Timeline for Automorphisms of $SL_n(\mathbb{Z})$
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 4, 2011 at 14:42 | comment | added | Jim Humphreys | @Guntram: As I mentioned to Keivan Karai, these sophisticated approaches including appeal to work of Margulis are not at all needed for the elementary study of the various automorphism groups here. The older concrete methods are fairly elementary and hard to improve on. | |
| Mar 3, 2011 at 11:40 | comment | added | Guntram | Any rational automorphism of $SL_n(\mathbf C)$ preserves the root datum associated to this semisimple algebraic group, see e.g. Springer's book on linear algebraic groups. In more down-to-earth terms, up to an inner automorphism, the diagonal matrices will be preserved, and the subgroups $e_{ij}$ of elementary matrices will be permuted. This permutation, if nontrivial, then can be shown to coincide with $A \mapsto (A^{-1})^T$. Looking at the induced map $e_{ij}(s) \mapsto e_{ij}(s)$ it follows that the automorphism must be in $GL_n(\mathbf C)$. Then there are rationality questions... | |
| Mar 3, 2011 at 11:00 | comment | added | HenrikRüping | I didn't get the last part. Okay any auromorphism extends to a automorphism of $SL_n(\mathbb{Q})$. But then some knowledge of the automorphisms of $SL_n(\mathbb{Q})$ must enter, right? What is this knowledge exactly ? | |
| Mar 3, 2011 at 10:30 | history | answered | Guntram | CC BY-SA 2.5 |