Timeline for Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
Current License: CC BY-SA 4.0
13 events
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| Apr 22, 2022 at 13:54 | answer | added | user481018 | timeline score: 0 | |
| Dec 14, 2020 at 14:29 | answer | added | Aurel | timeline score: 2 | |
| Dec 13, 2020 at 16:46 | history | edited | Mikhail Borovoi |
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| Dec 13, 2020 at 16:22 | answer | added | Geoff Robinson | timeline score: 4 | |
| Dec 13, 2020 at 8:21 | comment | added | YCor | @LSpice no, I should have said $\mathbf{R}$-anisotropic. I think a connected linear algebraic $\mathbf{Q}$-group $G\subset\mathrm{GL}_n$ has $G(\mathbf{Z})$ finite iff it's ($\mathbf{R}$-anisotropic)-by-($\mathbf{Q}$-split torus). | |
| Dec 13, 2020 at 1:44 | comment | added | LSpice | @YCor, compact = $\mathbf Q$-anisotropic? | |
| Dec 13, 2020 at 1:23 | comment | added | YCor | 1: this is the set of $\mathbf{Z}$-points in some explicit algebraic group. This is finite iff the latter is, I believe, compact-by-($\mathbf{Q}$-split torus). I guess this can algorithmically be checked from its Lie algebra (which can easily be described by a linear system of equations with rational coefficients). | |
| Dec 13, 2020 at 1:19 | history | edited | YCor |
edited tags
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| Dec 13, 2020 at 0:44 | history | edited | LSpice | CC BY-SA 4.0 |
Oops, messed up the title
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| Dec 13, 2020 at 0:39 | comment | added | LSpice | I did some proofreading, I believe without any change in meaning. \\ $\operatorname C'(K)$ is called the twisted centraliser of $K$ (twisted by the inverse-transpose automorphism $\sigma$ of $\operatorname{GL}(n, \mathbb Z)$). It is the intersection with $\operatorname{GL}(n, \mathbb Z)$ of the centraliser of the element $K \rtimes \sigma \in \operatorname{GL}(n, \mathbb Z) \rtimes \langle\sigma\rangle$. It has certainly been studied before, but I don't know what, if anything, is known about your specific questions. | |
| Dec 13, 2020 at 0:38 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Dec 13, 2020 at 0:25 | history | edited | YCor | CC BY-SA 4.0 |
formatting
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| Dec 13, 2020 at 0:14 | history | asked | en kuo | CC BY-SA 4.0 |