tailieunhanh - Báo cáo toán học: "A normalization formula for the Jack polynomials in superspace and an identity on partitions"

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@ Yvan Le Borgne CNRS LaBRI Universite de Bordeaux 1 351 Cours de la Liberation 33405 Talence Cedex France Philippe Nadeau Fakultat fur Mathematik Universitat Wien Nordbergstrafie 15 1090 Vienna Austria 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. . was partially supported by the French Agence Nationale de la Recherche projects SADA ANR-05-BLAN-0372 and MARS ANR-06-BLAN-0193. . 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 .