Geodesic
Jack Polynomials and Orientability Generating Series of Maps
Séminaire lotharingien de combinatoire, Tome 70 (2013-2014)

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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. 

@article{SLC_2013-2014_70_a9,
     author = {Maciej Do{\l}\k{e}ga and Valentin F\'eray and Piotr \'Sniady},
     title = {Jack {Polynomials} and {Orientability} {Generating} {Series} of {Maps}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {70},
     year = {2013-2014},
     url = {http://geodesic.mathdoc.fr/item/SLC_2013-2014_70_a9/}
}
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AU  - Maciej Dołęga
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AU  - Piotr Śniady
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%F SLC_2013-2014_70_a9
Maciej Dołęga; Valentin Féray; Piotr Śniady. Jack Polynomials and Orientability Generating Series of Maps. Séminaire lotharingien de combinatoire, Tome 70 (2013-2014). http://geodesic.mathdoc.fr/item/SLC_2013-2014_70_a9/