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Báo cáo toán học: "A normalization formula for the Jack polynomials in superspace and an identity on partitions"
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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A normalization formula for the Jack polynomials in superspace and an identity on partitions. | A normalization formula for the Jack polynomials in superspace and an identity on partitions Luc Lapointe Instituto de Matematica y Fisica Universidad de Talca Casilla 747 Talca Chile lapointe@inst-mat.utalca.cl Yvan Le Borgne CNRS LaBRI Universite de Bordeaux 1 351 Cours de la Liberation 33405 Talence Cedex France yvan.leborgne@labri.fr Philippe Nadeau Fakultat fur Mathematik Universitat Wien Nordbergstrafie 15 1090 Vienna Austria philippe.nadeau@univie.ac.at Submitted Jan 28 2008 Accepted May 27 2009 Published Jun 5 2009 Mathematics Subject Classification 05A15 05E05 Abstract We prove a conjecture of 3 giving a closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot. L. L. was partially supported by the Anillo Ecuaciones Asociadas a Reticulados financed by the World Bank through the Programa Bicentenario de Ciencia y Tecnologia and by the Programa Reticulados y Ecuaciones of the Universidad de Talca. ty.L.B. was partially supported by the French Agence Nationale de la Recherche projects SADA ANR-05-BLAN-0372 and MARS ANR-06-BLAN-0193. tp.N. was supported by the Austrian Science Foundation FWF grant S9607-N13 in the framework of the National Research Network Analytic Combinatorics and Probabilistic Number Theory . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R70 1 1 Introduction Let x 0 x1 XN 01 dN be a collection of 2N variables called respectively bosonic and fermionic or anticommuting or Grassmannian obeying the relations xiXj Xjxi xidj djxi and ỡiỡj djdi 02 0 . We call symmetric functions in superspace the ring of polynomials in these variables over the field Q that are invariant under the simultaneous interchange of xi Xj .