A new approach to permutation group representation

Abstract

The new approach to group representation theory is applied to the treatment of the permutation group. It is shown that obtaining (1) the primary characters and the fractional parentage coefficients, (2) the Yamanouchi bases and the Clebsch–Gordan coefficients, and (3) the irreducible matrix elements of the permutation group S ( f), are all simplified to a unified procedure — diagonalizing a certain operator in the corresponding representation. The operator to be diagonalized for the above three problems is (1) the 2-cycle class operator Cf of S ( f), (2) an appropriate linear combination of the f-1 2-cycle class operators Cf, Cf−1,...,C2 of the group chain S( f)⊇ S( f−1)⊇...⊇ S(2), and (3) an appropriate linear combination of 2f−3 2-cycle class operators Cf, Cf−1,...,C2, 𝒞f−1,...,𝒞2, 𝒞i being the 2-cycle class operator of the subgroup 𝒮(i) of the state permutation group 𝒮(f), respectively. This method, the eigenfunction method, is simpler in concept, yet more powerful in practical calculations.