Reduction theorems for relative Grothendieck rings

Abstract

Introduction.Relative Grothendieck rings arise naturally when one considers modular representations of a finite group G, and their restrictions to some fixed subgroup H of G.This article continues our earlier work on the subject [4], but can be read independently of that work.Let G be a finite group, and let O be a field of characteristic p, where we assume p^O to avoid trivial cases.By a "(/-module" we mean always a finitely generated left QG-module.Form the free abelian group ¿a? on the symbols [M], where M ranges over the isomorphism classes of G-modules ; let 38 be the subgroup of ¿é generated by all expressions [M1]-[M2]-[MZ], where M^M2 © M3.The factor group ¿á¡31 will be called the Green ring or representation ring of G.We shall denote it by a(G, G), in order to conform with notation to be introduced later.Define (2) This generalizes Lemma 2.6 of [4].