#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: |
---------------------------+------------------------------------------------
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.
>
> * 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.
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.
* David Roe suggested that the _coerce_impl method should be removed. I
have not looked at this yet.
* 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.
* Later I may add a method to compute intertwining elements which depend
on spectral parameters. These have applications to representations of
p-adic groups.
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.
--
Comment(by bump):
I posted a new version. This version works either before or after the
patch in #7718.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7729#comment:4>
Sage <http://www.sagemath.org>
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