Let $G$ be a finite group and let $Rep(G)$ be its representation ring (as a group it is the free $\mathbb{Z}$-module on the irreducible complex reps). The collection of permutation representations $\mathbb{C}[\mathcal{O}]$ for $\mathcal{O}\cong G/H$ a $G$-orbit generate a $\mathbb{Z}$-subalgebra which we will denote by $Per(G) \subset Rep(G)$.


If $Per(G) = Rep(G)$ then it follows that all complex representations of $G$ are defined over $\mathbb{Z}$. To see this note that under our assumption for every representation $V$ there exists a representation $U$ defined over $\mathbb{Z}$ s.t. $U \oplus V = W$ is defined over $\mathbb{Z}$. We can now take $V\_{\mathbb{Z}} = W\_{\mathbb{Z}} / U\_{\mathbb{Z}}$ as a $\mathbb{Z}$-form for $V$ (this is not really a precise proof, maybe a better arguemnt would be to explicitly write the projection operator which projectss onto the isotypic component of $V$ inside $\mathbb{Z}[G]$). 


Is the converse true?



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> **Question:** Suppose every $\mathbb{C}$-representation of $G$ has a $\mathbb{Z}$-form, does it follow that $Per(G) = Rep(G)$? If not what's a counter example?
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As an example when $G=S\_n$ both of the properties are satisfied and this is in fact the only non-trivial example I know of.



A finite group $G$ all of whose reps are defined over $\mathbb{Z}$ and yet $Rep(G)$ is not generated by permutation representations
The answer is no. Counterexamples include the Weyl groups of types E6, E7, and E8. For a proof that the representations of these Weyl groups are indeed all realisable over $\mathbb{Q}$ (equivalently over $\mathbb{Z}$), see M. Benard, [On the Schur Indices of Characters of the Exceptional Weyl Groups](https://www.jstor.org/stable/1970736), Annals of Mathematics, Vol. 94, No. 1 (Jul., 1971), pp. 89-107 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=297887)). For a proof of the statement that $Per(G)\neq Rep(G)$ for these groups, see D. Kletzing, [Structure and Representations of Q-Groups](https://link.springer.com/book/10.1007%2FBFb0103426), Lecture Notes in Mathematics 1084 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=765700)).


