Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: HARMONIC FUNCTIONS ON MULTIPLICATIVE GRAPHS AND INTERPOLATION POLYNOMIALS. | HARMONIC FUNCTIONS ON MULTIPLICATIVE GRAPHS AND INTERPOLATION POLYNOMIALS Alexei Borodin and Grigori Olshanski Department of Mathematics The University of Pennsylvania Philadelphia PA 19104-6395 . borodine@ Dobrushin Mathematics Laboratory Institute for Problems of Information Transmission Bolshoy Karetny 19 101447 Moscow GSP-4 RUSSIA. olsh@ olsh@ Submitted November 22 1999 Accepted May 15 2000. Abstract. We construct examples of nonnegative harmonic functions on certain graded graphs the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur s S and P functions and with Jack symmetric functions. As a by-product we compute certain Selberg-type integrals. Contents 0. Introduction 1. The general formalism 2. The Young graph 3. The Jack graph 4. The Kingman graph 5. The Schur graph 6. Finite dimensional specializations . Truncated Young branching . T-shaped Young branching 1991 Mathematics Subject Classification. Primary 05E10 Secondary 31C20 60C05. One of the authors . was supported by the Russian Foundation for Basic Research under grant 98-01-00303. Typeset by Ạ S-T .X 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 7 2000 R28 2 . Truncated Kingman branching . Truncated Schur branching 7. Appendix References 0. Introduction Let Y denote the lattice of Young diagrams ordered by inclusion. For fl A 2 Y we write A 1 if A covers 1 . A differs from 1 by adding a box. We consider Y as a graph whose vertices are arbitrary Young diagrams 1 and the edges are couples 1 A such that A fl. We shall call Y the Young graph. A function 1 is called a harmonic function on the Young graph VK if it satisfies the condition 1 n A 812 Y- .1 Ă A p. We are interested in nonnegative harmonic functions normalized at the empty diagram 1. Such functions form a convex set denoted as H Y . The .
