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

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 = ( σ 1 , σ 2 , …, σ p ) with σ 1 + σ 2 + … + σ p = 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 .