#7729: Iwahori Hecke algebras [with patch, needs review]
-----------------------------+----------------------------------------------
Reporter: bump | Owner: bump
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.3.1
Component: combinatorics | 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. It also works for affine Weyl groups in the
> most recently posted version.
>
> The following issues remain.
>
> * David Roe suggested that the _coerce_impl method should be removed. I
> have not looked at this yet.
>
> * 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
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. It also works for affine Weyl groups.
For some further discussion of this topic see
http://groups.google.com/group/sage-combinat-
devel/browse_thread/thread/78fc23f23cafe705?hl=en
--
Comment(by bump):
I posted a revised version. With this version, the base ring can be either
a field
containing q1 and q2, or a LaurentPolynomialRing. The previous version did
not
work with LaurentPolynomialRings.
Also, methods were added to
compute inverses of basis elements, a common task.
Finally, there is a bug fix in
sage.categories.pushout (import PolynomialRing when needed).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7729#comment:9>
Sage <http://www.sagemath.org>
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