Exterior power operations in the representation theory of the classical Weyl groups

Abstract

First, we introduce a class of operations, called ø-operations, on the representation rings of the classical Weyl groups and W(B k) and W(D k). These operations are shown to generate the exterior power operations in the representation rings R(W(B k)) and R(W(D k)). Given integers l h satisfying l+h =k, let β be a partition of l and α a partition of h. The main theorem shows that induced representations of the form where W βα is a product of Weyl groups, can be expressed as polynomials in the cooperations acting on the two canonical induced representations. , Next, we show that the set which consists of elements of the form. .is a basis of Q⊗R(W(B k)). Since the ø-operations generate the λ-operations, one can deduce that Q⊗R(W(B k)) is generated as a λ-ring over Q by the elements 1 ⊗ X k and 1 ⊗ Y k By applying a result of Lusztig which characterizes the irreducible representations of the Weyl groups W:(B k) and W(D k) it follows, as a corollary, that Q ⊗ R(W(D k)) is generated by two elements as λ-ring over Q.