This book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics.

G-invariant. □ The set of morphisms from ϕ to ρ has the additional structure of a vector space, as the following proposition reveals. Proposition 4.1.5. Let φ: G→GL(V ) and ρ: G→GL(W) be representations. Then Hom G(φ,ρ) is a subspace of Hom (V,W). Proof. Let T1, T2 ∈ HomG(ϕ, ρ) and . Then and hence c1T1 + c2T2 ∈ HomG(ϕ, ρ), as required. □ Fundamental to all of representation theory is the important observation, due to I. Schur, that roughly speaking morphisms between irreducible

deserve a name of their own. Definition 4.3.6 (Class function). A function is called a class function if f(g) = f(hgh − 1) for all g, h ∈ G, or equivalently if f is constant on conjugacy classes of G. The space of class functions is denoted Z(L(G)). In particular, characters are class functions. The notation Z(L(G)) suggests that the class functions should be the center of some ring, and this will indeed be the case. If is a class function and C is a conjugacy class, f(C) will denote the

retained the above notation. Proof. We already know that B is an orthonormal set by the orthogonality relations (Theorem 4.2.8). Since , it follows B is a basis. □ Next we show that χ1, …, χs is an orthonormal basis for the space of class functions. Theorem 4.4.7. The set χ 1 ,…,χ s is an orthonormal basis for Z(L(G)). Proof. In this proof we retain the previous notation. The first orthogonality relations (Theorem 4.3.9) tell us that the irreducible characters form an orthonormal set of

U to be a matrix whose columns are the elements of B. We proceed by induction on n, the case n = 1 being trivial. Assume the theorem is true for all dimensions smaller than n. Let λ be an eigenvalue of A with corresponding eigenspace Vλ. If , then A already is diagonal and there is nothing to prove. So we may assume that Vλ is a proper subspace; it is of course non-zero. Then by Proposition 2.2.3. We claim that Vλ ⊥ is A-invariant. Indeed, if v ∈ Vλ and w ∈ Vλ ⊥ , then and so Aw ∈ Vλ ⊥ . Note

then Tv ∈ W1 ∩ W2 = { 0} and so Tv = 0. But T is injective so this implies v = 0. Next, if v ∈ V , then some w1 ∈ W1 and w2 ∈ W2. Then . Thus V = V1 ⊕ V2. Finally, we show that V1, V2 are G-invariant. If v ∈ Vi, then . But Tv ∈ Wi implies ψgTv ∈ Wi since Wi is G-invariant. Therefore, we conclude that , as required. □ We have the analogous results for other types of representations, whose proofs we omit. Lemma 3.1.24. Let φ: G→GL(V ) be equivalent to an irreducible representation. Then φ is