Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: COMBINATORIAL PROOFS OF CAPELLI’S AND TURNBULL’S IDENTITIES FROM CLASSICAL INVARIANT THEORY. | COMBINATORIAL PROOFS OF CAPELLI S AND TURNBULL S IDENTITIES FROM CLASSICAL INVARIANT THEORY BY Dominique FOATA and Doron ZEILBERGER 0. Introduction. Capelli s C identity plays a prominent role in Weyl s W approach to Classical Invariant Theory. Capelli s identity was recently considered by Howe H and Howe and Umeda H-U . Howe H gave an insightful representation-theoretic proof of Capelli s identity and a similar approach was used in H-U to prove Turnbull s T symmetric analog as well as a new anti-symmetric analog that was discovered independently by Kostant and Sahi K-S . The Capelli Turnbulll and Howe-Umeda-Kostant-Sahi identities immediately imply and were inspired by identities of Cayley see T1 Garding G and Shimura S respectively. In this paper we give short combinatorial proofs of Capelli s and Turnbull s identities and raise the hope that someone else will use our approach to prove the new Howe-Umeda-Kostant-Sahi identity. 1. The Capelli Identity. Throughout this paper xi j are mutually commuting indeterminates positions as are Oij momenta and they interact with each other via the uncertainty principle pij xij xij pij h and otherwise xi j commutes with all the pk l if i j k l . Of course one can take pi j h d dxiĩj . Set X xj P pij 1 i j n . Capelli s Identity. For each positive integer n and for 1 i j n let n 1 1 Aij xkipkj h n i ij k 1 Then CAP sgn ơ Aơ1ị1 Aơn n det X. det P. Departement de mathematique Universite Louis-Pasteur 7 rue Rene Descartes F-67084 Strasbourg Cedex France foata@ . Supported in part by NSF grant DM8800663 Department of Mathematics Temple University Philadelphia PA 19122 . zeilberg@ . THE ELECTRONIC JOURNAL OF COMBINATORICS 1 1994 R1 2 Remark 1. The Capelli identity can be viewed as a quantum analog ot the classical Cauchy-Binet identity det XP det X. det P when the entries of X and P commute and indeed reduces to it when h 0. The matrix A is X P with quantum correction h n i Si j. Remark 2.
