Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich

By A. I. Kostrikin, I. R. Shafarevich

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Id where λ = |G| n f, χ . Proof. We claim that ρf is a G-map and use Schur’s Lemma to prove the result. For any g ∈ G we have, ρ(g)ρf ρ(g −1 ) = f (t)ρ(g)ρ(t)ρ(g −1 ) = t∈G f (t)ρ(gtg −1 ) = t∈G f (g −1 sg)ρ(s) = ρf . s∈G Hence ρf is a G-map. Id for some λ ∈ C. n = tr(ρf ) = f (t)χ(t−1 ) = |G| f, χ . f (t)tr(ρ(t)) = f (t)χ(t) = |G| |G| t∈G Hence we get λ = |G| n t∈G t∈G f, χ . 5. We need to prove that irreducible characters span H as being orthonormal they are already linearly independent. Let f ∈ H.

Be irreducible representations of G of dimension n1 , n2 , . . , nh , . . over C. Let χ1 , χ2 , . . , χh , . . are corresponding characters, called irreducible characters of G. We will fix this notation now onwards. We will prove that the number of irreducible characters and hence the number of irreducible representations if finite and equal to the number of conjugacy classes. In the last chapter we introduced an inner product , on C[G]. We also observed that character of any representation belongs to H, the space of class functions.

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. Suppose that |G| = d. Then for any g ∈ G, (ρ(g))d = I. Thus m(X), the minimal polynomial of ρ(g), divides the polynomial p(X) = X d − 1. Since p(X) has distinct roots so will m(X) and hence ρ(g) is diagonalisable. Let {e1 , · · · en } be an eigen basis for V and let {λ1 , · · · , λn } be the corresponding eigen values. 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}.

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