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Báo cáo toán học: " COMBINATORIAL PROOFS OF CAPELLI’S AND TURNBULL’S IDENTITIES FROM CLASSICAL INVARIANT THEORY"

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@math.u-strasbg.fr . Supported in part by NSF grant DM8800663 Department of Mathematics Temple University Philadelphia PA 19122 U.S.A. zeilberg@euclid.math.temple.edu . 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.