The irreducible six-dimensional complex representations of ${\rm Aut}(F\sb 2)$ that are nontrivial on $F\sb 2$

Abstract

Denote by $\Phi_2$ the automorphism group of the free group $F_2$ on two generators. We classify the irreducible 6-dimensional complex representations of $\Phi_2$ whose restriction to $F_2$ is nontrivial. J. Dyer, E. Formanek, and E. Grossman have shown how the Bürau representation of the braid group $B_4$ gives rise to a one-parameter family of irreducible 6-dimensional representations of $\Phi_2$. The faithfulness question for these and some other closely related representations of $\Phi_2$ is open. Our classification shows that all other 6-dimensional representations of $\Phi_2$ are not faithful.