Symmetry Classes: Functions of Three Variables

Abstract

The attempt to generalize this fact to functions of n variables led Alfred Young to develop his theory of symmetrizers, now subsumed in the theory of representations of the symmetric group. According to this theory, a function of n variables is uniquely expressible as the sum of Pn functions, each one belonging to a different symmetry class, where Pn is the number of partitions of the integer n. Unfortunately, a simple intuitive description of such symmetry classes has never been given except for n = 2. It is known that for functions of three variables, there is only one other symmetry class besides the two obvious symmetry classes of symmetric functions and of skew-symmetric functions. We give this third symmetry class a very simple characterization, one that seems to have been overlooked. We show that it consists of all cyclic-symmetric functions. We prove that every function of three variables is uniquely expressible as the sum of a symmetric function, a skew-symmetric function and a cyclic-symmetric function. To make this note self-contained, we have added a short derivation of some known formulas. We believe that the underlying idea of this note will extend to functions of n variables. We hope the present note will at least entice the reader to further study of the vast theory of symmetry classes.