Which groups have strictly rational representations? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:49:28Z http://mathoverflow.net/feeds/question/95106 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95106/which-groups-have-strictly-rational-representations Which groups have strictly rational representations? Grant Rotskoff 2012-04-25T02:06:28Z 2012-04-25T11:04:42Z <p>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}\$).</p> <p>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: </p> <p>Are there other "classes" of groups such that every irreducible representation is realizable over \$\mathbb{Q}\$?</p> <p>Take "classes" to mean whatever you think appropriate (so long as it doesn't mean the collection of all groups with only rational irreps).</p> http://mathoverflow.net/questions/95106/which-groups-have-strictly-rational-representations/95115#95115 Answer by Bruce Westbury for Which groups have strictly rational representations? Bruce Westbury 2012-04-25T04:46:36Z 2012-04-25T04:46:36Z <p>This is closely related to two previous questions.</p> <p>See the answer by M.Z. to <a href="http://mathoverflow.net/questions/47952" rel="nofollow">http://mathoverflow.net/questions/47952</a>. This gives a necessary and sufficient condition for the entries of the character table to be integers. This is David Speyer's remark above.</p> <p>The following question then discusses the Schur index <a href="http://mathoverflow.net/questions/47009" rel="nofollow">http://mathoverflow.net/questions/47009</a>. However the conclusion is that this is difficult to determine.</p> http://mathoverflow.net/questions/95106/which-groups-have-strictly-rational-representations/95149#95149 Answer by Robert Heffernan for Which groups have strictly rational representations? Robert Heffernan 2012-04-25T11:04:42Z 2012-04-25T11:04:42Z <p>Finite groups all of whose ordinary complex representations have rational-valued characters are sometimes called <em>Q-groups</em> (and sometimes called <em>rational groups</em>). There is a monograph by Denis Kletzing (<em>Structure and Representations of Q-Groups</em>, 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.</p>