Classifying irreducible representations in characteristic zero
Alexandre Turull · Cambridge University Press eBooks · 2003
We discuss techniques which allow us to classify the representations of some finite groups over non-algebraically closed fields of characteristic zero. We propose the use of Clifford classes as a convenient and effective way to describe and calculate our answers. Furthermore, we argue for the use of global, rather than local, calculations for the classical groups. Introduction Given a finite group G and a field F in characteristic zero, a basic problem is to understand the representations of G as automorphisms of finite dimensional vector spaces over F . Maschke's Theorem allows us to concentrate on the irreducible representations of G . The use of characters is a convenient and useful way to work with isomorphism classes of representations. However, the computation of the isomorphism types of irreducible representations is often difficult. This problem amounts to the calculation of the Schur index m F (χ) for each complex irreducible character χ ∈ Irr( G ). Richard Brauer contributed two important tools for the solution of these problems. First, instead of calculating the Schur indices of χ over F , for all F , one can now calculate an element of the Brauer group [χ] over a suitable field, and one deduces from it all the Schur indices of χ for all fields F . Second, his Characterization of Characters Theorem, and related ideas, show that one could calculate Schur indices by understanding the restriction of the character to certain solvable subgroups, and the Schur indices of the irreducible characters of these solvable subgroups.