Brian Bi
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Exercise 4.2.3 Let \(V_1, \ldots, V_n\) be the irreducible representations of \(G\). Let \(f : G \to k\) be the function that sends the identity element to 1 and all other elements to 0. Then \(f \in F_c(G, k)\). Let \(P \in k[G]\) be the element \(\sum_{g\in G} g\). Then \(gP = P\) for all \(g \in G\), therefore \(P^2 = |G|P = 0\), so \(\rho(P)^2 = \rho(P^2) = 0\) in all representations of \(G\). Since \(\rho(P)\) is nilpotent, it is also traceless, that is, \(\chi_V(P) = 0\) for all representations \(V\). However, \(f(P) = 1\). Therefore \(f \notin \span(\chi_{V_1}, \ldots, \chi_{V_n})\). But the \(\chi_{V_1}, \ldots, \chi_{V_n}\) are linearly independent according to Theorem 3.6.2, so their span is an \(n\)-dimensional subspace of \(F_c(G, k)\) which is a strict subset of \(F_c(G, k)\). Therefore \(n\) is strictly less than \(\dim F_c(G, k)\), which is the number of conjugacy classes of \(G\).