Determinantal method and the Littlewood-Richardson rule

Abstract

Determinantal method is a powerful tool to study invariants and representations of classical groups. Let R, = Z![x,,,],, , = 1, 2 ,,,,, n be the integral coordinate ring of the module of all n x n integral matrices. In [7], Doubilet, Rota, and Stein gave a beautiful theory on combinatorial approach to invariant theory using a nice basis of R, called standard basis whose elements are simply parametrized by means of “standard tableaux.” This standard basis also occurs in the theory of flag manifolds (cf. [12], [17]). As application of this theory, Procesi and DeConcini gave many results on the characteristic free approach to the invariants and represen- tation spaces of classical groups [4], [S J, 161. In particular, a representation space of each finite dimensional irreducible representation of GL(n, C) is realized in R, @I C canonically. This representation space plays a key roll in application to physics, since each weight vector in this representation space is actually an element of standard basis. Now let us consider the tensor product representation of two arbitrarily irreducible representations pi, and pfl of GL(n, @). Tensor product representations have deep relation with the interaction of particles (cf. [Z]). The branching rule of the tensor product representation pA @ pp into its irreducible constituents is described using combinatorial methods on Young diagrams. This rule, the Littlewood-Richardson rule, was first found by Littlewood [14] and proved rather recently (cf. [15]). However, in application to physics, not only the branching rule but also more detailed information of the representation spaces of tensor product representations is required. In case of GL(2, C), the decomposition of pj.@pp is multiplicity free. So in this case, each weight vector of each irreducible constituent of