#7729 closed enhancement (fixed)
Iwahori Hecke algebras on the T basis
Reported by: | Daniel Bump | Owned by: | Daniel Bump |
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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 )
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)
Change History (24)
comment:1 Changed 13 years ago by
Summary: | Iwahori Hecke algebras → Iwahori Hecke algebras [with patch, needs review] |
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comment:2 Changed 13 years ago by
Description: | modified (diff) |
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comment:3 Changed 13 years ago by
Description: | modified (diff) |
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comment:4 Changed 13 years ago by
Description: | modified (diff) |
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comment:5 Changed 13 years ago by
Component: | algebra → combinatorics |
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comment:6 Changed 13 years ago by
Status: | new → needs_work |
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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
Description: | modified (diff) |
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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
Status: | needs_work → needs_review |
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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
Description: | modified (diff) |
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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
Description: | modified (diff) |
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comment:11 Changed 13 years ago by
Description: | modified (diff) |
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Changed 13 years ago by
Attachment: | iwahori.patch added |
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Iwahori Hecke algebra patch, including revisions from
comment:12 Changed 13 years ago by
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
Attachment: | trac_7729_iwahori-hecke-algebra.patch added |
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IwahoriHeckeAlgebraT now takes a Coxeter group + moved method to ModulesWithBasis?. Replaces the previous patch.
comment:13 Changed 13 years ago by
Summary: | Iwahori Hecke algebras [with patch, needs review] → Iwahori Hecke algebras on the T basis |
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Changed 13 years ago by
Attachment: | trac_7729_iwahori-hecke-algebra.2.patch added |
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comment:14 Changed 13 years ago by
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
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
Attachment: | trac_7729_iwahori-hecke-algebra.3.patch added |
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Implements Iwahori Hecke algebras
comment:16 Changed 13 years ago by
Reviewers: | → Anne Schilling |
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comment:17 Changed 13 years ago by
Authors: | Daniel Bump → Daniel Bump, Nicolas M. Thiery |
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Status: | needs_review → 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 13 years ago by
Merged in: | → 4.3.1.alpha2 |
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Resolution: | → fixed |
Status: | positive_review → closed |
comment:19 Changed 13 years ago by
Merged in: | 4.3.1.alpha2 → sage-4.3.1.alpha2 |
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comment:20 Changed 6 years ago by
Authors: | Daniel Bump, Nicolas M. Thiery → Daniel Bump, Nicolas M. Thiéry |
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I posted a new version. This version works either before or after the patch in #7718.