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 .