Three-pion states and a new approach to the permutation groupS3. I

Abstract

A new method is developed for picking a given irreducible representation of the permutation group $S_3$ out of the product of many representations. It is then used to construct three-pion wave functions of arbitrary spin and parity. Applications of the method to three-fermion systems are briefly described. The essence of the method is to represent the two-dimensional representation of $S_3$ as a "complex number" in an Argand diagram; because the action of $S_3$ involves simple rotations and reflections in the diagram, the behavior of products of many two-dimensional representations is easy to analyze.