Explicit expressions for the central characters of the symmetric group

Abstract

A conjecture concerning the construction of explicit expressions for the central characters of the symmetric group S n is presented. The expression for the central characters corresponding to a class with a given cycle structure (1) l1 (2) l2 … (n) l n is a polynomial in the symmetric power sums over the “contents” of the Young diagram specifying the irreducible representation of interest. Each term in this polynomial is a product of symmetric power sums multiplied by a polynomial in n. Both the degrees of the polynomials in n and the powers of the various symmetric power sums appearing in each term are specified by an algorithm involving sets of partitions of integers associated with the lengths of the nontrivial cycles specifying the class of interest and combinations thereof. The coefficients of the polynomials in n are obtained by solving a system of linear equations which arise from the evaluation of the proposed expression for the central characters with respect to Young diagrams whose number of boxes is less than ∑ n i=2 ib i,, and equating to zero.