Yes, http://wiki.sagemath.org/HeckeAlgebras
is exactly what we need. These are so fundamental that they should
be in Sage.
> 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.
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.)
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?
Dan
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