On the Schur Indices of Characters of the Exceptional Weyl Groups

Abstract

We will be concerned in this paper with the irreducible characters of the Weyl groups of type E6, E7, and E8. Character tables for these groups have been given by J. S. Frame [6, 7]. In all cases, the characters are rationalvalued. It is fairly easy to show that each character has a real splitting field. The object of this paper is to show that each character has a rational representation. Other authors have shown that the same results hold for other irreducible Weyl groups. A. Young [13] studied the families of groups of type A, and B, as permutation groups. He devised a method for constructing all representations and showed that they could be constructed over the rational field Q. W. Specht [11] did the same for the family of groups of type D. This result is easily shown for the group of type G2, which is dihedral of order 12. T. Kondo [9] presented a character table for the group of type F4 and showed that each character has a rational representation. Our present results complete the proof that any representation of a Weyl group of a finite dimensional semi-simple complex Lie algebra is equivalent to a rational representation. Our methods are character-theory methods and we restate our results in terms of the Schur index. Let X be an irreducible character of a group G and let F be a field. The Schur index m,(X) of X over F is the smallest positive integer m such that mX is a character afforded by an F(X)-representation of G.