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Let G be a finite group and K
an arbitrary field. We denote by K(G) the group algebra of G over K. Let G be the
direct product of finite groups G1 and G2,G = G1 ×G2, and let Mi be an irreducible
K(Gi)-module, i = 1,2. In this paper we study the structure of M1,M2, the outer
tensor product of M1 and M2.
While M1,M2 is not necessarily an irreducible K(G)-module, we prove below
that it is completely reducible and give criteria for it to be irreducible. These
results are applied to the question of whether the tensor product of division
algebras of a type arising from group representation theory is a division
algebra.
We call a division algebra D over K K-derivable if D≅HomK(G)(M,M) for some
finite group G and irreducible K(G)-module M. If B(K) is the Brauer group of K,
the set B0(K) of classes of central simple K-algebras having division algebra
components which are K-derivable forms a subgroup of B(K). We show also that
B0(K) has infinite index in B(K) if K is an algebraic number field which is not an
abelian extension of the rationals.
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