The Dimensions of Irreducible Tensor Representations of the Orthogonal and Symplectic Groups

Abstract

It is well known [13] that the irreducible tensor representations (IRs) of the unitary, orthogonal, and symplectic groups in an n-dimensional space may be specified by means of Young tableaux associated with partitions (σ)s = (σ1, σ2, …, σp) with σ1 + σ2 + … + σp = s. Formulae for the dimensions of the corresponding representations have been established [1; 8; 9; 13] in terms of the row lengths of these tableaux. It has been shown [12] for the unitary group, U(n), that the formula may be written as a quotient whose numerator is a polynomial in n containing s factors, and whose denominator is a number independent of n, which likewise may be expressed as a product of s factors. This formula is valid for all n.