On the integral Tate conjecture for finite fields and representation theory

TL;DR

This paper introduces new counterexamples to the integral Tate conjecture over finite fields, using representation theory and classifying spaces of groups of type A_n, revealing limitations of existing conjectures.

Contribution

It presents the first counterexamples involving groups of type A_n and employs representation theory to analyze classes where Milnor's operations vanish.

Findings

Identified new counterexamples to the integral Tate conjecture

Demonstrated the role of representation theory in constructing these counterexamples

Showed that Milnor's operations vanish on the constructed classes

Abstract

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.