I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand this automorphism group, $Aut(GL_n(\mathbb{Z}))$, explicitly?
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4$\begingroup$ I'm not sure of the meaning of your first sentence, but for the second one, the automorphism group for $n\ge 3$ is reduced to the "obvious" one, namely $\mathrm{PGL}_n(\mathbf{Z})\rtimes\{\tau\}$ where $\tau$ is the inverse-transpose involutive automorphism. (Essentially, Mostow rigidity reduces to compute the normalizer in the real automorphism group, i.e., to show that $\mathrm{SL}_n(\mathbf{Z})$ equals its own normalizer in $\mathrm{SL}_n(\mathbf{R})$.) However it's very plausible that this was known earlier in this case by a direct algebraic approach. $\endgroup$– YCorCommented Jan 6, 2019 at 17:31
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$\begingroup$ @YCor Thank you! My first sentence was just a motivation in case it was helpful. Would you like to rewrite this as an answer so I can select it? It would be helpful if you added a reference or added a few more details about the semi-direct product with tau. $\endgroup$– cheyneCommented Jan 6, 2019 at 18:24
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$\begingroup$ I'm lazy at the moment to fill in the details, this is why a comment is maybe better to start with. $\endgroup$– YCorCommented Jan 6, 2019 at 18:26
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$\begingroup$ I completely understand. Thanks again! $\endgroup$– cheyneCommented Jan 6, 2019 at 18:45
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1 Answer
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Hua and Reiner described this in their paper "Automorphisms of the unimodular group" (for them "unimodular" means determinant of absolute value 1, hence it's ${\rm GL}_n(\mathbf Z)$ rather than just ${\rm SL}_n(\mathbf Z)$) that appeared in 1951 here. The end result (their Theorem 4) is that the automorphisms of ${\rm GL}_n(\mathbf Z)$ are generated by inner automorphisms, the conjugate-transpose map $A \mapsto (A^{-1})^\top$, the map $A \mapsto (\det A)A$ (they include it "for even $n$ only," which must mean they have a way of getting it from the other generators for odd $n$ -- I haven't read the paper closely to see where that is indicated [Edit: see comment below]), and one additional automorphism when $n=2$.
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2$\begingroup$ For odd $n$ the endomorphism $A\mapsto \det(A)A$ is not injective (it maps $-I$ to $I$). $\endgroup$– YCorCommented Jan 7, 2019 at 0:54
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