On the integral Tate conjecture for finite fields and representation theory

We describe a new source of counterexamples to the so-called integral Hodge and integral Tate conjectures. As in the other known counterexamples to the integral Tate conjecture over finite fields, ours are approximations of the classifying space of some group BG. Unlike the other examples, we find groups of type A_n, our proof relies heavily on representation theory, and Milnor's operations vanish on the classes we construct.