Proposition 10

There exists a unique mapping $\varphi_G$ from $\mathscr{S}_K(G)$ to $\mathscr{S}_{K'}(G)$ such that $\lambda$ and $\varphi_G(\lambda )$ are related through $\varphi$ for every $\lambda$ in $\mathscr{S}_K(G)$. Moreover, $\varphi_G$ is bijective.

The uniqueness of $\varphi_G$ follows from property a) above.

A) Suppose that the field K has characteristic 0.

The group $\mu_n(K)$ is cyclic (V, §11, No. 2, p. 78, Theorem 1); choose a generator $\zeta$ of this group. Consider the ring homomorphism $\rho :\mathbf{Z}[X]\rightarrow \mathscr{O}_n$ that sends X to $\zeta$. It is surjective. The cyclotomic polynomial $\Phi_n(X)$ is irreducible in $\mathbf{Q}[X]$ (V, §11, No. 5, p. 84, Theorem 2); it is therefore the minimal polynomial of $\zeta$ over $\mathbf{Q}$. The polynomial $\Phi_n$ is monic with integer coefficients (V, §11, No. 4, p. 81). Let $P\in \mathbf{Z}[X]$ be a polynomial such that $P(\zeta ) = 0$; by Euclidean division of polynomials (IV, §1, No. 6, p. 10), there exist two polynomial Q and R in $\mathbf{Z}[X]$ such that P = QΦ$_n+ R$ and deg(R) $<$ deg(Φ$_n$). We have $R(\zeta ) = 0$, and therefore R = 0 because $\Phi_n$ is the minimal polynomial of $\zeta$. Consequently, the kernel of $\rho$ is the ideal $\Phi_n\mathbf{Z}[X]$ of $\mathbf{Z}[X]$, and $\rho$ induces a ring isomorphism from $\mathbf{Z}[X]/\Phi_n\mathbf{Z}[X]$ to $\mathscr{O}_n$.

Set $\zeta '=\varphi (\zeta )$; it is a primitive $n$-th root of unity in $K'$, and we therefore have $\Phi_n(\zeta ') = 0$ (V, §11, No. 5, p. 83, Lemma 3). Consequently, there exists a homomorphism $\varphi_0$ from the ring $\mathscr{O}_n$ to the field $K'$ that transforms $\zeta$ into $\zeta '$; it extends the mapping $\varphi$ from $\mu_n(K)$ to $\mu_n(K')$. Let $\mathscr{O}$ be the subring of K consisting of the elements $\frac{a}{n^r}$ with $a\in \mathscr{O}_n$ and $r\in \mathbf{N}$. Since $n\cdot 1$ is invertible in $K'$, the homomorphism $\varphi_0$ extends to a homomorphism $\varphi_1$ from $\mathscr{O}$ to $K'$.

We identify the algebra $\mathscr{O}[G]$ of the group G over $\mathscr{O}$ with a subring of the algebra K[G] and define a ring homomorphism Φ from $\mathscr{O}[G]$ to $K'[G]$ by the formula

Denote by $\mathscr{C}$ the set of conjugacy classes of G. For C in $\mathscr{C}$, denote by $u_C$ the element $\sum_{g\in C}g$ of $\mathscr{O}[G]$; the family $(u_C)_{C\in\mathscr{C}}$ is a basis over $\mathscr{O}$ of the center $Z(\mathscr{O}[G])$ of the algebra $\mathscr{O}[G]$ (VIII, p. 398). For any element $\lambda$ of $\mathscr{S}_K$(G), denote by $\chi_{\lambda}$ its character, by $d_{\lambda}$ its dimension, and by $e_{\lambda}$ the element of K[G] defined by

The family $(e_{\lambda})$ is a basis over K of the center Z(K[G]) of the ring K[G] (VIII, p. 408). We have

with

For $C\in \mathscr{C}$ and $\lambda \in \mathscr{S}_K$(G), set $\beta_{C,\lambda}=|G|d^{-1}_{\lambda}d(C)^{-1}\chi_{\lambda}(C)$. The entries of the matrix $(\alpha_{\lambda ,C})$ are in $\mathscr{O}$, and it follows from formula (31) of VIII, p. 411 that its inverse matrix is the matrix $(\beta_{C,\lambda})$, whose entries are also in $\mathscr{O}$ by Proposition 9 of VIII, p. 415. Consequently, the family $(e_{\lambda})$ is a basis of the $\mathscr{O}$-module $Z(\mathscr{O}[G])$.

The elements $\Phi (u_C) =\sum_{g\in C}g$ of $K'[G]$ form a basis over $K'$ of the center $Z(K'[G])$ of the ring $K'[G]$. We have

and the matrix with entries $\varphi_1(\alpha_{\lambda ,C})$ is invertible. The family of the $\Phi (e_{\lambda})$ is therefore a basis of $Z(K'[G])$. The family $(e_{\lambda})$ is a partition of the idempotent 1 in Z(K[G]) (VIII, p. 146 and p. 408); in other words, we have

$\sum_{\lambda}e_{\lambda}= 1,e^2_{\lambda}=e_{\lambda},e_{\lambda}e_\mu= 0$ if $\lambda \not=\mu$.

It follows that the family of the $\Phi (e_{\lambda})$ is a partition of the idempotent 1 in $Z(K'[G])$; since this family is a basis of $Z(K'[G])$ over $K'$, its elements are the indecomposable idempotents in $Z(K'[G])$ (VIII, p. 148, Remark 4).

For $\lambda '$ in $\mathscr{S}_{K'}$(G), define $\chi_{\lambda'},d_{\lambda'}$, and $e_{\lambda'}$ as above. By VIII, p. 408, the elements $e_{\lambda'}$ are the indecomposable idempotents in $Z(K'[G])$. Hence there exists a bijection $\varphi_G$ from $\mathscr{S}_K(G)$ to $\mathscr{S}_{K'}(G)$ such that $\Phi (e_{\lambda}) =e_{\varphi_G(\lambda)}$ for every $\lambda$ in $\mathscr{S}_K(G)$.

Let $\lambda$ in $\mathscr{S}_K(G)$; set $\lambda '=\varphi_G(\lambda )$. Let $(V_{\lambda}, \pi_{\lambda})$ (resp. $(V_{\lambda'}, \pi_{\lambda'})$) be a linear representation of G whose associated K[G]-module (resp. $K'$[G]-module) has class $\lambda$ (resp. $\lambda '$). Let us prove that $\lambda$ and $\lambda '$ are related. Let $g$ be an element of G. Let $\delta (T)$ be the determinant of the endomorphism $1 + T\pi_{\lambda}(g)$ of the K[T]-module $K[T]\otimes_KV_{\lambda}$. Let $\omega_1, . . . , \omega_{d_{\lambda}}$ be the eigenvalues of $\pi_{\lambda}(g)$; we have

We define $\delta '(T)$ likewise, and we denote the eigenvalues of $\pi_{\lambda'}(g)$ by $\omega '_1, . . . , \omega '_{d_{\lambda'}}$. The $\mathscr{O}$-module $\mathscr{O}[G]$ is free with basis G, and the K-vector space K[G] has basis G. Denote by Δ(T) the determinant of multiplication by $1 +e_{\lambda}gT$ in the $\mathscr{O}$[T]-module $\mathscr{O}[T]\otimes_{\mathscr{O}}\mathscr{O}[G]$. It is also the determinant of multiplication by $1 +e_{\lambda}gT$ in the K[T]-module $K[T]\otimes_KK[G]$. Let $\overline{\varphi}_1$ be the homomorphism from $\mathscr{O}[T]$ to $K'[T]$ that extends $\varphi_1$ and sends T to T. Since G is a basis of the $K'$-vector space $K'[G]$, the polynomial $\overline{\varphi}_1(\Delta (T))$ is equal to the determinant $\Delta '(T)$ of multiplication by $1 +e_{\lambda'}gT$ in the $K'$-vector space $K'[G]$.

The algebra K[G] is the direct sum of its simple components $e_\mu K[G]$ for $\mu$ running through $\mathscr{S}_K(G)$. For $\mu$ different from $\lambda$, the element $e_{\lambda}g$ annihilates $e_\mu K[G]$. Moreover, multiplication by $e_{\lambda}g$ coincides with multiplication by $g$ in $e_{\lambda}K[G]$. In view of VIII, p. 409 and Example 6 of VIII, p. 400, the representation of G in $e_{\lambda}K[G]$ is the direct sum of $d_{\lambda}$ representations of class $\lambda$. We consequently have $\Delta (T) =\delta (T)^{d_{\lambda}}$.

Analogously, we have $\Delta '(T) =\delta '(T)^{d_{\lambda'}}$.

From the relation $\Delta '(T) =\overline{\varphi}_1$(Δ(T)), we deduce first that $d^2_{\lambda}=d^2_{\lambda'}$, and therefore $d_{\lambda}=d_{\lambda'}$, and then that the sequence $\varphi (\omega_1), . . . , \varphi (\omega_{d_{\lambda}})$ can be deduced from the sequence $(\omega '_1, . . . , \omega '_{d'_{\lambda'}})$ by a permutation of the set of indices.

Since this is true for every element $g$ of G, the representations $\lambda$ and $\lambda '$ are related. We have therefore proved Proposition 10 when the field K has characteristic 0.

B) General case.

Let L be an algebraically closed field of characteristic 0 (for example, an algebraic closure of $\mathbf{Q}$). Denote by $\mathscr{S}_L(G)$ the set of classes of simple L[G]-modules. Choose an isomorphism $\eta$ from the group $\mu_n(L)$ to the group $\mu_n$(K), and set $\eta '=\varphi \circ \eta$. By part A) of the proof, there exist bijections

with the following property: for every $\lambda$ in $\mathscr{S}_L$(G), the representations $\lambda$ and $\eta_G(\lambda )$ are related through $\eta$, and the representations $\lambda$ and $\eta '_G(\lambda )$ are related through $\eta '$. The bijection $\varphi_G=\eta '_G\circ \eta_G^{-1}$ has the desired properties.

The bijection $\varphi_G$ from $\mathscr{I}_K(G)$ to $\mathscr{I}_{K'}(G)$ extends to an isomorphism, also denoted by $\varphi_G$, from the Grothendieck group $R_K(G)$ to the group $R_{K'}(G)$.