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I want to calculate the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$ when its generators $G_i$ (whose number is finite) are known.

For any matrix $S$ that commutes with the group: $G_iS$ = $SG_i$, and I get a system of linear equations. Any commuting element then can be written as $S = \sum_k x_k S_k, x_k \in \mathbb{Z} $, where $S_k$ are matrices with integer entries, and the commuting matrices form a ring with a finite basis.

But how do I calculate from these equations the group of invertible matrices? Particularly since the centralizer can have infinite elements.

Some special cases for $n=4$ are addressed here:

https://www.ams.org/mcom/1973-27-121/S0025-5718-1973-0333025-7/S0025-5718-1973-0333025-7.pdf

but is there a general method of solving the problem?

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  • $\begingroup$ You say "calculate the centralizer", but part of the question requires to specify what kind of input is expected. Actually, it follows from Borel–Harish-Chandra that the centralizer is finitely generated. Hence a possible output is a finite generating subset of the centralizer. (One could ask, as an output, to just enumerate the centralizer, but this is kind of trivial and mostly useless — namely one enumerates the centralizer in the rational (or integral) matrix algebra and then checks when one gets an element of $\mathrm{GL}_n(\mathbf{Z})$.) $\endgroup$
    – YCor
    Commented Jun 2, 2023 at 13:01
  • $\begingroup$ @YCor The input should be the generators $G_i$ and the output the generators of the centralizer. That the output is a complete should be verifiable. To find the generators by enumeration I came across this: semanticscholar.org/paper/… . But I don't think he proves the set is complete. $\endgroup$
    – Jim
    Commented Jun 2, 2023 at 13:39

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There is an algorithm to do this (and also to test two matrices in ${\rm GL}(n,{\mathbb Z})$ for conjugacy) described in the paper:

The conjugacy problem in ${\rm GL}(n,{\mathbb Z})$ Bettina Eick, Tommy Hofmann, E. A. O'Brien, Journal of the London Mathematical Society, Volume 100, Issue 3, December 2019 Pages 731-756.

It is implemented in Magma as the function $\mathtt {GLCentraliser}$.

Later edit: In view of the comments, I should have mentioned that the algorithm returns a finite generating set of the centralizer.

It is also worth pointing out that, given a finitely generated subgroup $G$ of ${\rm GL}(n,{\mathbb Z})$, there is a surprisingly fast algorithm to decide whether $G$ is finite and, if so, to find an isomorphic image of $G$ in ${\rm GL}(n,{\mathbb F}_q)$ for some finite field $q$, which facilitates further investigation of the structure of $G$.

If $G$ is infinite, then you can decide which of the two Tits' alternatives it satisfies. There is not much more you can do if it has a free subgroup (although you can often find a free subgroup). If it is virtually solvable, then you can do further structural computations.

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  • $\begingroup$ I guess in the non-virtually-solvable case, that one can compute the amenable radical (which is then locally cyclic). $\endgroup$
    – YCor
    Commented Jun 2, 2023 at 20:47

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