Exercise 29, § 21

Keep the assumptions and notation of Exercises 27 and 28. Choose a primitive $m$-th root of unity $\zeta$. Let $\mathscr{O}_m$ be the ring $\mathbf{Z}[X]/(\Phi_m)$ (VIII, p. 415); we denote the class of X in $\mathscr{O}_m$ by $x$. Let $\varphi :\mathscr{O}_m\rightarrow F$ be the homomorphism that sends $x$ to $\zeta$. It induces an isomorphism from the subgroup of $\mathscr{O}_m$ generated by $x$ to the group $\mu_m(F)$; we denote the inverse isomorphism by $\tau$.

a) Let $\pi$ be a linear representation of G over F, of finite dimension $d$. For $g\in G$, let $P_g(X) =\prod^d_{i=1}(X-\lambda_i)$ be the characteristic polynomial of $\pi (g)$; denote by $\beta_{\pi}(g)$ the element $\sum_i\tau (\lambda_i)$ of $\mathscr{O}_m$. The resulting function $\beta_{\pi}: G\rightarrow \mathscr{O}_m$ is called the Brauer character of $\pi$; we have $\varphi \circ \beta_{\pi}=\chi_{\pi}$ and $\beta_{\pi}(x) =\beta_{\pi}(x_s)$ for every $x$, so that $\beta_{\pi}$ is determined by its restriction to $G^{reg}$.

b) Let $\widehat{G}$ be the set of isomorphism classes of simple F[G]-modules. Prove that the functions $(\beta_{\lambda})_{\lambda\in\widehat{G}}$ are linearly independent over $\mathbf{Z}$. (If there exists a nontrivial relation $\sum n_{\lambda}\beta_{\lambda}= 0$, then we may assume that one of the $n_{\lambda}$ is prime to $p$ and apply $\varphi$ to arrive at a contradiction using Exercise 27, f).)

c) Let $\mathscr{B}$ be the set of functions from $G^{reg}$ to $\mathscr{O}_m$; the homomorphism $R_F(G)\rightarrow \mathscr{B}$ that sends the class of $\pi$ to $\beta_{\pi}$ is injective. Deduce that two semisimple F[G]-modules are isomorphic if and only if their Brauer characters are equal.