On new multiplicity formulas of weights of representations for the classical groups

Abstract

This is a continuation of [6]. In this paper we give the multiplicity formulas of weights in an irreducible representation of the classical groups SO(n, C) and Sp(2n, C) (c GL(2n, C)) using only Young diagrams. Roughly speaking, for each series of classical groups the weight-multiplicities can be expressed as a polynomial function in the ranks. In this direction R. C. King and S. P. 0. Plunkett have presented the formulas of this kind using multinomial coefficients in [S]. (Also see [4].)’ In this paper we present another explicit expression of these polynomial functions. Moreover we characterize the set of weights (dominant integral weights) in each irreducible representation only in terms of Young diagrams and also give the degrees of these polynomial functions and their leading coefficients. Our formulas are based on the known result (concerning the so-called Kostka coefficients) in the case of GL(n) and are a natural generalization from the general linear groups to the other classical groups. As in the case of GL(n), the dominant integral weights of SG(2n + 1) and Sp(2n) (c GL(2n)) can be identified with the Young diagrams. (See Sect. 1.) In case of SG(2n) the situation is a little bit delicate and for this case see Section 1. For each series of the classical groups, starting from these identifications we are able to regard the multiplicities of the dominant integral weight q in the irreducible representation with the highest weight 1 as the function of the rank n for fixed Young diagrams 2 and q. Here “each series of the classical groups” means one of the following: