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Jack polynomials and orientability generating series of maps
- Source :
- S\'em. Lothar. Combin., 70:Art. B70j, 50 pp., 2014 (electronic)
- Publication Year :
- 2013
-
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 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 on Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.<br />Comment: v2: change of title, substantial changes of the content v3: substantial changes in the presentation
- Subjects :
- Mathematics - Combinatorics
05E05 (Primary), 05C10, 05C30, 20C30 (Secondary)
Subjects
Details
- Database :
- arXiv
- Journal :
- S\'em. Lothar. Combin., 70:Art. B70j, 50 pp., 2014 (electronic)
- Publication Type :
- Report
- Accession number :
- edsarx.1301.6531
- Document Type :
- Working Paper