Classification of upper motives of algebraic groups of inner type A_n

TL;DR

This paper characterizes when upper motives of certain algebraic varieties associated with central simple algebras are isomorphic, revealing a dichotomy for algebraic groups of inner type A_n based on p-adic valuations and Brauer group classes.

Contribution

It provides a complete criterion for isomorphism of upper motives of flag varieties linked to central simple algebras, connecting algebraic and motivic properties.

Findings

Upper motives are isomorphic iff p-adic valuations of gcds are equal.

Classes of p-primary components generate the same subgroup in the Brauer group.

Results reveal a dichotomy in motives of algebraic groups of inner type A_n.

Abstract

Let A, A' be two central simple algebras over a field F and \mathbb{F} be a finite field of characteristic p. We prove that the upper indecomposable direct summands of the motives of two anisotropic varieties of flags of right ideals X(d_1,...,d_k;A) and X(d'_1,...,d'_s;A') with coefficients in \mathbb{F} are isomorphic if and only if the p-adic valuations of gcd(d_1,...,d_k) and gcd(d'_1,..,d'_s) are equal and the classes of the p-primary components A_p and A'_p of A and A' generate the same group in the Brauer group of F. This result leads to a surprising dichotomy between upper motives of absolutely simple adjoint algebraic groups of inner type A_n