Opened 13 years ago

Closed 13 years ago

Last modified 6 years ago

#7729 closed enhancement (fixed)

Iwahori Hecke algebras on the T basis

Reported by: Daniel Bump Owned by: Daniel Bump
Priority: major Milestone: sage-4.3.1
Component: combinatorics Keywords: Iwahori Hecke Algebra
Cc: Merged in: sage-4.3.1.alpha2
Authors: Daniel Bump, Nicolas M. Thiéry Reviewers: Anne Schilling
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

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Description (last modified by Daniel Bump)

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

Attachments (4)

iwahori.patch (20.4 KB) - added by Daniel Bump 13 years ago.
Iwahori Hecke algebra patch, including revisions from
trac_7729_iwahori-hecke-algebra.patch (21.3 KB) - added by Nicolas M. Thiéry 13 years ago.
IwahoriHeckeAlgebraT now takes a Coxeter group + moved method to ModulesWithBasis?. Replaces the previous patch.
trac_7729_iwahori-hecke-algebra.2.patch (21.1 KB) - added by Daniel Bump 13 years ago.
trac_7729_iwahori-hecke-algebra.3.patch (21.2 KB) - added by Daniel Bump 13 years ago.
Implements Iwahori Hecke algebras

Download all attachments as: .zip

Change History (24)

comment:1 Changed 13 years ago by Daniel Bump

Summary: Iwahori Hecke algebrasIwahori Hecke algebras [with patch, needs review]

comment:2 Changed 13 years ago by Daniel Bump

Description: modified (diff)

comment:3 Changed 13 years ago by Daniel Bump

Description: modified (diff)

comment:4 Changed 13 years ago by Daniel Bump

Description: modified (diff)

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

comment:5 Changed 13 years ago by Daniel Bump

Component: algebracombinatorics

comment:6 Changed 13 years ago by Nicolas M. Thiéry

Status: newneeds_work

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.

comment:7 Changed 13 years ago by Daniel Bump

Description: modified (diff)

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.

comment:8 Changed 13 years ago by Daniel Bump

Status: needs_workneeds_review

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.

comment:9 Changed 13 years ago by Daniel Bump

Description: modified (diff)

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).

comment:10 Changed 13 years ago by Daniel Bump

Description: modified (diff)

comment:11 Changed 13 years ago by Daniel Bump

Description: modified (diff)

Changed 13 years ago by Daniel Bump

Attachment: iwahori.patch added

Iwahori Hecke algebra patch, including revisions from

comment:12 Changed 13 years ago by Daniel Bump

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

Changed 13 years ago by Nicolas M. Thiéry

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

comment:13 Changed 13 years ago by Nicolas M. Thiéry

Summary: Iwahori Hecke algebras [with patch, needs review]Iwahori Hecke algebras on the T basis

Changed 13 years ago by Daniel Bump

comment:14 Changed 13 years ago by Daniel Bump

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

comment:15 Changed 13 years ago by Daniel Bump

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

Changed 13 years ago by Daniel Bump

Implements Iwahori Hecke algebras

comment:16 Changed 13 years ago by Daniel Bump

Reviewers: Anne Schilling

comment:17 Changed 13 years ago by Anne Schilling

Authors: Daniel BumpDaniel Bump, Nicolas M. Thiery
Status: needs_reviewpositive_review

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.

comment:18 Changed 13 years ago by Robert Miller

Merged in: 4.3.1.alpha2
Resolution: fixed
Status: positive_reviewclosed

comment:19 Changed 13 years ago by Minh Van Nguyen

Merged in: 4.3.1.alpha2sage-4.3.1.alpha2

comment:20 Changed 6 years ago by Frédéric Chapoton

Authors: Daniel Bump, Nicolas M. ThieryDaniel Bump, Nicolas M. Thiéry
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