Yahoo Tuning Groups Ultimate Backup tuning-math Question about classifying lattices

I've sort of, in a roundabout way, asked this before, but now I'll
just ask directly. Lattices are strange and mysterious to me and I'd
like to know how to classify them. Does anyone know?

Here's specifically what I want to know:
An unordered pair of two vectors can be said to define a single,
unique subspace consisting of all linear combinations of those
vectors. This representation of a subspace is also not unique, because
there's more than one unordered pair which gives you the same
subspace. One way to get a unique representation is to take this pair
of vectors, put it in a matrix, and reduce it to rref form, which acts
as a sort of "key" or "fingerprint" for the subspace I was talking
about. So for any two unordered pairs of vectors, you can simply
compare the rref forms of both and see how if they're the same to see
if they lie in the same subspace.

An approach that's so much more elegant than this that it blows my
mind is that instead, you can use the exterior product, which is a
simple operator with a few simple axioms, and it does almost all of
that for you and gets you halfway there. You can wedge the vectors in
your unordered pair together to get a bivector. Then, after messing
with the sign and modding things out by the GCD and all that, you get
one unique bivector which uniquely identifies the entire subspace and
which can be identified with a point on the Grassmannian for your
vector space. The whole thing is tied together and connected in a way
that I sort of get, but that I don't really understand completely yet.

So, you all know that. My question is:
Now let's say you take this unordered pair of two vectors to define,
instead, a single unique lattice consisting of all -integral- linear
combinations of those vectors. This representation of a lattice is,
again, not unique. This makes it difficult to figure out which lattice
two vectors define, and how to compare two of these unordered pairs to
see if they reside on the same lattice.

For these, instead of rref, you can use something like Hermite form.
And for lattices, as with subspaces, I'd really, really, really like
to figure out some ultra-elegant solution, similar to the exterior
product, which will do the exact same thing as Hermite form, given
some similar messing with the sign. Something which is similarly all
fleshed out, some analogue to the Grassmannian which deals with
"lattices through the origin" instead of subspaces through the origin.
Something which, like the exterior product, just has a few simple
axioms which encapsulate this -underlying property- that sets of
vectors share if they define the same lattice, just like the
anticommutative property of the exterior product somehow leads to it
encapsulating a lot of the same properties that sets of vectors share
if they define the same subspace.

If anyone has any idea where to look for this, I'd reeeeeallllly
appreciate the help. I've been doing a ton of reading on Bravais
lattices and Miller indices and I still don't get it. There's so much
stuff I've learned but I still don't get this. It seems like there's
even a nice Pontryagin dual relationship here, because if you
"contort" one of these lattices in one space, the corresponding
reciprocal lattice in the dual space gets split in half...! It's like
it's all there, directly in front of me and I just don't get the basic
mathematical property that I'm after that makes it all tick. That's
where I keep getting stuff. I keep running into arcane, inelegant
solutions like LLL reduction and Hermite form and I know there's
something simpler. Maybe something like the wedge product, but which
can also represent something like the "angle" the bivector makes with
the origin, so that a^2b != 2a^b?

Halp?

-Mike

Question about classifying lattices

🔗Mike Battaglia <battaglia01@gmail.com>

🔗Mike Battaglia <battaglia01@gmail.com>

🔗clamengh <clamengh@yahoo.fr>

🔗clamengh <clamengh@yahoo.fr>