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Suppose that v ⊗ w ∈ Sym2 (V ) and g ∈ G then we have θ(ρ(g)(v ⊗ w)) = θ ρ(g) λij (vi ⊗ vj + vj ⊗ vi ) λij (ρ(g)vi ⊗ ρ(g)vj + ρ(g)vj ⊗ ρ(g)vi ) = θ = λij [θ(ρ(g)vi ⊗ ρ(g)vj ) + θ(ρ(g)vj ⊗ ρ(g)vi )] = λij [ρ(g)vj ⊗ ρ(g)vi + ρ(g)vi ⊗ ρ(g)vj )] λij (vj ⊗ vi + vi ⊗ vj ) = ρ(g) = ρ(g)(v ⊗ w). We also see that Sym2 (V ) ∩ Λ2 (V ) = {0}. Also their individual dimensions add up to n(n+1) + n(n−1) = n2 = dim(V ⊗ V ), hence we have V ⊗ V = Sym2 (V ) ⊕ Λ2 (V ). 2. The characters of Sym2 (V ) and Λ2 (V ) are χS and χA respectively given by 1 2 χ (g) + χ(g 2 ) χS (g) = 2 1 2 χA (g) = χ (g) − χ(g 2 ) 2 Proof.

Then from the proof of previous theorem it follows that {(ei ⊗ ej − ej ⊗ ei ) | i < j} is an eigen basis for Λ2 (V ) with corresponding eigen values {λi λj | i < j}. We now have χA (g) = T r(ρ ⊗ ρ)(g) = λi λj = i

Then χ(g) = i aii (g) and χ (g) = j bjj (g). 6 part 4. Using the similar argument we get χ, χ = (χ, χ) = ( i aii , j bjj ) = n 1 i=1 n = 1 where n is the dimension of the representation ρ. 2. The number of irreducible characters are finite. Proof. Since the irreducible characters form an orthonormal set they are linearly independent. Hence their number has to be less than the dimension of C[G] which is |G|, hence finite. Once we prove that two representations are isomorphic if and only if their characters are same this corollary will also give that there are finitely many non-isomorphic irreducible representations.

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Representation Theory of Finite Groups [Lecture notes] by Anupam Singh


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