Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Asymptotics of Young Diagrams and Hook Numbers. | Asymptotics of Young Diagrams and Hook Numbers Amitai Regev Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot 76100 Israel and Department of Mathematics The Pennsylvania State University University Park PA 16802 . Anatoly Vershik St. Petersburg branch of the Mathematics Institute of the Russian Academy of Science Fontanka 27 St. Petersburg 191011 Russia and The Institute for Advanced Studies of the Hebrew University Givat Ram Jerusalem Israel Submitted August 22 1997 Accepted September 21 1997 Abstract Asymptotic calculations are applied to study the degrees of certain sequences of characters of symmetric groups. Starting with a given partition fl we deduce several skew diagrams which are related to fl. To each such skew diagram there corresponds the product of its hook numbers. By asymptotic methods we obtain some unexpected arithmetic properties between these products. The authors do not know finite nonasymptotic proofs of these results. The problem appeared in the study of the hook formula for various kinds of Young diagrams. The proofs are based on properties of shifted Schur functions due to Okounkov and Olshanski. The theory of these functions arose from the asymptotic theory of Vershik and Kerov of the representations of the symmetric groups. Work partially supported by . Grant . t Partially supported by Grant INTAS 94-3420 and Russian Fund 96-01-00676 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 1997 R22 2 1. Introduction and the main results Asymptotic calculations are applied to study the degrees of certain sequences of characters of symmetric groups Sn n 1. We obtain some unexpected arithmetic properties of the set of the hook numbers for some special families of fixed skew-Young diagrams . The problem appeared in the study of the hook formula for various kinds of Young diagrams. The proof of is based on the properties of shifted Schur functions which appeared in the asymptotic
