Báo cáo toán học: "A Pfaffian–Hafnian Analogue of Borchardt’s Identity"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Pfaffian–Hafnian Analogue of Borchardt’s Identity. | A Pfaffian-Hafnian Analogue of Borchardt s Identity Masao ISHIKAWA Faculty of Education Tottori University Koyama Tottori Japan ishikawa@ Hiroyuki KAWAMUKO Faculty of Education Mie University Tsu Mie Japan kawam@ Soichi OKADA Graduate School of Mathematics Nagoya University Chikusa-ku Nagoya Japan okada@ Submitted Sep 13 2004 Accepted Jun 6 2005 Published Jun 14 2005 Mathematics Subject Classifications 05E05 Abstract We prove Pf Xi - Xj Xi Xj 2y i ij 2n Xi - Xj Xi Xj n 1 i j 2n Hf f--1--- Xi Xj 1 ij 2n and its variants by using complex analysis. This identity can be regarded as a Pfaffian-Hafnian analogue of Borchardt s identity and as a generalization of Schur s identity. 1 Introduction Determinant and Pfaffian identities play a key role in combinatorics and the representation theory see for example 4 5 6 8 10 11 . Among such determinant identities the central ones are Cauchy s determinant identities 2 I i i j n xj xi yj yi nAi Xi Vj Ĩỉ1 i j n xj - xi Vj - Vi ntj i i - XiVj 1 2 THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N9 1 C. W. Borchardt 1 gave a generalization of Cauchy s identities dpt 1 A Hi i i n xi xi vi Vi e i 1 A e ta. Vi 2 i i j n II x Vi permVi Vi i ij 3 1 1 ni i i n xi - xi vi - yi Í 1 1 -xiVi 2Ji i n QZ i 1 - XVi permv -xivJi i i n 4 Here perm A is the permanent of a square matrix A dehned by perm A m 2Ơ 2 anơ n . 2 Sn This identity 3 is used when we evaluate the determinants appearing in the 0-enumeration of alternating sign matrices see 11 . I. Schur 12 gave a Pfaffian analogue of Cauchy s identity 1 in his study of projective representations of the symmetric groups. Schur s Pfaffian identity and its variant 9 14 are Pf xi - xi xi xw i i i 2n n i i i 2n xi - xi xi xi xi - xi 1 - xixi J i ii 2n n i i i 2n xi - xi 1 - xixi 5 6 In this note we give identities which can be regarded as Pfaffian analogues of Borchardt s identities 3 4 and as generalizations of Schur s identities 5 6 . Theorem .