Representations of Finite and Lie Groups

The following sections are included:

• Preface

• Contents

Our topic is the representation theory of finite and, more generally, of compact topological groups. The latter will be defined formally later; for the moment the reader can think of a topological group as a set carrying both a topology and a group structure, which are compatible in the sense that multiplication and inversion are continuous. Examples are SL₂(ℝ) (noncompact) and the special unitary groups SU_n (compact), both of which are important in theoretical physics. A representation of G is a homomorphism of G into Aut_ℂ(V), the group of linear automorphisms of a finite dimensional vector space over the complex number ℂ…

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The following sections are included:

• Representations of G₁ × G₂

• Real Representations

• Exercises

We start with a representation Π : G → Aut(V), and write i^⋆p : H → Aut(V) for its restriction to some subgroup H ⊆ G. When it is necessary to specify the subgroup concerned we will replace i^⋆ by or Res_H. Let W be a subspace of V stable under the action of H, and so corresponding to a subrepresentation θ : H → Aut(W). For each left coset gH, represented by g, it makes sense to define W_g ⊆ V to be ρ(g)W, where this subspace depends on the coset and not on the representative chosen. Action by the elements of G permutes the subspaces W_g among themselves, and their sum is a G-subrepresentation of V…

The following sections are included:

• Alternating and Symmetric Products

• The Representation Ring R(G) and its λ-structure

• * Representations of SL₂(픽_p) in Characteristic p

• Exercises

The following sections are included:

• Induced Representations

• Irreducible Representations of SU₂

• Exercises

The following sections are included:

• Representations of the Lie algebra sl₂

• Exercises

The following sections are included:

• Principal Series for SL₂(_p)

• Discrete Series for SL₂(_p)

• The Non-compact Lie Group SL₂(ℝ)

• Principal Series

• Discrete Series

• Exercises

The following sections are included:

• Appendix A Integration over Topological Groups

• Appendix B Rings with Minimal Condition

• Appendix C Modular Representations

• Solutions and Hints for the Exercises

• Bibliography

• Index