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Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let $$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$ and $$SO_n(k,f):=O_n^+(k,f):=O_n(k,f) \cap SL_n(k)$$ be the associated (special) orthogonal group, i.e. linear transformations which preserve the form (with determinant 1). According to a paper by María J. Wonenburger ("The automorphisms of the group of rotations and its projective group corresponding to quadratic forms of any index", Canad. J. Math. 15 (1963), 302–303), building upon the works of Dieudonné and Rickart, the following is true:

Theorem: Suppose $\operatorname{char} k\neq 2$ and $n\geq 5$. Then for every $\phi\in \operatorname{Aut}(SO_n(k,f))$ there exists a homomorphism $\chi:SO_n(k,f)\to \{\pm I_n\}$ and a semi-similitude $T$ of $f$ such that $$\phi(g)=\chi(g)\ TgT^{-1}.$$

Here, $I_n$ is the identity matrix, and a semi-similitude of $k^n$ is a permutation of $K^n$ for which there are $\sigma\in \operatorname{Aut}(k)$ and $\lambda \in k\setminus\{0\}$ such that for all $x,y\in k^n$ one has $$f(Tx,Ty) = \lambda\ f(x,y)^\sigma.$$

If we call automorphisms that can be written as in the theorem ``standard'', then the theorem shows that for $n\geq 5$, all automorphisms are standard automorphisms.

Question: What is known when $n=3$ and $n=4$? Are there any ``non-standard'' automorphisms? If yes, when do they exist and what do they look like?

Both (counter)examples as well as partial affirmative answers (e.g. with restrictions on the Witt index or on the the field) are of interest.

EDIT: removed nonsensical statement about algebraic groups and types

UPDATE: I just discovered a paper by Li Zunxian, "Quaternion algebra and automorphisms of $\rm PO^+_4(V),\;\rm PO'_4(V)$ and $\rm P\Omega_4(V)$" (direct link), which seems to construct ``exceptional'' automorphisms of some projective orthogonal groups for $n=4$. I have not yet had time to study it in detail or figure out if this lifts to the non-projective case.

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1 Answer 1

This is at least a partial answer. There are two distinct viewpoints here: the concrete one involving forms and automorphism groups (which came first historically and usually requires characteristic $\ne 2$ to avoid tricky points) and the much more general one involving simple algebraic groups over a field $k$ (where Borel and Tits covered a great deal of territory when the groups are $k$-isotropic). But the cases $n=3,4$ you raise are degenerate from either viewpoint. When $n=3$ you don't get Lie type $B_2$ (that corresponds to $n=5$) but rather $B_1 = A_1$ along with a line on which nothing happens. When $n=4$ what you get is Lie type $D_2 = A_1 \times A_1$, also a degenerate case. In these low dimensions the automorphism group problem comes down to a form of $\mathrm{SL}_2$ over the field.

For convenience, it might be useful to link directly to the online version of Wonenburger's short paper here.

ADDED: Though it gets far away from the theme of groups preserving bilinear forms, there is a thorough treatment of the rank 1 degenerate cases (in concrete style) by O.T. O'Meara, Lectures on linear groups. Expository Lectures from the CBMS Regional Conference held at Arizona State University, Tempe, Ariz., March 26–30, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 22. American Mathematical Society, Providence, R.I., 1974. See especially his section 5.6, along with the comments and references in 5.8. The nature of the field $k$ inevitably comes into play here. As in other sources he mentions, O'Meara discusses more generally the isomorphisms between various linear groups and not just the automorphisms of a single group.

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Thanks! And oops on the Lie types. That explains why things are worse for $n=3,4$. As for the more general algebraic groups view point: While it might be more general and modern, treatments of that tend to focus on the simple / adjoint / simply-connected cases, but it appears to be difficult to go from this to $SO_n$, which is in general none of the above -- and while knowing the automorphisms of the derived group already is useful, it's still not quite the same as understanding those of $SO_n$ ... So I'd still be interested in concrete ``non-standard'' automorphisms for $n=3,4$. –  Max Horn Jul 31 at 9:36
    
@Max: I'm certainly not aware of anything non-standard happening in those dimensions, but I haven't explored the literature in a long time. What O'Meara did has the advantage of being down-to-earth but also somewhat more refined than the older methods of Dieudonne and others. Still, the nature of the field is a big factor in what forms of the groups exist, and then what automorphisms look like. –  Jim Humphreys Jul 31 at 13:38
    
Thanks again! I absolutely would expect the field to play a role. I also did try to find anything on this in O'Meara's works (in fact before posting this question), but I failed -- possibly because I looked at the wrong places, I was not even aware of the lecture notes you mentioned, and will now try to get hold of a copy. But I did have a look at "The Classical Groups and K-Theory", in particular chapter 8, but it seems to either require $n\geq 5$, or an isotropic form, or both... –  Max Horn Jul 31 at 14:08
    
I understand you as saying: "There may be no 'simple and nice' answer and the field may play a heavy role". That may be so, but in that case, I'd still be interested in seeing a concrete example where a non-standard automorphism pops up, or at least something else goes haywire... :-). I guess instead of bugging people here I should just sit down and try to come up with one myself :-). –  Max Horn Jul 31 at 14:21
    
@Max: I don't see offhand why a "non-standard" automorphism should exist, but the CBMS lectures by O'Meara are a useful starting point. –  Jim Humphreys Aug 7 at 13:25

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