Brian Bi

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 $$G$$.