Opened 8 years ago

Closed 8 years ago

Last modified 15 months ago

#7729 closed enhancement (fixed)

Iwahori Hecke algebras on the T basis

Reported by: bump Owned by: 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:

Description (last modified by 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 bump 8 years ago.
Iwahori Hecke algebra patch, including revisions from
trac_7729_iwahori-hecke-algebra.patch (21.3 KB) - added by nthiery 8 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 bump 8 years ago.
trac_7729_iwahori-hecke-algebra.3.patch (21.2 KB) - added by bump 8 years ago.
Implements Iwahori Hecke algebras

Download all attachments as: .zip

Change History (24)

comment:1 Changed 8 years ago by bump

  • Summary changed from Iwahori Hecke algebras to Iwahori Hecke algebras [with patch, needs review]

comment:2 Changed 8 years ago by bump

  • Description modified (diff)

comment:3 Changed 8 years ago by bump

  • Description modified (diff)

comment:4 Changed 8 years ago by bump

  • Description modified (diff)

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

comment:5 Changed 8 years ago by bump

  • Component changed from algebra to combinatorics

comment:6 Changed 8 years ago by nthiery

  • Status changed from new to needs_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 8 years ago by 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 8 years ago by bump

  • Status changed from needs_work to needs_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 8 years ago by 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 8 years ago by bump

  • Description modified (diff)

comment:11 Changed 8 years ago by bump

  • Description modified (diff)

Changed 8 years ago by bump

Iwahori Hecke algebra patch, including revisions from

comment:12 Changed 8 years ago by 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 8 years ago by nthiery

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

comment:13 Changed 8 years ago by nthiery

  • Summary changed from Iwahori Hecke algebras [with patch, needs review] to Iwahori Hecke algebras on the T basis

Changed 8 years ago by bump

comment:14 Changed 8 years ago by bump

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

comment:15 Changed 8 years ago by 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 8 years ago by bump

Implements Iwahori Hecke algebras

comment:16 Changed 8 years ago by bump

  • Reviewers set to Anne Schilling

comment:17 Changed 8 years ago by aschilling

  • Authors changed from Daniel Bump to Daniel Bump, Nicolas M. Thiery
  • Status changed from needs_review to positive_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 8 years ago by rlm

  • Merged in set to 4.3.1.alpha2
  • Resolution set to fixed
  • Status changed from positive_review to closed

comment:19 Changed 8 years ago by mvngu

  • Merged in changed from 4.3.1.alpha2 to sage-4.3.1.alpha2

comment:20 Changed 15 months ago by chapoton

  • Authors changed from Daniel Bump, Nicolas M. Thiery to Daniel Bump, Nicolas M. Thiéry
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