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

Abstract

First, we introduce a class of operations, called ⊘-operations, on the repre-sentation rings of the classical Weyl groups 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 Wely groups, can be expressed as polynomials in the ⊘ poerations 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 ⊘-operations generate the λ-operations, one can deduce that Q ⊗ R(W( _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 Wely groups W(B_k ) and W(D_k ) ir follows, as a corollary, that Q ⊗ R(W(B_k )) is generated by two elements as a λ-ring over Q.