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) = (λ 1 , λ 2 , … , λ i ) is a partition of p , the representation of the symmetric group on n symbols corresponding to the partition ( n — p , λ 1 , … , λ i ) 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}).