  Fix a base field F, a finite field K and consider a sequence of central
simple F-algebras A_1,...,A_n. In this note we provide some results toward a
classification of the indecomposable motives lying in the motivic
decompositions of projective homogeneous varieties under the action of
PGL(A_1)x...xPGL(A_n) with coefficients in K. We give a complete classification
of those motives if n=1 and derive from it the motivic dichotomy of projective
linear groups. We then provide several classification results as well as
counterexamples for arbitrary n showing that the situation is less rigid. These
results involve a neat study of rational maps between generalized Severi-Brauer
varieties which is certainly of independent interest.

Algebraic Geometry