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

On the integral Tate conjecture for finite fields and representation theory