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It can be shown, via the construction of the representations of the symmetric group, that every representation of $S_n$ is equivalent to a representation with values in $\mathbb{Q}.$ Presumably, this is a fairly rare phenomenon: it clearly doesn't hold for cyclic groups ($\mathbb{Z}/p\mathbb{Z}$ has one-dimensional representations given by the $p$th roots of unity, hence $p-1$ of its representations lie outside of $\mathbb{Q}$).

Moreover, there is a formula which constructs the rational characters of a group (due to Artin: see Curtis and Reiner section 15), but it doesn't seem to give an answer to the following question:

Are there other "classes" of groups such that every irreducible representation is realizable over $\mathbb{Q}$?

Take "classes" to mean whatever you think appropriate (so long as it doesn't mean the collection of all groups with only rational irreps).

flag	asked Apr 25 2012 at 2:06 Grant Rotskoff 353●3●9

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At the minimum each element must be conjugate to its inverse. – Benjamin Steinberg Apr 25 2012 at 2:27

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More than that: If $g$ has order $n$, and $GCD(m,n)=1$, then $g$ must be conjugate to $g^m$. But neither of these are sufficient conditions. See also groupprops.subwiki.org/wiki/… . – David Speyer Apr 25 2012 at 2:42

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To subsume the list in David Speyer's comment: any direct product of symmetric groups and copies of $D_4$ (also called $D_8$) – Will Sawin Apr 25 2012 at 3:53

$S_6(2)$, perhaps? It's index 2 in $W(E_7)$, and its characters are rational. – S. Carnahan♦ Apr 25 2012 at 10:00

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@Will: In fact every Weyl-Group and therefore every product of Weyl groups has this property. The Dieder group of order 8 is just the Weyl group $W(B_2)$. – Johannes Hahn Apr 25 2012 at 14:01

Robert Heffernan · Answer 1 · 2012-04-25 11:04:42Z UTC

Finite groups all of whose ordinary complex representations have rational-valued characters are sometimes called Q-groups (and sometimes called rational groups). There is a monograph by Denis Kletzing (Structure and Representations of Q-Groups, Springer Lecture Notes in Mathematics 1084, 1984) which might be of interest. Symmetric groups are Q-groups, as you mention, as are Weyl groups. I can't remember if Kletzing constructs any other infinite families although I do remember that $D_n$ is a Q-group only when $n=1,2,3,4$ or $6$. It's also worth mentioning that homomorphic images and direct products of Q-groups are also Q-groups.

Bruce Westbury · Answer 2 · 2012-04-25 04:46:36Z UTC

This is closely related to two previous questions.

See the answer by M.Z. to http://mathoverflow.net/questions/47952. This gives a necessary and sufficient condition for the entries of the character table to be integers. This is David Speyer's remark above.

The following question then discusses the Schur index http://mathoverflow.net/questions/47009. However the conclusion is that this is difficult to determine.

Which groups have strictly rational representations?

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