#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 )
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 8 years ago by
- Summary changed from Iwahori Hecke algebras to Iwahori Hecke algebras [with patch, needs review]
comment:2 Changed 8 years ago by
- Description modified (diff)
comment:3 Changed 8 years ago by
- Description modified (diff)
comment:4 Changed 8 years ago by
- Description modified (diff)
comment:5 Changed 8 years ago by
- Component changed from algebra to combinatorics
comment:6 Changed 8 years ago by
- 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
- 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
- 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
- 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
- Description modified (diff)
comment:11 Changed 8 years ago by
- Description modified (diff)
comment:12 Changed 8 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 8 years ago by
IwahoriHeckeAlgebraT now takes a Coxeter group + moved method to ModulesWithBasis?. Replaces the previous patch.
comment:13 Changed 8 years ago by
- Summary changed from Iwahori Hecke algebras [with patch, needs review] to Iwahori Hecke algebras on the T basis
Changed 8 years ago by
comment:14 Changed 8 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 8 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
comment:16 Changed 8 years ago by
- Reviewers set to Anne Schilling
comment:17 Changed 8 years ago by
- 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
- 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
- Merged in changed from 4.3.1.alpha2 to sage-4.3.1.alpha2
I posted a new version. This version works either before or after the patch in #7718.