On a Result in Young's Quantitative Substitutional Analysis

In (4) Young investigates the representation in terms of his exact seminormal units m^ of substitutional expressions which are unchanged on premultiplication by any permutation of a given set of consecutive letters, or which are changed in sign on premultiplication by an odd permutation of those letters. He illustrates his results by deducing the forms of the matrices representing the positive and negative symmetric groups on a set of consecutive letters. We shall derive Young's results on the positive and negative symmetric groups in a slightly different form, and in terms of his orthogonal representation of the symmetric group £fn of degree n corresponding to a partition (X) of n. The standard tableaux of shape (A) will be denoted by Slt S2, ..., Sf.