Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: BIJECTIONS FOR HOOK PAIR IDENTITIES. | BIJECTIONS FOR HOOK PAIR IDENTITIES C. Krattenthaler Institut fur Mathematik der Universitat Wien Strudlhofgasse 4 A-1090 Wien Austria. e-mail KRATT@ WWW http People kratt Submitted March 4 2000 Accepted April 10 2000. Abstract. Short bijective proofs of identities for multisets of hook pairs arm-leg pairs of the cells of certain diagrams are given. These hook pair identities were originally found by Regev. 1. Introduction. In their work 7 on asymptotic analysis of degrees of sequences of symmetric group characters Regev and Vershik obtained some hook formulas which led them to conjecture surprising identities for multisets of hooks. These identities were shortly thereafter proved independently by Bessenrodt 1 Janson 2 and Regev and Zeilberger 8 . Another such identity was added by Postnikov and Regev 4 . Moving one step ahead Regev 5 observed that in fact all these identities are not only true as identities for multisets of hooks but even as identities for multisets of the corresponding arm-leg pairs. He called the latter and we follow this convention hook pairs. As is shown in 5 these identities imply several nice formulas for special evaluations of Schur and Jack polynomials. All the aforementioned identities feature hooks and arm-leg pairs of regions which are built out of nonshifted Ferrers diagrams. Finally in 6 Regev provided similar identities for regions resulting from shifted diagrams. Regev proves his multiset identities in 5 6 by inductive arguments. The proofs in 2 4 8 are also inductive only Bessenrodt s argument in 1 is combinatorial. The purpose of this paper is to provide short bijective proofs of all these identities. In fact what I am going to demonstrate is that there is just one master bijection out of which all the identities result straightforwardly. In the next section we provide all the relevant definitions and formulate in Theorems 1-3 three key identities from 5 6 which straightforwardly imply all .
