On a paper of fox about a method for calculating the ordinary irreducible characters of symmetric groups

Abstract

R. F. Fox considered in his paper [2] a method for calculating the ordinary irreducible characters of the symmetric group S n from the so-called Frobenius compound characters, a method which is based on the fact that the matrix of the multiplicities of irreducible constitutents of the compound characters is a triangular one for a certain order of rows and columns. This matrix is discussed here. That this matrix is a triangular one Fox derives from relations between idempotents. However this can be seen directly by an easy application of the so-called Littlewood-Richardson rule, which also permits evaluation of the entries of this matrix. Further remarks can be made which come from this rule, too, and which facilitate and shorten the computation. They also answer some questions of Fox concerning the general form of this matrix. First column and main diagonal consist of 1's, and the last row is the row of the degrees of the irreducible representations of S n . An additional remark concerns rows which can be used for those matrices for higher degrees n again. For every degree n 0 a number m(n 0) is derived so that the non-vanishing part of the i-th row for every i≤m(n 0) is the same in every i-th row for all n≥ n 0. Fox observed that the first five rows of this matrix for S 5 are the first five rows of the partition number table. An example is given that this cannot be generalized to higher degrees n.