tailieunhanh - Báo cáo toán học: "A unified view of determinantal expansions for Jack Polynomials"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A unified view of determinantal expansions for Jack Polynomials. | A unified view of determinantal expansions for Jack Polynomials Leigh Roberts 1 School of Economics and Finance Victoria University P. O. Box 600 Wellington New Zealand Submitted September 28 2000 Accepted December 29 2000 AMS Subject Classification 05E05 Abstract Recently Lapointe et. al. 3 have expressed Jack Polynomials as determinants in monomial symmetric functions m . We express these polynomials as determinants in elementary symmetric functions 6 showing a fundamental symmetry between these two expansions. Moreover both expansions are obtained indifferently by applying the Calogero-Sutherland operator in physics or quasi Laplace Beltrami operators arising from differential geometry and statistics. Examples are given and comments on the sparseness of the determinants so obtained conclude the paper. 1 Introduction Notation The partition A has weight w A . The conjugate partition to A is denoted by A while the transpose of a matrix A is denoted by AT. Variates are denoted by x1 x2 . xn with Di d dxi and w w A w k . throughout. The dominance partial ordering is denoted by thus K k1 k2 . A l1 l2 . k1 k2 . ki l1 l2 . li for all i. The conventional total 1 Thanks are due to Peter Forrester for commenting on an early draft and an anonymous referee who simplified the structure of the paper and suggested improved proofs of Lemma 7 and Theorem 8. Any errors remain the responsibility of the author. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R3 1 R ordering of partitions viz. the reverse lexicographic ordering RLO is denoted by hence 4 3 1 . The mx functions are stacked into a column vector M in RLO so that the ordering of the indices of the vector elements from the top is w w 1 1 w 2 2 w 2 1 1 . . There is an analogous stacking of the ex functions into a column vector E. Overview In 3 the Calogero-Sutherland operator is given by Lapointe et. al. as . n 1 n 7 . H- H av 2 i 1 iDi XjDj . l ij 1 x i j Following Stanley 6 p. 84 we define .