# The Lambda-structure Of The Representation Rings Of The Classical Weyl Groups

1991

Dissertation

## Degree Name

Doctor of Philosophy

## Abstract

First, we introduce a class of operations, called {dollar}\phi{dollar}-operations, on the representation rings of the classical Weyl groups {dollar}{lcub}\cal W{rcub}(B\sb{lcub}k{rcub}){dollar} and {dollar}{lcub}\cal W{rcub}(D\sb{lcub}k{rcub}){dollar}. These operations are shown to generate the exterior power operations in the representation rings {dollar}R({lcub}\cal W{rcub}(B\sb{lcub}k{rcub})){dollar} and {dollar}R({lcub}\cal W{rcub}(D\sb{lcub}k{rcub})).{dollar} Given integers l, h satisfying {dollar}l + h=k{dollar}, let {dollar}\beta{dollar} be a partition of l and {dollar}\alpha{dollar} be a partition of h. The main theorem shows that induced representations of the form {dollar}{dollar}Ind\sbsp{lcub}{lcub}\cal W{rcub}\sb{lcub}\beta,\alpha{rcub}{rcub}{lcub}{lcub}\cal W{rcub}(B\sb{lcub}k{rcub}){rcub}1,{dollar}{dollar}where {dollar}{lcub}\cal W{rcub}\sb{lcub}B,a{rcub}=\prod{lcub}\cal W{rcub}(B\sb{lcub}B{rcub})\times\prod{lcub}\cal W{rcub}(A\sb{lcub}a{rcub}),{dollar} can be expressed as an algebraic combination of {dollar}\phi{dollar}-operations acting on the two canonical induced representations {dollar}{dollar}\eqalign{lcub}X\sb{lcub}k{rcub}&= Ind\sbsp{lcub}{lcub}\cal W{rcub}(B\sb{lcub}k-1{rcub})\times{lcub}\cal W{rcub}(B\sb1){rcub}{lcub}{lcub}\cal W{rcub}(B\sb{lcub}k{rcub}){rcub}1\cr\cr Y\sb{lcub}k{rcub}&= Ind\sbsp{lcub}{lcub}\cal W{rcub}(B\sb{lcub}k-1{rcub}){rcub}{lcub}{lcub}\cal W{rcub}(B\sb{lcub}k{rcub}){rcub}1.\cr{rcub}{dollar}{dollar};Next, we show that the set {dollar}{dollar}\left\{lcub}1 \otimes Ind\sbsp{lcub}{lcub}\cal W{rcub}\sb{lcub}\beta,a{rcub}{rcub}{lcub}{lcub}\cal W{rcub}(B\sb{lcub}k{rcub}){rcub}1\right\{rcub}{dollar}{dollar}is a basis of {dollar}\doubq \otimes R({lcub}\cal W{rcub}(B\sb{lcub}k{rcub})){dollar}. Since the {dollar}\phi{dollar}-operations generate the {dollar}\lambda{dollar}-operations, one can deduce that {dollar}\doubq\otimes R({lcub}\cal W{rcub}(B\sb{lcub}k{rcub})){dollar} is generated as a {dollar}\lambda{dollar}-ring over {dollar}\doubq{dollar} by the elements {dollar}1 \otimes X\sb{lcub}k{rcub}{dollar} and {dollar}1 \otimes Y\sb{lcub}k{rcub}{dollar}. By applying a result of Lusztig which characterizes the irreducible representations of the Weyl groups {dollar}{lcub}\cal W{rcub}(B\sb{lcub}k{rcub}){dollar} and {dollar}{lcub}\cal W{rcub}(D\sb{lcub}k{rcub}){dollar} it follows, as a corollary, that {dollar}\doubq\otimes R({lcub}\cal W{rcub}(D\sb{lcub}k{rcub})){dollar} is generated by two elements as a {dollar}\lambda{dollar}-ring over {dollar}\doubq{dollar}.

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