula leads to some conclusions about the structure constants of Jack characters. We also show that our formula may be understood as an analogue of Leonov--Shiraev's formula.
Record ID
UAM90b3cc3b466947b48731b0a4e567baa4
Diploma type
Doctor of Philosophy
Author
Adam Burchardt Adam Burchardt,, Wydział Matematyki i Informatyki [nowa struktura organizacyjna] (SNŚ/WMiI)Szkoła Nauk Ścisłych [nowa struktura organizacyjna] (SNŚ)
Title in Polish
Stałe strukturalne charakterów Jacka
Title in English
Structure constants of Jack characters
Language
(en) English
Certifying Unit
Faculty of Mathematics and Computer Science (SNŚ/WMiI/FoMaCS)
Discipline
mathematics / (mathematics domain) / (physical sciences)
Scientific discipline (2.0)
6.3 mathematics
Status
Finished
Defense Date
12-06-2018
Title date
12-06-2018
Supervisor
Piotr Śniady Piotr Śniady,, Wydział Matematyki i Informatyki [nowa struktura organizacyjna] (SNŚ/WMiI)Szkoła Nauk Ścisłych [nowa struktura organizacyjna] (SNŚ)
URL
http://hdl.handle.net/10593/23557 Opening in a new tab
Keywords in English
Jack, ennumerative combinatorics, maps, matchings
Abstract in English
In 1996 Goulden and Jackson introduced a family of coefficients $(c_{\mu, \nu}^{\lambda})$ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $J^{(\alpha )}_\pi$. Goulden and Jackson suggested that there is a combinatorics of matchings hidden behind the coefficients $c_{\mu,\nu}^{\lambda}$. This \emph{Matchings-Jack Conjecture} remains open. Jack characters are a generalization of the characters of the symmet\-ric groups, they provide a kind of dual information about the Jack polynomials. We investigate the structure constants $g_{\mu,\nu}^{\lambda}$ for Jack characters. They are a generalization of the connection coefficients for the symmetric groups. We give formulas for the top-degree part of $g_{\mu,\nu}^{\lambda}$ and $c_{\mu,\nu}^{\lambda}$. We present those results in context of Matchings-Jack Conjecture of Goulden and Jackson. 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 expresses a given nested product with respect to those two multiplications as a sum of products of the cumulants. This formula leads to some conclusions about the structure constants of Jack characters. We also show that our formula may be understood as an analogue of Leonov--Shiraev's formula.
Thesis file
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Uniform Resource Identifier
https://researchportal.amu.edu.pl/info/phd/UAM90b3cc3b466947b48731b0a4e567baa4/
URN
urn:amu-prod:UAM90b3cc3b466947b48731b0a4e567baa4

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