Determinantal Expression and Recursion for Jack Polynomials

We describe matrices whose determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of such a matrix is a list of monomial functions, the entries of the sub-diagonal are of the form -(r + s), with r and s N+, the entries above the sub-diagonal are nonnegative integers, and below all entries are 0. The quasi-triangular nature of these matrices gives a recursion for the Jack polynomials allowing for efficient computation. A specialization of these results yields a determinantal formula for the Schur functions and a recursion for the Kostka numbers. 1 

• Type: Journal
• Year: 2000
• Where: COMBINATORICS
• Authors: Luc Lapointe, Alain Lascoux, Jennifer Morse
• Keywords: COMBINATORICS 2000, Jack Polynomials, Monomial Basis, Monomial Functions