Abstract

The Dade Conjectures, which relate the numbers of irreducible ordinary or projective characters with given defects and given inertia groups of certain local subgroups of a finite group are verified for the sporadic simple McLaughlin Group McL.

AMS (MOS) subject classification Numbers: 20C15 (20C20 20C25).

Keywords: Dade Conjectures, characters, blocks, defects.

Introduction

In the Representation Theory of Finite Groups there is a large number of open problems and long-standing conjectures. Many of these originate from a famous lecture [4] of R. Brauer, in which he described the subject by listing its most interesting and natural open problems and which stimulated an enormous amount of further research. About 10 years later an observation of J. McKay [17] on character degrees of finite groups led to a series of conjectures, notably the Alperin-McKay-Conjecture, and the Alperin Weight Conjecture [1], which has been reformulated in several ways, see for example [15]. The latest in this series seem to be the conjectures of Dade given in [6], [7], and [8]. Any of these conjectures implies the Alperin Weight Conjecture. The strongest form of Dade's conjectures, the “Inductive Conjecture”, which implies all others, has the big advantage, that it can be shown to hold for all finite groups provided that it could be verified for all finite simple groups.