Okay, I hope we've cleared that up. But as I understand the notation (as in < http://en.wikipedia.org/wiki/Projective_linear_group >), PGL(n+1, R) denotes the group of linear transformations of R^(n+1) factored out by the scalars {cI | c in R}, which is the very group I meant acting on 1-dimensional subspaces of R^(n+1) and thereby inducing an action on S^n [sic]. (Which action is indeed the double cover of the action of PGL(n+1,R) on P^n.) Its dimension would be dim(GL(n+1, R)) - 1 = (n+1)^2 - 1 = n^2 + 2n, as Fred has mentioned. Or restricting to orientation-preserving linear maps, we have PGL+(n+1,R) with the same dimension, and more reasonable to compare with Conf(S^n). If we're only considering orientation-preserving conformal maps of S^n, then we can ignore spherical inversions and just say every such map can be described as a composition of rotations, dilatations, and translations on R^n (corresponding to S^n - {pt} by a stereographic projection). --Dan On Aug 31, 2014, at 11:03 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
You're describing PGL(n, R), not PGL(n+1, R).
I was talking about transformations of real projective (n+1)-space, in which S^n is the unit sphere.
You were talking about transformations of real projective n-space, in which S^n is the double cover of the entire space RP^n.
Apologies for the misunderstanding.
Sent: Sunday, August 31, 2014 at 6:20 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
Yes -- I think you used the word "preserve", which means exactly what you just said -- so I think what you said was clear.
Maybe we are using different terminology.
Because with the terms I'm using, the group Conf(S^n) is generated by the inversions in circles (okay, even numbers of inversions in circles).
The orientation preserving projective group on S^n would be PGL+(n+1,R), which just permutes the lines through the origin in R^(n+1) and so induces a map of S^n to itself.
But (take S^2 for instance, where Conf(S^2) == PSL(2,C)) most elements of Conf(S^n) do not take antipodal pairs to antipodal pairs, whereas elements of PGL(n+1,R) always take antipodal pairs to antipodal pairs.
With this view, I don't understand:
-----
The [orientation-preserving] elements of the projective group PSL(n+1) acting on S^n are precisely the elements of the conformal group Conf(S^n).
--Dan
On Aug 31, 2014, at 9:32 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
I meant `fix as a whole' rather than `fix pointwise'.
Sent: Saturday, August 30, 2014 at 6:32 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
Adam, I thought you referred to elements of PGL(n+1) that *preserve* S^n, which is not the same thing.
--Dan
On Aug 30, 2014, at 1:52 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
In fact, I wonder if the elements of the conformal group Conf(S^n) that happen to also be elements of [the projective group PLG(n+1) acting on S^n] are just the rotations. (This is certainly true for n = 2, where
Conf(S^2) = Aut(S^2) = PSL(2,C),
the holomorphic automorphism group of S^2.
The [orientation-preserving] elements of the projective group PSL(n+1) acting on S^n are precisely the elements of the conformal group Conf(S^n), as I mentioned in my previous e-mail.
The group SO(n) of rotations is considerably smaller.
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