THE CLEBSCH-GORDAN DECOMPOSITION AND THE COEFFICIENTS FOR THE SYMMETRIC GROUP

Abstract

Publisher Summary This chapter discusses the decomposition of a tensor product of two irreducible representations of the symmetric group into a direct sum of irreducible representations, which provides the formalism for the calculation of the Clebsch–Gordan coefficients. The chapter further presents the development of a computer program using this formalism. The defining space for the left regular representation of the symmetric group is the group ring. The group ring elements are operators, each given by a linear combination of permutations. There are several standard constructions for a basis of the group ring. The coefficients in the expansion of an element of the group in terms of the basis vectors give the entries of the matrices of the left regular representation of the symmetric group. The irreducible representations of the group are found by the reduction of the left regular representation. In this reduction, the irreducible representations fall into classes of equivalent representations.