Lecture notes for Introduction to Representation Theory

If V, W are representations of G, then V W is also a representation, via
Therefore, ν V W (g) = ν V (g)ν W (g).
δ V W (g) = δ V (g) δ W (g).
An interesting problem discussed below is to decompose V W (for irreducible V, W ) into the
direct sum of irreducible representations.
3.5 Orthogonality of characters
We define a positive definite Hermitian inner product on F c (G, C) (the space of central functions)
by
1
(f 1 , f 2 ) =
f 1 (g)f 2 (g).
|G| gG
The following theorem says that characters of irreducible representations of G form an orthonormal
basis of F c (G, C) under this inner product.
Theorem 3.8. For any representations V, W
(ν V , ν W ) = dim Hom G (W, V ),
and 1, if V ∪ = W,
(ν V , ν W ) =
0, if V W
if V, W are irreducible.
Proof. By the definition
(ν V , ν W ) =
1 ν V (g)ν W (g) =
|G| gG
1 ν V (g)ν W (g)
|G| gG
1
⎨
where P = |G| gG
= 1 ν V W (g) = Tr | V W (P ),
|G| gG
g Z(C[G]). (Here Z(C[G]) denotes the center of C[G]). If X is an irreducible
representation of G then Id, if X = C,
P | X =
0, X = ⇒ C.
Therefore, for any representation X the operator P | X is the G-invariant projector onto the subspace
X G of G-invariants in X. Thus,
Tr | V W (P ) = dim Hom G (C, V W ⊕ )
= dim(V W ⊕ ) G = dim Hom G (W, V ).
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