Timeline for Conjugacy for $p$-adic matrices of finite order
Current License: CC BY-SA 4.0
13 events
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| Jan 25, 2023 at 0:12 | comment | added | reuns | If $A,B\in GL_n(\Bbb{Z}_p)$ have the same characteristic polynomial which is separable $\bmod p$ then $A=C B C^{-1}$ with $C\in GL_n(\Bbb{Z}_p)$. This is because $A\bmod p$ has a cyclic vector $\Bbb{F}_p^n = \bigoplus_{m=0}^{n-1} \Bbb{F}_pA^m v$, if $v$ is the reduction of $w$ then $\Bbb{Z}_p^n = \bigoplus_{m=0}^{n-1} \Bbb{Z}_pA^m w$ so $A$ is conjugate to its companion matrix. | |
| Jan 24, 2023 at 17:25 | history | edited | YCor | CC BY-SA 4.0 |
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| Aug 10, 2011 at 19:36 | comment | added | Frieder Ladisch | Afterthought: Since the answer to your first question is "No", it may also be useful to know that conjugacy mod $p^r$ implies conjugacy in $GL_n(\mathbb{Z}_p)$, when $r$ is big enough, i. e. such that $p^r$ doesn't divide $m$. This follows from the treatment of Maranda's theorem in Curtis-Reiner, Methods. | |
| Aug 8, 2011 at 4:43 | answer | added | Junkie | timeline score: 2 | |
| Aug 7, 2011 at 10:20 | history | edited | Tim Dokchitser | CC BY-SA 3.0 |
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| Aug 7, 2011 at 9:30 | answer | added | Jeff Adler | timeline score: 2 | |
| Aug 7, 2011 at 9:24 | vote | accept | Tim Dokchitser | ||
| Aug 6, 2011 at 11:21 | answer | added | Alex B. | timeline score: 13 | |
| Aug 5, 2011 at 14:46 | answer | added | Frieder Ladisch | timeline score: 6 | |
| Aug 5, 2011 at 13:22 | answer | added | Geoff Robinson | timeline score: 10 | |
| Aug 5, 2011 at 13:01 | comment | added | Geoff Robinson | I think that the name of Alfredo Jones is relevant here. Try the 1962 Curtis and Reiner, and their later "Methods of Representation Theory" tome. | |
| Aug 5, 2011 at 11:43 | answer | added | Gjergji Zaimi | timeline score: 7 | |
| Aug 5, 2011 at 9:49 | history | asked | Tim Dokchitser | CC BY-SA 3.0 |