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

Upper motives of products of projective linear groups