The fields generated by the values of the characters of the finite classical groups
John Columba Kelly · Warwick Research Archive Portal (University of Warwick) · 1975
The absolutely irreducible characters of a finite group G take their values in a cyclotomic extension of the field of rational numbers, and so are defined in some cyclotomic extension L of any field K of characteristic zero. The purpose of this thesis is to determine the smallest such extension L where K is either the rational field or a certain p-adic field, and G is one of the classical groups defined over a finite field with q elements, Fq. In particular it is shown that there is at least one group defined over Fq in each classical "family" whose characters take values in the ring of Witt vectors W(Fq). Section 5 of the thesis deals with a different problem. A theorem of Frobenius, expressing all the irreducible characters of the symmetric groups as integral linear combinations of the characters of certain permutation representations, is generalised to the Weyl group W(Cn), and some consequences for the representation theory of finite groups with (B,N)-pairs are deduced.