Representation groups for the Schur index

Abstract

Fix an algebraic number field K and an irreducible (complex) character x of some finite group G. Assume K = K(X) contains the values of x so that the simple component A = .4(x, K) of the group algebra KG determined by 1 is central over K. It is a basic consequence of the Brauer-Witt theorem that A is a so-called cyclotomic algebra (see [lS]). We exhibit here a slightly different point of view, namely that of constructing a “representation group” for A, similar to the concept used in Clifford theory. From class field theory it is known that the order of the class [A] of A in the Brauer group Br(K) equals its index m,(x). It suffices to study the primary components of [A]. So fix a prime p and let [A], denote the pcomponent of [A] in Br(K). Let K, be the field obtained from K by adjoining the nth roots of unity. If n = exp(G) then by Brauer K, is a splitting field for A (and G). We shall see that there is a field L s K, with p-power degree over K such that