Maybe there is someone who is able to
answer some of the following questions :
1. Does the character table of a group with n
conjugacy classes only contain
character values which are algebraic of degree
strictly smaller than n ?
(Clearly, this is not a consequence of the fact that
character values of a group G are sums of
Exponent(G)'th roots of unity)
2. Let d be the 'determinant' of the character table
of a group G of order n
(in GAP : d := DeterminantMat(List(Irr(G),ValuesOfClassFunction))).
- Is d always different from zero ?
- Is d^2 always an integer which is divisible by n ?
(Obviously, d is determined up to the sign by the group G,
hence d^2 is uniquely determined by G)
Thanks in advance,