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Jack polynomials and orientability generating series of maps


Dołęga, Maciej; Féray, Valentin; Śniady, Piotr (2014). Jack polynomials and orientability generating series of maps. Seminaire Lotharingien de Combinatoire (SLC), 70(50):online.

Abstract

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture the existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular, it gives a positive answer for Jack characters specialized to Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.

Abstract

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture the existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular, it gives a positive answer for Jack characters specialized to Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.

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Additional indexing

Item Type:Journal Article, not_refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:29 Jan 2015 16:38
Last Modified:14 Feb 2018 22:30
Publisher:Institut de Recherche Mathematique Avancee, Strasbourg
ISSN:1286-4889
OA Status:Closed
Free access at:Official URL. An embargo period may apply.
Publisher DOI:https://doi.org/10.1142/S0129167X1450038
Official URL:http://www.emis.de/journals/SLC/wpapers/s70sniady.pdf

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