Iwahori Hecke algebras on the T basis

[dwbump]

The attached patch implements Iwahori Hecke algebras. Given a Cartan Type (finite or affine), 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

For some further discussion of this topic see
http://groups.google.com/group/sage-combinat-devel/browse_thread/thread/78fc23f23cafe705?hl=en

Component: combinatorics

Keywords: Iwahori Hecke Algebra

Author: Daniel Bump, Nicolas M. Thiéry

Reviewer: Anne Schilling

Merged: sage-4.3.1.alpha2

Issue created by migration from https://trac.sagemath.org/ticket/7729

[dwbump]

comment:4

I posted a new version. This version works either before or after the patch in #7718.

[nthiery]

comment:6

Hi Dan!

Thanks much for implementing this very useful feature!

Do you mind renaming it into IwahoriHeckeAlgebraT or TBasis, so that we can
later use IwahoriHeckeAlgebra for the abstract Iwahori Hecke algebra with its other bases?

Other than that, the code looks good, except for the duplication of the CombinatorialFreeModule code. Do you mind if I refactor it to use the category framework and CombinatorialFreeModule?
I am not sure when I'll be able to do that though, so maybe it's best to first get this patch
into Sage. Unless you are tempted by the adventure.

I don't expect particular problems with affine weyl groups or other coxeter groups.

Ah one thing: please make the algebra generators into a family indexed by the index set of the Dynkin diagram (so that T[1], ... ) will do what we expect.

[dwbump]

comment:7

I've revised it so that it works with affine Weyl groups.

I don't mind renaming it IwahoriHeckeAlgebraT but it seems to
me that perhaps other presentations can be handled within this
framework. I think I should leave the refactoring to the
category framework to you.

I will make the algebra generators into a family. When I've
done that I will change the status back to needs review.

[dwbump]

comment:8

I've addressed two out of three of Nicolas' requests, and his
message indicates that the refactoring issue can be postponed.

• The name is now IwahoriHeckeAlgebraT

• self.algebra_generators() now returns a finite family.

I've changed the status back to needs review.

Nicolas wrote:

Do you mind renaming it into IwahoriHeckeAlgebraT or TBasis, so that we can later > use IwahoriHeckeAlgebra? for the abstract Iwahori Hecke algebra with its other bases?

what other bases do we need? There is the Bernstein Zelevinsky presentation.

[dwbump]

comment:9

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).

[dwbump]

Attachment: iwahori.patch.gz

Iwahori Hecke algebra patch, including revisions from

[dwbump]

comment:12

I qfolded two patches from the trac server and re uploaded the patch.

trac_7729-iwahori-hecke-fixdoctests-nt.patch
trac_7729-iwahori-hecke-reviewer-nt.patch

[nthiery]

IwahoriHeckeAlgebraT now takes a Coxeter group + moved method to ModulesWithBasis. Replaces the previous patch.

[nthiery]

comment:13

Attachment: trac_7729_iwahori-hecke-algebra.patch.gz

[dwbump]

Attachment: trac_7729_iwahori-hecke-algebra.2.patch.gz

[dwbump]

comment:14

I made minor revisions to the docstring and reposted the patch as
trac_7729_iwahori-hecke-algebra-2.patch.

[dwbump]

comment:15

The patch trac_7729_iwahori_hecke_algebra_3.patch implements Anne Schilling's
comments from:

http://groups.google.com/group/sage-combinat-devel/msg/e2abca2135c73e33?hl=en

It also adds Nicolas as an author, and fixes the copyright year.

Dan

[dwbump]

Attachment: trac_7729_iwahori-hecke-algebra.3.patch.gz

Implements Iwahori Hecke algebras

[dwbump]

Reviewer: Anne Schilling

[anneschilling]

Changed author from Daniel Bump to Daniel Bump, Nicolas M. Thiery

[anneschilling]

comment:17

This patch implements the much desired Iwahori Hecke algebras in sage. It is very well documented with explanations on usage and references to the literature. All methods have doctests and all tests pass. In particular, I tested several special cases (like the nilCoxeter case q_1=q_2=0) and everything seemed to work fine.

[rlmill]

Merged: 4.3.1.alpha2

[sagetrac-mvngu]

Changed merged from 4.3.1.alpha2 to sage-4.3.1.alpha2

[fchapoton]

Changed author from Daniel Bump, Nicolas M. Thiery to Daniel Bump, Nicolas M. Thiéry