Elements of the Green ring related to Alperin's weight conjecture

Abstract

J. L. Alperin introduced his celebrated weight conjecture, a major question in the modular representation theory of finite groups, which has now been studied for several decades. Alperin viewed this conjecture as the beginning of a ``Lie theory for the general finite group", as the title of the article indicates. This paper is intended as a tribute to Alperin's insight. In this note, we place an emphasis on projective modules and putting the numerical questions arising from the weight conjecture into a ``module form" within the appropriate Green ring of RG, when G is a finite group, and R is a suitable complete discrete valuation ring with residue field of prime characteristic p. We believe that this work moves in the direction of a structural approach to Alperin's weight conjecture.