Timeline for Do representations of finite groups of Lie type preserve diagonalizable elements?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| May 21, 2010 at 11:52 | comment | added | Someone | No. Unfortunately Jim's answer is gone. Half of it was an even better (compared to Xani's) explanation, when the matrices are diagonalizable. But at the end Jim made a claim that was simply wrong. | |
| May 21, 2010 at 9:07 | comment | added | Guntram | OK, but this doesn't contradict Xandi, right? A representation preserves diagonalizable iff $k$ contains $\mathbb F_q$. | |
| May 21, 2010 at 8:29 | comment | added | Someone | As with Jim's answer also the comment I added there vanished, I'll repeat the simple counterexample here: take the embedding $SL_2(q) \to GL_{2e}(p)$ by simply considering a $2$-dimensional $F_q$-vector space as $2e$-dimensional $F_p$-vector space. | |
| May 20, 2010 at 11:54 | comment | added | Victor Protsak | Sorry, I was carried away by your algebraic group language - I meant "semisimple". Indeed, semisimple matrices aren't diagonalizable in general. Nonetheless, an element of order dividing $q-1$ will a fortiori be diagonalizable over any field containing $\mathbb{F}_q$ (and if it doesn't, it cannot possibly be true), by a root of unity argument. But it looks like Jim has already answered your question without the lifting assumption. | |
| May 20, 2010 at 11:29 | comment | added | Guntram | Thank you for your answer. I don't see, though, where you used the specific group $SL_2$. For example, a cyclic group has a representation over the reals which is not diagonalizable over the reals. | |
| May 20, 2010 at 11:04 | comment | added | Victor Protsak | If the representation lifts to characteristic zero, the answer is "yes", because the group is finite and every finite order matrix in char 0 is semisimple. | |
| May 20, 2010 at 11:00 | answer | added | Jim Humphreys | timeline score: 7 | |
| May 20, 2010 at 10:58 | answer | added | Xandi Tuni | timeline score: 6 | |
| May 20, 2010 at 9:48 | history | asked | Guntram | CC BY-SA 2.5 |