Which groups have strictly rational representations? - MathOverflow

Which groups have strictly rational representations?

2012-04-25T02:06:28Z

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).

Answer by Bruce Westbury for Which groups have strictly rational representations?

2012-04-25T04:46:36Z

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.

Answer by Robert Heffernan for Which groups have strictly rational representations?

2012-04-25T11:04:42Z

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.