Timeline for Variation of centraliser in $\operatorname{GL}(n,\mathbb{Z})$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 17, 2021 at 3:17 | comment | added | en kuo | Thanks for you guys. I have checked that the computer program Magma can compute this group C'(K) for a given K only when the $K$ is positive or negative definite. But I have some K which is symmetric but indefinite. Is it any way to compute them? | |
| Dec 14, 2020 at 15:21 | comment | added | Aurel | Well that's why it's a partial answer (or a long comment if you prefer). But (perhaps naively) this intersection does not look too bad to me. For instance I would guess one should be able to reduce to the case of a nondegenerate quadratic form intersected with the usual symplectic group. | |
| Dec 14, 2020 at 14:58 | comment | added | YCor | But how do you describe "explicitly" this intersection (of stabilizers of stabilizers of a quadratic form and an alternating form) in the general case? I have little idea what it looks like in general (except precisely in the particular cases you're emphasizing). | |
| Dec 14, 2020 at 14:48 | comment | added | Aurel | @YCor I agree, but I thought the OP was expecting a more explicit answer than an algorithm. For instance it is not clear to me how general the algebraic group appearing can be, and whether you have to use the most general algorithm you describe, which should work on any algebraic group defined over $\mathbf{Q}$. Otherwise, you could always use the one-liner "apply the Grunewald-Segal algorithm to compute generators of this arithmetic group, take the kernel to $GL_n(\mathbf{Z}/3\mathbf{Z})$ and check whether there's anything left". | |
| Dec 14, 2020 at 14:41 | comment | added | YCor | (...) a necessary condition is that such a Lie algebra is direct product of its derived subalgebra and its center, which is checkable. Whether the derived subalgebra is $\mathbf{R}$-anisotropic should be easy computing its Killing form. Hence we're reduced, given a Lie $\mathbf{Q}$-subalgebra that is Lie algebra of a torus, to determine whether this torus is $\mathbf{R}$-anisotropic by $\mathbf{Q}$-split. This should be practically doable although I don't see exactly right away how. | |
| Dec 14, 2020 at 14:37 | comment | added | YCor | My comments also addressed a bit the general case. Algorithmically one should be able, given an input rational Lie subalgebra of $\mathfrak{gl}_n$, which is known to be Lie algebra of an algebraic group, to determine whether the connected part of this algebraic group is reductive ($\mathbf{R}$-anisotropic)-by-($\mathbf{Q}$-split torus). | |
| Dec 14, 2020 at 14:29 | history | answered | Aurel | CC BY-SA 4.0 |