On Fri, Jun 05, 2009 at 07:47:05PM -0700, William Stein wrote:
>
> Hi Sage-Devel,
>
> This email will focus entirely on functionality present in Magma that
> is not available in Sage. We will almost entirely ignore questions of
> efficiency for this email.
...
> VIII. Finite Groups; Finitely-Presented Groups
>
On the margin:
VIII bis. Finite semigroups / monoids
PROJECT: wrap/integrate/port Semigroupe by Jean-Éric Pin. He has shown
much interest in this:
http://www.liafa.jussieu.fr/~jep/semigroupes.html
See also the preliminary AutomaticMonoid patch on Sage-Combinat.
PROJECT: computation of the representation theory of a semigroup
I have some preliminary stuff on this in the upcoming SemiGroups category.
Franco has expertise. This does not really exist in other software.
> X. Coxeter Groups; Lie Groups
>
> * I have the impression that our Coxeter code is extremely good,
> and is under active development (e.g., by Dan Bump).
Well, I am happy with the flexibility of the infrastructure which
should scale well:
PROJECT: generalize to non crystallographic root systems, and to
pseudo reflection groups
On the other hand, the whole thing needs speed. This will come from
optimizing the underlying data structures:
PROJECT: make CombinatorialFreeModule ***fast*** and then merge with
the existing FreeModule.
> * I'm not sure about the situation for Lie groups.
AFAIK, nothing yet, except computing with their characters.
> XII. Algebras
>
> * Generic algebras -- we have them, but they are slow and not used
> much. Little of interest is supported.
>
> PROJECT: Create a fast optimized Cython implementation of free
> algebras and their quotients. Basically, rewrite sage/algebras/free*.
> Make sure that all functionality in Magma for these is also present in
> Sage.
Should be for a good part a corollary of the above project
(CombinatorialFreeModule) + support from the categories
(AlgebrasWithBasis and friends). Remains good algorithms for computing
in quotients (non or skew commutative Groebner, ...).
> * Ideal theory of quaternion algebras over *number fields*. This is
> extremely high on my todo list. We have very fast basic
> arithmetic, but need ideal theory over Dedekind domains, etc.
>
> PROJECT: Implement quaternion algebras over over number fields,
> following the algorithms and strategy John Voight used for Magma.
>
> * Quantum groups, Lie algebras, Universal enveloping algebras --
> pretty sure we don't have this in Sage.
Nope. We have crystals though, which are very much related. There
certainly will be people working on this kind of stuff in the
Sage-Combinat team.
> XIII. Representation Theory
>
> * Characters of finite groups; GAP has this, of course, but I don't
> think Sage nicely wraps it.
>
> PROJECT: Wrap more of GAP's functionality, in particular characters of
> finite groups.
On the margin:
* Representations of finite dimensional algebras
PROJECT: port Florent's code in MuPAD-Combinat for computing the
simple and projective modules and the Cartan matrix.
Most of the work will be to have good support for subspaces and
quotients in CombinatorialFreeModules and in the categories.
> XIV. Commutative Algebra
>
> * Modules over multivariate rings. This is a big Gap in
> functionality in Sage; presumably Singular can help fill it in?
>
> PROJECT: If Singular can do this well, wrap it. If not, consider
> Macaulay 2 which certainly can.
>
> * Differential rings: I think we have nothing in Sage.
>
> PROJECT: Add support for differential rings. Again, Singular or Macaulay2?
>
> * Invariant theory: I think we have nothing in Sage.
>
> PROJECT: Add support for invariant theory. Maybe this in GAP +
> Singular as a package.
There is a small piece of the Singular finvar package exposed in
MatrixGroup_gap_finite_field.
My PhD student Nicolas Borie is working on invariant theory for
permutation groups. I had implemented a library on this topic for my
own PhD (PerMuVAR, integrated in MuPAD-Combinat), based on F4-style
SAGBI-Groebner basis. He is investigating a new Groebner free
algorithm. This is a long run project, so don't hold your breath.
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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