A Complex Irreducible Representation of the Quaternion Group and a Non-free Projective Module over the Polynomial Ring in Two Variables over the Real Quaternions

It is a classical result of Frobenius and Schur ([F], p. 20–22 or [Se], p. 121–124) that any finite dimensional complex irreducible representation of a finite group, whose character is real, either descends to a real representation or can be extended to a representation of the group over the real quaternion algebra. The simplest example where the latter phenomenon holds is the standard 2-dimensional complex representation of the group of integral quaternions. An application of some general results of Barth and Hulek shows that this representation leads to a canonical rank 2 (stable) vector bundle over the complex projective plane. It can be shown that the restriction of this bundle to the affine plane gives rise to a non-free projective module of H [X, Y], isomorphic to the one constructed in [OS] in another context in a different manner. (The existence of this projective module led, incidentally, to the construction in ([P1]) of non diagonalisable, (in fact indecomposable), non singular symmetric 4×4 matrices of determinant one over the polynomial ring in two variables over the field of real numbers, producing remarkable counter examples to the so called quadratic analogue of Serre’s conjecture and opening up a new and fruitful area of research (cf. [P3]). On the other hand, it was shown in ([KPS]) that any non-free projective module over D[X, Y], where D is a finite dimensional division algebra over a field, extends (and essentially uniquely) to a vector bundle over the projective plane over this field, with a D-structure.