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.
Publié le : 2000-05-14
Classification:
free group on two generators,
braid group on four strings,
irreducible finite dimensional representations,
Gröbner basis routine,
20C15,
20C40,
20E05,
20F36
@article{1045604679,
author = {\DJ okovi\'c, Dragomir \v Z.},
title = {The irreducible six-dimensional complex representations of ${\rm Aut}(F\sb 2)$ that are nontrivial on $F\sb 2$},
journal = {Experiment. Math.},
volume = {9},
number = {3},
year = {2000},
pages = { 457-465},
language = {en},
url = {http://dml.mathdoc.fr/item/1045604679}
}
Đoković, Dragomir Ž. The irreducible six-dimensional complex representations of ${\rm Aut}(F\sb 2)$ that are nontrivial on $F\sb 2$. Experiment. Math., Tome 9 (2000) no. 3, pp. 457-465. http://gdmltest.u-ga.fr/item/1045604679/