Classification of upper motives of algebraic groups of inner type A_n

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