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

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Proof. Fix any inner product ·, · on Cn . Now, each element g of G defines an inner product ·, · g on Cn by setting v, w g = gv, gw for all v and w in V . We define v, w = g∈G v, w g . One can check that ·, · is an inner product on Cn . Moreover, for any h ∈ G and any v and w in V , we have hv, hw = ghv, ghw = g∈G kv, kw = v, w . ) So, the representation V is unitary with respect to ·, · . 5, it is completely reducible. 4 Representation Theory Notes Character Theory We know that representations can be thought of as group homomorphisms G → GL(V ).

Then, we say that the representation is unitary if the action of G on V preserves the inner product: that is, for all g in G and all v, w ∈ V , gv, gw = v, w . Remark. Thinking of a representation as a homomorphism G → GL(V ), the statement that a representation ϕ : G → GL(V ) is unitary is precisely the statement that the elements of g act by unitary matrices on V , or in other words, that ϕ(G) ⊆ U(V ), the subgroup of GL(V ) consisting of unitary matrices. The main significance of a unitary representation for representation theory is given by the following theorem.

34 Katherine Christianson Representation Theory Notes Notice that the dimension of C[G] (as a C-vector space) is |G| and the dimenk sion of Matni (C) is n2i . Because C[G] ∼ = i=1 Matni (C), the dimensions of these two vector spaces must be equal, so we get k n2i . |G| = i=1 We would also like to compare the dimensions of the centers of these C-algebras. We require two facts: first, for any rings R1 and R2 , Z(R1 × R2 ) = Z(R1 ) × Z(R2 ); and second, the center of any matrix algebra consists only of multiple of the identity.

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

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