Upper motives of products of projective linear groups

TL;DR

This paper classifies indecomposable motives of projective homogeneous varieties under products of projective linear groups, revealing a dichotomy for single factors and more complex behavior for multiple factors.

Contribution

It provides a complete classification for n=1 and explores the more intricate structure for n>1, including counterexamples and rational map analysis.

Findings

Complete classification of motives for n=1

Motivic dichotomy for projective linear groups

Counterexamples showing less rigidity for n>1

Abstract

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.