A note on the algebra of S-functions

Abstract

1. The particular set of symmetric polynomials known as S-functions has recently been shown to be of importance in a variety of algebraic problems. Many of the applications of these functions depend upon the properties of the operation which Littlewood terms ‘new multiplication’, by which, from two given S-functions {μ}, {ν} of respective degrees m and n, is constructed a symmetric function {μ} ⊗ {ν} of degree mn. Littlewood has devoted a considerable part of his paper (2) to explaining various methods by which this function can be expressed in terms of S-functions of degree mn. None of these methods is really simple (in the sense that the rule for expressing the ordinary product {μ} {ν} in terms of S-functions of degree m + n is simple); and, indeed, it would be unreasonable to expect any simple rule of general validity for writing the resulting expression down since, for instance, a knowledge of the explicit expression for {μ} ⊗ {n} would yield immediately a knowledge of all the linearly independent concomitants, of degree n, of an algebraic form of type {μ} in an arbitrary number of variables. Nevertheless, the evaluation of {μ} ⊗ {ν} in particular cases is often necessary. Of the methods suggested by Littlewood for performing this evaluation the one which seems to be normally the simplest (his ‘third method’) involves a process which is not shown to be free from ambiguity, and which can actually be shown by examples to give, in certain cases, alternative solutions, the choice between which must be made by other considerations. It is therefore perhaps worth while putting on record a quite different method of procedure, which, apart from any intrinsic interest which it may possess, seems to be quite practicable for most of the actual evaluations performed by Littlewood.