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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
- 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)
- Status
- Finished
- Defense Date
- 12-06-2018
- Title date
- 12-06-2018
- Supervisor
- 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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- File: 1
- Structure constants of Jack characters, File PhD Adam Burchardt.pdf / 1 MB
- PhD Adam Burchardt.pdf
- publication date: 02-01-2020
- Structure constants of Jack characters, File PhD Adam Burchardt.pdf / 1 MB
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- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/phd/UAM90b3cc3b466947b48731b0a4e567baa4/
- URN
urn:amu-prod:UAM90b3cc3b466947b48731b0a4e567baa4