Opening more doors: multi-V-maps, subspaces of multivectors

Svals can be thought of as cosets of vals. We can extend this same
principle to consider "smultivals", which are cosets of multivals.

The V-map admits a natural generalization sending multivals ->
smultivals; I'll call this construct a multi-V-map, or an n-V-map for
a specific grade of n-multivector. This map can be represented as a
matrix in which the columns are "smultimonzos" representing the basis
of smultivals you want to go to.

As an example, say you're in the 7-limit and you want to go to the
2.5/3.7/5 limit. You can construct a map sending 7-limit bivectors to
2.5/3.7/5 bivectors as follows:
1) Figure out the bimonzo basis for the 2.5/3.7/5 subgroup. In this
case, it's (2^5/3).(2^7/5).(5/3^7/5).
2) Construct a matrix in which the columns are the above bimonzo
basis; an example will be given below.
3) Left-multiply this matrix by bivals to send them to s-bivals.

For instance, (2^5/3) = ||-1 1 0 0 0 0>>, (2^7/5) = ||0 -1 1 0 0 0>>,
and (5/3^7/5) = ||0 0 0 1 -1 1>>. Therefore, 2-V-map for the example
above is the matrix with these multimonzos as columns. If we
left-multiply this matrix by the bival representing myna, for example,
which is <<10 9 7 -9 -17 -9||, we get the multival <<-1 -2 -1||, which
represents 2.5/3.7/5 starling temperament.

If we left-multiply it by a "mapping matrix of bivals" rather than a
single bival, we get a new mapping matrix in the subgroup instead. One
use for mapping matrices of bivals is in representing Gene's Fokker
groups. However, I haven't tested this to make sure there's no hitch
in transforming Fokker group matrices to s-Fokker group matrices, so
be careful.

Much like V-maps "temper out vals", multi-V-maps temper out multivals.
For instance, the multivals eliminated by the above 2-V-map are <<0 0
0 -1 0 1||, <<0 0 0 1 1 0||, and <<1 1 1 0 0 0||.

As a last note, you need to be careful, however, because it's easy to
screw up and construct bogus multi-V-maps which don't actually mean
anything. For instance, you can come up with a map sending R^6 to R^4,
but this space loses the interpretation of being an exterior power of
anything. You can also easily screw up and set the columns of your
multi-V-map as not being totally-decomposable multivectors too. Don't
let this happen to you!

-Mike

On Sat, Jul 21, 2012 at 5:48 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Svals can be thought of as cosets of vals. We can extend this same
> principle to consider "smultivals", which are cosets of multivals.
>
> The V-map admits a natural generalization sending multivals ->
> smultivals; I'll call this construct a multi-V-map, or an n-V-map for
> a specific grade of n-multivector. This map can be represented as a
> matrix in which the columns are "smultimonzos" representing the basis
> of smultivals you want to go to.

In fact, it just occurred to me that for any individual V-map, you can
derive the whole host of multi-V-maps. If I want to go from 2.3.5.7 to
2.5/3.7/5, for instance, that implies an associated 1-V-map, 2-V-map,
etc. In other words, for any V-map on a vector space X, we immediately
have n-V-maps on the nth exterior power /\^n(X) for all n.

I'm just going to use the term "V-map" for the whole shebang then, the
master map defined on the entire exterior algebra /\(X), which sends
n-grade multivals to n-grade smultivals. If I ever want to talk about
just the V-map on vals, and it's not clear from context what I mean,
I'll call that a 1-V-map.

-Mike