Timeline for linear independence of orbits via a set of transformations in char p
Current License: CC BY-SA 3.0
14 events
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| Feb 6, 2013 at 17:36 | history | edited | Tim | CC BY-SA 3.0 |
Updated question to a more specific one.
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| Feb 6, 2013 at 17:28 | history | edited | Tim |
edited tags
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| Feb 6, 2013 at 17:04 | comment | added | Tim | @Abhinav: The cyclicity of $E \otimes \mathbb{F}_p$ can indeed be attacked with Hilbert's theorem 90, and I have thought about this a good deal. But I'm not sure if all that applies to the question at hand, namely, does the cyclicity of $E \otimes k$ imply the cyclicity of $E \otimes \mathbb{F}_p$. The only hope of attacking this problem seems to a be through linear algebra and/or modular representation theory. It couldn't hurt to get my hands dirty, as you suggest though. | |
| Feb 6, 2013 at 16:57 | comment | added | Tim | @Jason: Nice, I think you're right. Thanks. | |
| S Feb 6, 2013 at 16:53 | vote | accept | Tim | ||
| S Feb 6, 2013 at 16:53 | vote | accept | Tim | ||
| S Feb 6, 2013 at 16:53 | |||||
| Feb 6, 2013 at 16:26 | comment | added | Jason Starr | @Abhinav: Your counterexample for $p=2$ extends to every $p$. I think you can repeat factors in other diagonal entries to get counterexamples for every $n\geq p$. | |
| Feb 6, 2013 at 15:05 | comment | added | Abhinav Kumar | I don't know if there's a principled way to make counterexamples. Keeping in mind what Jason said, you probably need to keep $p$ small. You could try $p = 2$ and $n = 3$, with the non-identity elements of a group isomorphic to $\mathbb{Z}/2 \oplus \mathbb{Z}/2$ (so try to find two idempotent transformations which commute). I don't know if this will work, but it should be easy enough to write a computer program to search. In the other direction, I assume you're already tried rephrasing the problem in a form where you could hit it with Hilbert 90? | |
| Feb 6, 2013 at 13:49 | vote | accept | Tim | ||
| S Feb 6, 2013 at 16:53 | |||||
| Feb 6, 2013 at 13:46 | answer | added | Jason Starr | timeline score: 3 | |
| Feb 6, 2013 at 13:38 | vote | accept | Tim | ||
| Feb 6, 2013 at 13:39 | |||||
| Feb 6, 2013 at 5:18 | comment | added | Tim | I was hoping something very general like the above would be true. Truth be told, I'm interested in something a bit more specific: the $T_i$ are the non-identity elements of an abelian group of order $n+1$. Also, if it helps, we may assume $\sum T_i = -1$. This is coming from the following number theoretic place: Let $E$ be the group of units of a real abelian number field with Galois group G. If $E \otimes k$ is a cyclic $k[G]$-module, is it true that $E \otimes \mathbb{F}_p$ is a cyclic $\mathbb{F}_p[G]$-module? | |
| Feb 6, 2013 at 4:50 | answer | added | Abhinav Kumar | timeline score: 5 | |
| Feb 6, 2013 at 2:54 | history | asked | Tim | CC BY-SA 3.0 |