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In article <32A35298.10C1@ix.netcom.com>,
Michael Thayer   wrote:
>John Baez wrote:
>> In particular, thanks to the cannonball trick of Lucas, the vector
>> 
>>                v = (70,0,1,2,3,4,...,24)
>> 
>> is "lightlike".  In other words,
>> 
>>                     v.v = 0
>I don't see what is so significant about the vector v.  for instance,
>the 10 dimensional vector (3,1,1,1,1,1,1,1,1,1) is also light like, and
>you make no big deal about that.  Is there some reason why the ascending
>values in v are important?
Yikes!  Thanks for catching that massive hole in the exposition.
You're right that there's no shortage of lightlike vectors in 
the even unimodular Lorentzian lattices of other dimensions 8n+2; there 
are also lots of other lightlike vectors in the 26-dimensional one.
Any one of these gives us a lattice in 8n-dimensional Euclidean space.
In fact, we can get all 24 even unimodular lattices in 24-dimensional
Euclidean space by suitable choices of lightlike vector.   The lightlike
vector you wrote down happens to give us the E8 lattice in 8 dimensions.
So what's so special about I wrote, which gives the Leech lattice?  Of 
course the Leech lattice is itself special, but what does this
have to do with the nicely ascending values of the components of v?
Alas, I don't know the real answer.  I'm not an expert on this
stuff; I'm just explaining it in order to try to learn it.  Let me
just say what I know, which all comes from Chap. 27 of Conway and
Sloane's book "Sphere Packings, Lattices, and Groups".
If we have a lattice, we say a vector r in it is a "root" if
the reflection through r is a symmetry of the lattice.   Corresponding
to each root is a hyperplane consisting of all vectors perpendicular
to that root.  These chop space into a bunch of "fundamental regions".
If we pick a fundamental region, the roots corresponding to the 
hyperplanes that form the walls of this region are called "fundamental
roots".  The nice thing about the fundamental roots is that the
reflection through any root is a product of reflections through these
fundamental roots.  
[For more stuff on reflection groups and lattices see "week62" and
the following weeks.]
In 1983 John Conway published a paper where he showed various 
amazing things; this is now Chapter 27 of the above book.  First, 
he shows that the fundamental roots of the even unimodular Lorentzian 
lattices in dimensions 10, 18, and 26 are the vectors r with r.r = 2
and r.v = -1, where the "Weyl vector" v is
(28,0,1,2,3,4,5,6,7,8)
(46,0,1,2,3,......,16)
and
(70,0,1,2,3,......,70)
respectively.  
They all have this nice ascending form but only in 26 dimensions
is the Weyl vector lightlike!   
Howerver, Conway doesn't seem to explain *why* the Weyl vectors have 
this ascending form.  So I'm afraid I really don't understand how 
all the pieces fit together.  All I can say is that for some reason 
the Weyl vectors have this ascending form, and the fact that the Weyl 
vector is also lightlike makes a lot of magic happen in 26 dimensions.
For example, it turns out that in 26 dimensions there are *infinitely
many* fundamental roots, unlike in the two lower dimensional cases.
Just to add mystery upon mystery, Conway notes that in higher dimensions
there is no vector v for which all the fundamental roots r have
r.v equal to some constant.  So the pattern above does not continue.  
I find this stuff fascinating, but it would drive me nuts to try
to work on it.  It's as if God had a day off and was seeing how many
strange features he could build into mathematics without actually
making it inconsistent.

In article <583hi7$mkj@b.stat.purdue.edu>,
Herman Rubin  wrote:
>I believe that matrices of the form I+K, K compact, also occur in
>quantum mechanics, and that the determinant is sometimes needed.
Yes.  But physicists, having no sense of restraint, also work with 
det(A) where A is not of the above form.  For example, in quantum 
field theory they often talk about the determinant of the 
Laplacian!  To make sense of this, people have developed a large 
battery of tools for regularizing determinants of operators, using 
zeta function tricks to make sense of divergent sums.  These 
are especially handy in applications of quantum field theory to
topology and vice versa --- the sort of thing Atiyah, Singer, Witten
and company are famous for.

Eric Billault wrote:
> 
> Let SU(n) the special unitary group.
> Let SO(2) the group of rotations of R^2
> Let p=E(n/2)
> The product T = SO(2) x ... x S0(2) (p factors) is naturally embedded in
> SU(n)
> Question:
> 
>   Which space SU(n)/T is diffeomorphic to ?
Well, T is a torus, so is conjugate to a sub-torus of the standard
maximal torus.  I confess to not knowing what E(n/2) is supposed to
mean, but it would seem to be [n/2].  I also don't know how you intend
to imbed (R^2)^p into C^n, but all such are going to have to map this
torus into a conjugate of the maximal torus
S={diag(lambda_1,...,lambda_n) | |lambda_i|=1, lambda_1...lambda_n = 1}.
SU(n)/S is a flag manifold, of all complex lines in complex planes in...
up to the whole space.  For a subtorus T of rank p < n-1, SU(n)/T would
be a principle SU(n-p) bundle over the partial flag manifold
SU(n)/(TxSU(n-p)).  
If you are embedding each R^2 as C, then you can take n-1 factors.  This
doesn't seem right, so I presume you are embedding SO(2) inside SO(2,C)
first, as the complex 2 x 2 matrices 
[cos t   sin t]
[             ]
[-sin t  cos t],   
then putting it in SU(n).  I don't think there is anything special about
the way that the torus is embedded, up to conjugation.
-- 
David L. Johnson         		dlj0@lehigh.edu, dlj0@netaxs.com
Department of Mathematics		http://www.lehigh.edu/~dlj0/dlj0.html
Lehigh University
14 E. Packer Avenue			(610) 758-3759
Bethlehem, PA 18015-3174