The Calculation of Clebsh-Gordan Coefficients for the Permutation Group by the Eigenfunction Method

TL;DR

This paper details a pedagogical approach using the eigenfunction method to compute Clebsch-Gordan coefficients for the permutation group, addressing degeneracy removal with additional class operators.

Contribution

It elaborates a detailed, step-by-step eigenfunction method for calculating permutation group Clebsch-Gordan coefficients, including degeneracy resolution techniques.

Findings

Eigenfunctions yield the basis vectors for the permutation group

Degeneracy is resolved using subgroup class operators or state permutation operators

Complete basis vectors for the permutation group are obtained

Abstract

We use the eigenfunction method to calculate the Clebsh-Gordan coefficients for the permutation group . This method is well-established by Jin-Quan Chen. Here we elaborate the detailed procedures for the pedagogical purpose. Due to the nature of the symmetry, one may get the degeneracy from the solution of eigenfunctions for given one class operator. In order to remove the degeneracy we use extra class operators, which may be the subgroup class operator or even the state permutation operator. In doing so, a variety of eigenvalues come out. Every eigenfunction is therefore obtained, and basis vectors are completely found.