NOTES ON REPRESENTATIONS OF FINITE GROUPS Contents 1 ...

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2 BEN MCKAY2. Examples(1) Let G be the group of permutations of the numbers 1, 2, . . . , n. Let V bean n-dimensional vector space with a basis v 1 , v 2 , . . . , v n . Associate to eachpermutation p the linear map ρ(p) given byρ(p)v i = v p(i) .(2) Let G be the group Z n and V = C. Each linear map V → V is justmultiplication by some complex number. Letρ(m) = e 2πim/n ∈ C × .(3) For any group G, let V = C, we can just let ρ(g) = I for every g ∈ G. Thisρ is called the trivial representation.(4) Suppose that G is a finite group, and that V is a vector space whose dimensionequals the number of elements in G. Suppose that V has a basiswhich is indexed by the elements of G, say{v i } i∈G.Then for each g ∈ G, define ρ(g) byρ(g)v i = v gi .This representation is called the regular representation.(5) We can generalize the notion of representation.Definition 4. If G is a group and X is a set, an action of G on X is a choice,for each g ∈ G, of permutation p(g): X → X, so that p(gh) = p(g)p(h). Ifthe choice of permutation p(g) is understood, we will write p(g)x as gx.(6) For example, the group of permutations G of the numbers 1, 2, . . . , n actsin the obvious way onX = {1, 2, . . . , n} .(7) If G acts on X, take V to be any vector space with a basis{v i } i∈Xindexed by the elements of the set X. Then letρ(g)v x = v gx ,for each x ∈ X. We call ρ the permutation representation associated to theaction on X.(8) Recall that if V 1 and V 2 are two vector spaces, they have a direct sumV 1 ⊕ V 2 consisting of the pairs (v 1 , v 2 ) for v 1 ∈ V 1 and v 2 ∈ V 2 , withpairwise addition and scaling. If ρ 1 : G → GL (V 1 ) and ρ 1 : G → GL (V 2 )are two representations, then V 1 ⊕ V 2 is a representation, via the ruleρ(g) (v 1 , v 2 ) = (ρ 1 (g)v 1 , ρ 2 (g)v 2 ) .(9) Recall that if V 1 and V 2 are two vector spaces, they have a tensor productV 1 ⊗ V 2 . It is tricky to say what the elements of V 1 ⊗ V 2 are, but we cansay that ifu 1 , u 2 , . . . , u pis a basis of V 1 andw 1 , w 2 , . . . , w q
NOTES ON REPRESENTATIONS OF FINITE GROUPS 3is a basis of V 2 , then there are some elementsu 1 ⊗ w 1 , u 1 ⊗ w 2 , . . . , u 1 ⊗ w q ,u 2 ⊗ w 1 , u 2 ⊗ w 2 , . . . , u 2 ⊗ w q ,.,u p ⊗ w 1 , u p ⊗ w 2 , . . . , u p ⊗ w q ,forming a basis of V 1 ⊗V 2 (called the product basis). If ρ 1 : G → GL (V 1 ) andρ 1 : G → GL (V 2 ) are two representations, then V 1 ⊗ V 2 is a representation,via the ruleρ(g) (u i ⊗ w j ) = ρ 1 (g)u i ⊗ ρ 2 (g)w j .3. InvariantsDefinition 5. If ρ: G → GL (V ) is a representation, a fixed vector of the representationis a vector v ∈ V so that ρ(g)v = v for every g ∈ G. We denote by V G theset of all fixed vectors. For any vector v ∈ V , we denote by p(v) the vectorp(v) = 1|G|∑ρ(g)v,the average of all of the things that G can do to v. We call p the averaging operator,and sometimes write p(v) as v G .Lemma 1. The averaging operator is linear, and G-invariant, and is a projectionto V G , i.e. p ◦ p = p, and p is the identity on V G .Proof. Linearity is clear. Clearly since we averaged over all of the different elementsof G, p is G-invariant. If v is fixed, each ρ(g) drops out, and we get p(v) = v. Clearlythe p(v) is G-invariant, i.e. the image of p is contained in V G so p ◦ p = p. □g∈G4. SubrepresentationsLemma 2. Suppose that ρ: G → GL (V ) is a representation of a finite group G.Take any inner productv, w ∈ V ↦→ 〈v, w〉 ∈ C.From this inner product, define a new operation, which we will write asand is defined byv, w ∈ V ↦→ 〈v, w〉 G ∈ C,〈v, w〉 G = 1|G|∑〈ρ(g)v, ρ(g)w〉 .g∈G(Average over all ways of getting G to act on the pair v, w of vectors). Then thisnew operation is a G-invariant inner product, i.e.〈ρ(g)v, ρ(g)w〉 G = 〈v, w〉 G ,for any g ∈ G and v, w ∈ V .In particular, V has a G-invariant inner product.
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