  The spinor representation of the Lorentz group does not accept simple
generalization with the group GL(4,R) of general linear coordinate
transformations. The Dirac equation may be written for an arbitrary choice of a
coordinate system and a metric, but the covariant linear transformations of the
four-component Dirac spinor exist only for isometries. For usual diagonal
Minkowski metric the isometry is the Lorentz transformation. On the other hand,
it is possible to define the Dirac operator on the space of anti-symmetric
(exterior) forms, and in such a case the equation is covariant for an arbitrary
general linear transformation. The space of the exterior forms is
16-dimensional, but usual Dirac equation is defined for four-dimensional
complex space of Dirac spinors. Using suggested analogy, in present paper is
discussed possibility to consider the space of Dirac spinors as some
"subsystem" of a bigger space, where the group GL(4,R) of general relativity
acts in a covariant way.
  For such purposes in this article is considered both Grassmann algebra of
complex anti-symmetric forms and Clifford algebra of Dirac matrices. Both
algebras have same dimension as linear spaces, but different structure of
multiplication. The underlying sixteen-dimensional linear space also may be
considered either as space of complex 4 x 4 matrices, or as space of states of
two particles: the initial Dirac spinor and some auxiliary system. It is shown
also, that such approach is in good agreement with well known idea to consider
Dirac spinor as some ideal of Clifford algebra. Some other possible
implications of given model are also discussed.
