Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric Groups

Murnaghan (9) has proposed the following method of analyzing the Kronecker product of two symmetric group representations. If (λ) = (λ1, λ2, … , λi) is a partition of p, the representation of the symmetric group on n symbols corresponding to the partition (n — p, λ1 , … , λi) is denoted by [λ] and is said to be of depth p. If [λ] is of depth p and [μ] of depth q, then the terms in the Kronecker product [λ] X [μ] of depth p + q are terms which correspond to the terms in the product of S-functions {λ} {μ}).