Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities. | Noncommutative determinants Cauchy-Binet formulae and Capelli-type identities I. Generalizations of the Capelli and Turnbull identities Sergio Caracciolo Dipartimento di Fisica and INFN Universita degli Studi di Milano via Celoria 16 I-20133 Milano ITALY Alan D. Sokal Department of Physics New York University 4 Washington Place New York NY 10003 USA sokal@ Andrea Sportiello Dipartimento di Fisica and INFN Universita degli Studi di Milano via Celoria 16 I-20133 Milano ITALY Submitted Sep 20 2008 Accepted Aug 3 2009 Published Aug 7 2009 Mathematics Subject Classification 15A15 Primary 05A19 05A30 05E15 13A50 15A24 15A33 15A72 17B35 20G05 Secondary . Abstract We prove by simple manipulation of commutators two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull s Capelli-type identities for symmetric and antisymmetric matrices. Key Words Determinant noncommutative determinant row-determinant columndeterminant Cauchy-Binet theorem permanent noncommutative ring Capelli identity Turnbull identity Cayley identity classical invariant theory representation theory Weyl algebra right-quantum matrix Cartier-Foata matrix Manin matrix. Also at Department of Mathematics University College London London WC1E 6BT England. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R103 1 1 Introduction Let R be a commutative ring and let A aij nj- 1 be an n X n matrix with elements in R. Define as usual the determinant n det A sgn ơ- ỊỊ aiơ i . One of the first things one learns about the determinant is the multiplicative property det AB det A detB . More generally if A and B are m X n matrices and I and J are subsets of n 1 2 . n of cardinality I J r then one has the Cauchy-Binet formula det AtB ij det AT iL det Blj L c m L r det Ali det blj L c m L r where
