#7729: Iwahori Hecke algebras [with patch, needs review]
---------------------------+------------------------------------------------
Reporter: bump | Owner: bump
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.3.1
Component: algebra | Keywords: Iwahori Hecke Algebra
Work_issues: | Author: Daniel Bump
Upstream: N/A | Reviewer:
Merged: |
---------------------------+------------------------------------------------
Description changed by bump:
Old description:
> The attached patch implements Iwahori Hecke algebras. Given a Cartan
> Type, the Iwahori Hecke algebra is a deformation of the group algebra
> over the Weyl group. It has generators in bijection with the simple
> reflections of the Weyl group that satisfy simple quadratic relations of
> the form (T_i-q1)*(T_i-q2) = 0. Often we default q2=-1, q1=q in which
> case the relation is of the form T_i^2=(q-1)T_i+q. The generators also
> satisfy the braid relations.
>
> {{{
> sage: R.<q>=PolynomialRing(QQ)
> sage: H = IwahoriHeckeAlgebra("A3",q)
> sage: [T1,T2,T3]=H.algebra_generators()
> sage: T1*(T2+T3)*T1
> T1*T2*T1 + (q-1)*T3*T1 + q*T3
> }}}
>
> This code is very tested for type A and is almost certainly correct for
> Weyl groups of finite type. I have not tried it for any affine Weyl
> groups.
>
> The following issues remain.
>
> * It may require some revision in order to follow Sage's coercion model.
> David Roe suggested that the _coerce_impl method should be removed.
>
> * The get_action method is a kludge to avoid the crash reported in #7725.
> That crash is fixed by David Roe's patch in #7718, but this patch does
> not work with the patch in #7718.
>
> For some further discussion of this topic see
> http://groups.google.com/group/sage-combinat-
> devel/browse_thread/thread/78fc23f23cafe705?hl=en
>
> It is well tested for type A and is probably correct for all Cartan Types
> of finite type. I have not tried it with
New description:
The attached patch implements Iwahori Hecke algebras. Given a Cartan Type,
the Iwahori Hecke algebra is a deformation of the group algebra over the
Weyl group. It has generators in bijection with the simple reflections of
the Weyl group that satisfy simple quadratic relations of the form
(T_i-q1)*(T_i-q2) = 0. Often we default q2=-1, q1=q in which case the
relation is of the form T_i^2=(q-1)T_i+q. The generators also satisfy the
braid relations.
{{{
sage: R.<q>=PolynomialRing(QQ)
sage: H = IwahoriHeckeAlgebra("A3",q)
sage: [T1,T2,T3]=H.algebra_generators()
sage: T1*(T2+T3)*T1
T1*T2*T1 + (q-1)*T3*T1 + q*T3
}}}
This code is very tested for type A and is almost certainly correct for
Weyl groups of finite type. I have not tried it for any affine Weyl
groups.
The following issues remain.
* It may require some revision in order to follow Sage's coercion model.
David Roe suggested that the _coerce_impl method should be removed.
* The get_action method is a kludge to avoid the crash reported in #7725.
That crash is fixed by David Roe's patch in #7718, but this patch does not
work with the patch in #7718.
* It should be made to work with Affine Weyl groups. I have not checked
whether this requires further modification.
* Subjectively, it seems a little slow compared with a previous
implementation for type A only. This is probably a limitation of the
{{{WeylGroup()}}} class on which it depends. My earlier implementation was
based on Permutation. If it proves unacceptably slow it may be possible to
speed it up by a caching scheme.
For some further discussion of this topic see
http://groups.google.com/group/sage-combinat-
devel/browse_thread/thread/78fc23f23cafe705?hl=en
It is well tested for type A and is probably correct for all Cartan Types
of finite type.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7729#comment:2>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/sage-trac?hl=en.
