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When studying behaviour of linear representations of finite groups under extension of fields, I came up across two natural questions, which I couldn't solve (Reference: Representations of finite groups: Algebra and Arithmatic by S. Weintraub).

Let $R$ be a ring which is $F$-algebra ($F$ is a field), $E$ be an extension of $F$, set $R'=E\otimes_F R$. If $M$ is an $R$-module, them $M'=E\otimes_F M$ is an $R'$-module.

Definition: If $M'$ is an $R'$ module that is isomorphic to $E\otimes_F M$ for some $R$-module $M$, then $M'$ is said to be defined over $F$.

(If I have understood this definion correctly, then in terms of matrix representations of groups, it says: if $\rho\colon G\rightarrow GL(n,E)$ is a matrix representation of a finite group $G$, we say that it is defined over $F$ if there is $A\in GL(n,E)$ such that $A\rho(g)A^{-1}\in GL(n,F), \forall g\in G$.)

With this set up, an irreducible representation of $G$ over $F$ may decompose over $E$. Assume $G$ is finite group, and $char(F)>0$, $(char(F),|G|)=1$, $E$ is an extension of $F$.

Question 1: If $\rho_1, \rho_2, \cdots, \rho_r$ is the set of all inequivalent irreducible representations of $G$ over $F$, and if $\rho_i$ decomposes over $E$ as $\rho_{i1}\oplus \rho_{i2} \oplus\cdots \oplus \rho_{it_{i}}$ ($1\leq i\leq r$), is the set $\{ \rho_{ij} \colon 1\leq i\leq r, 1\leq j\leq t_i\}$ is complete set of irreducible representations of $G$ over $E$?

Question 2: If $M'$ is an irreducible $R'$ module, that is not defined over $R$, does some multiple of $M'$, say $dM'$ ($d\in \mathbb{N}$) is defined over $R$?


The second question, I tried, seems to have negative answer; I am not sure:

for example, consider a complex $1$-dimensional representation of $C_4=\langle x\colon x^4\rangle$ given by $x\mapsto [\sqrt -1]$. This representation is clearly not defined over $\mathbb{R}$. Some multiple of this representation (say $d$) will be the map $x\mapsto \sqrt-1 I_d$; still this representation (of dimension $d$) is not defined over $\mathbb{R}$, since this matrix $\sqrt-1 I_d$ is central, its conjugate in $GL(d,\mathbb{C})$ can not be in $GL(d,\mathbb{R})$.

In the paragraph before Prop: 4.17 in the book mentioned above (p.33), author may want to say positive answer to question 2, but I couldn't understood.

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The idea here is that representations defined over a smaller field may split to a sum of conjugate representations over a bigger field. You get all the irreducible ones in this way. For example the rep of the cyclic group that you describe is a component (together with its complex conjugate) of the 2-dimensional rep over the reals, where the generator acts on the real plane by a 90 degree rotation. IOW the answer to your Q1 is "Yes", and the answer to your Q2 is "No" (for the reason you gave). For example Curtis & Reiner explain this very thoroughly. –  Jyrki Lahtonen Oct 10 '11 at 15:26

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