Difference between every pair of these: S"++++-", V"++++-" and "++++-"

[Hongying]

Hi,

• On notation:
  What is the difference between every pair of the three: @basis S"++++-", @basis V"++++-" and @basis "++++-"?
  I'm a little confused in trying the examples in README.md. It seems they all give me the same results.

• On algebra
  In Grassmann.js, is V"++++-" also the conformal model G^{4,1} of R^3?
  (I noticed in the section "Null-basis of the conformal split" that we can declare a space with signature ∞ and ∅, but did not figure out how I can construct a signature such that it is in correspondence with "++++-".

Thank you,
Hongying

[Dream Scatter]

Thanks for your question. For the first example, there is no difference between them. There is a generic constructor V"..." which automatically tries to select the correct sub type of the abstract Manifold type, while the S"..." constructor is specific for the Signature instance. When S"..." is used it is specific for signatures of +/- values, you also get the option of specifying null-basis by using the @basis S"∅∞+++" command. Then the origin always must be specified in the first position and the point at infinity is placed right after the origin.

For the second example: No, V"++++-" is not the null-basis split for spacetime, you need to use S"∅∞+++" for that. These two algebras are isomorphic to each other, but have different basis. Specifically, instead of an extra basis of + and - in Signature you have a null-basis. This is described in my paper, but I recommend reading other books/sources to learn more.

Also related to this is chakravala/DirectSum.jl#2 regarding the generic constructors V"...".