On Fri, Dec 11, 2009 at 07:11:37AM -0800, Daniel Bump wrote:
> Yes, http://wiki.sagemath.org/HeckeAlgebras is exactly what we
> need. These are so fundamental that they should be in Sage.
Definitely.
> > I think rewriting from scratch an implementation of the generic Hecke
> > algebra in the T_w basis, with two parameters q1 and q2 would be a
> > good starting point. With the current category and root system stuff,
> > it should be about 20 lines of code. Starting with the example in
> > AlgebrasWithBasis, I would add the following functions:
>
> I'll give it a try.
Thanks.
> One point is that there is quite a bit of Hecke algebra
> code in SAGE which has nothing to do with this, since
> Hecke algebras means something different to people
> who work in modular forms. (Going back of course to
> Hecke himself.)
Yes.
> For example in hecke.py we find:
>
> > In Sage a "Hecke algebra" always refers to an algebra of
> > endomorphisms of some explicit module, rather than the
> > abstract Hecke algebra of double cosets attached to a
> > subgroup of the modular group.
>
> This sentence points up how overloaded the term
> Hecke algebra is in mathematics. Hecke
> algebras as in hecke.py and generic Hecke
> algebras as we want are related to each
> other, but the relation is not so close. The
> "abstract Hecke algebra of double cosets"
> mentioned in this quote would be the
> spherical (maximal abelian) subalgebra in the
> case where W=WeylGroup("A1"), q1=p and q2=1.
> Or for modular forms with Nebentypus p you
> (these must be unramified or Steinberg at p) you
> would get the full Hecke algebra, again with
> q1=p and q2=1. The general Iwahori Hecke
> algebra similarly has applications in
> automorphic forms but it also has
> applications everywhere else in mathematics.
>
> Is there any way to mitigate the name clash?
Would it be natural to use IwahoriHeckeAlgebra(W, q1, q2)?
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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