Duality of a Young diagram describing a representation and dimensionality formulas

Abstract

We discuss the standard dimensionality formulas for irreducible representations of continuous Lie groups SU(n), O(n), and Sp(n). Every Young diagram describes not only an irreducible representation of any of the groups SU(n), O(n), and Sp(n) of dimension n, but it also describes an irreducible representation of the symmetric group SN, where N is the number of boxes in the diagram. This property of a Young diagram leads to the factorization of the dimensionality formulas for irreducible representations of any of these continuous groups into two factors. One factor which depends upon the dimension n of the group appears as a polynomial (with integral roots) of degree N. If we choose the leading coefficient of this polynomial as unity, the other factor turns out to be just 1/N! times the dimensionality of the associated irreducible representation of the symmetric group on N symbols. The polynomial depends upon the type of the group under study, i.e., whether it is an SU(n) or O(n) or Sp(n). We also give simple recipes to read the dimensionality of the representations of these groups from the Young diagram following our formulas.