Representation fields for orders of small rank
Arenas Carmona, Luis
A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.