Timeline for centralizer of the order 2^k cyclic permutation matrix over F_2
Current License: CC BY-SA 3.0
15 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 31, 2014 at 21:18 | vote | accept | Dima Pasechnik | ||
| Jul 31, 2014 at 21:18 | answer | added | Dima Pasechnik | timeline score: 0 | |
| Aug 26, 2013 at 17:32 | history | edited | Dima Pasechnik | CC BY-SA 3.0 |
corrected the explanation of the formula, added a link
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| Aug 26, 2013 at 17:29 | comment | added | Dima Pasechnik | oops, indeed, the formula I gave is for the quotient of the centralizer over $\mathbb{Z}/2^k\mathbb{Z}$. I'll update the body of the question. | |
| Aug 26, 2013 at 16:09 | comment | added | Frieder Ladisch | There must be a misprint in your formula, since we get no factor $(\mathbb{Z}/2^k)$. The result in Bass yields the formula $\bigoplus_{i=1}^k (\mathbb{Z}/2^i)^{2^{k-i}}$. | |
| Aug 24, 2013 at 7:59 | comment | added | Dima Pasechnik | here I asked a more general question: mathoverflow.net/questions/140280/… | |
| Aug 19, 2013 at 8:32 | comment | added | Dima Pasechnik | Murray's paper doesn't really give you a closed form answer, as far as I can see, but Prop. XI(5.7) of Bass indeed does. I wonder if the more general case of $2^km$-cycle, for $m$ odd, has been done before (We also have a result for this case too). | |
| Jul 31, 2013 at 8:21 | comment | added | Tim Dokchitser | There is a paper by S. H. Murray, Conjugacy classes in Maximal Parabolic Subgroups of General Linear Groups, J. Algebra 233, 135-155 (2000). In Section 4 (Centralizers in general linear groups) he works out the centralizers of arbitrary elements in $GL_n(k)$. | |
| Jul 29, 2013 at 7:29 | comment | added | Jeremy Rickard | You're calculating the group of units of the group algebra ${\mathbb F}_2C$ of a cyclic group $C$ of order $2^k$ (your matrix describes how a generator of $C$ acts on the regular representation, and so the centralizer is the group of units in $\operatorname{End}_{{\mathbb F}_2C}({\mathbb F}_2C)\cong\mathbb{F}_2C$). In this guise, it's calculated in Prop. XI(5.7) of Bass's Algebraic K-Theory. | |
| Jul 28, 2013 at 22:05 | history | edited | Dima Pasechnik | CC BY-SA 3.0 |
added 3 characters in body
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| Jul 28, 2013 at 20:36 | comment | added | user6976 | @Geoff: You are right. | |
| Jul 28, 2013 at 20:31 | comment | added | Geoff Robinson | The matrix is similar to a matrix in Jordan normal form with a single Jordan block and eigenvalue 1,so its centralizer is conjugate to the centralizer of that matrix in Jordan form. | |
| Jul 28, 2013 at 20:28 | comment | added | Geoff Robinson | @Mark: That argument works in characteristic zero, but not here. The only eigenvalue in this case is 1. | |
| Jul 28, 2013 at 19:52 | history | asked | Dima Pasechnik | CC BY-SA 3.0 |