Re: discriminating SO3 invariance from general O3 invariance

In article <dc650d$4i1$1@xxxxxxxxxxxxxxxxx>, rusin@xxxxxxxxxxxxxxxxxxxxx
(Dave Rusin) wrote:

> In article <1121957328.412673.23400@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Ofek
> Shilon" <ofek@xxxxxxxxxxxxx> wrote:
>
> > Given the space S of symmetric real 3x3 matrices, we operate on it with
> > O3 via conjugation.
> > I wish to discriminate SO3 conjugation classess from O3 conjugation
> > classess. that is - i'm looking for a function of a matrix E which is
> > invariant under rotations, but not under reflections (in general, at
> > least).
> > I believe i'm able to prove, with the aid of some earlier advice in
> > this group, that such an invariant function cannot be a polynomial in
> > the entries of E, of any degree. Still, I hope, the search is not lost.
> > is anyone familiar with such invariants?

[...]

> Question: what about if we replace "3" by "4" (or any other even number)?
>
> For example, SO_2 is the circle group, which is abelian,
> whereas O_2 includes as the matrix ((-1,0),(0,1)), conjugation by
> which sends each element of SO_2 to its inverse; so some of the
> (O_2)-conjugacy classes split into two (SO_2)-conjugacy classes.
>
> This much I understand. Less clear to me is what I consider to be
> the physicists' point of view here:
> > that is - i'm looking for a function of a matrix E which is
> > invariant under rotations, but not under reflections (in general, at
> > least).
> Wnat does this mean, for O_2 or O_4 say? In what sense is this
> function "non-polynomial"? (Polynomial in what variables?)

I can't answer your question, which the OP might be able to shed some
light on, but FWIW here are some related observations.

If we confine ourselves to symmetric matrices, then the O(n) conjugacy
classes are the same as the SO(n) conjugacy classes, regardless of whether
n is odd or even. If two real symmetric matrices have the same
eigenvalues, then you can always find a rotation that makes their
eigenspaces coincide; there is no "handedness" to the geometry.

If we allow non-symmetric matrices then this no longer holds, with
rotations in R^n with even n providing counterexamples. In R^2, it's
simple: every rotation is either clockwise or counterclockwise, and no
rotation can turn one into the other. For R^4 it's a bit more
complicated: rotations in a single plane can be swapped, by turning the
plane over just as in R^3, but simultaneous rotations in two planes will
have larger conjugacy classes under O(4) than under SO(4). For example,
the rotation by A in the xy plane and B in the zw plane can be mapped by
conjugation with diag(-1,1,1,1) to a rotation by (-A,B) in the same two
planes, but this can't be achieved by conjugation with any element of
SO(4).

--
Greg Egan

Email address (remove name of animal and add standard punctuation):
gregegan netspace zebra net au
.

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