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Let $R$ be an integral domain, $G$ be a group, $\rho_1, \rho_2 : G \to \mathrm{GL}_n(R)$ two semisimple representations (i.e., the corresponding left-$R[G]$-modules $M_1, M_2$ are direct sums of simple modules).

Assume that for all $g \in G$, the characteristic polynomials $\rho_1(g)$ and $\rho_2(g)$ coincide as elements of $R[T]$. Does it imply that $\rho_1 \cong \rho_2$?

By Semisimple representation is determined by characteristic polynomials? and the links, it is true that the representations $\tilde{\rho_i} : G \to \mathrm{GL}_n(K)$ are isomorphic, where $K = \mathrm{Frac}(R)$ (even if the characteristic is positive; but we need assumption on the characteristic if we only assume that traces coincide --- see also Do characters distinguish non-isomorphic representations of any semisimple algebra?). But what about an isomorphism over $R$?

(The motivation is to study integral Tate modules of elliptic curves, where $R = \mathbb{Z}_{\ell}$).

Alphonse
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  • This definition of semisimple doesn't look right to me if $R$ isn't a field; if $R$ has any nonzero proper ideal $I$ then an action of $G$ on $R^n$ has a nontrivial invariant submodule, namely $I^n$. So I think there aren't any simple representations in this sense, and the only semisimple representation is zero. – Qiaochu Yuan Nov 03 '22 at 16:54

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