Iwahori-Hecke algebra with several bases

[sagetrac-brant]

Set up the algebra to handle multiple bases; implemented the Kazhdan--Lusztig basis.

This is a follow up to ticket #7729.

See http://wiki.sagemath.org/HeckeAlgebras for some design discussion.

---

Apply:

• attachment: trac_14261-iwahori_hecke-ts.patch

Depends on #13735
Depends on #14014
Depends on #14678
Depends on #14516
Depends on #15257

CC: @sagetrac-sage-combinat @anneschilling @AndrewAtLarge @samclearman @zabrocki

Component: combinatorics

Keywords: Iwahori Hecke algebra, days45

Author: Brant Jones, Travis Scrimshaw, Andrew Mathas

Reviewer: Andrew Mathas, Brant Jones, Travis Scrimshaw

Merged: sage-5.13.beta1

Issue created by migration from https://trac.sagemath.org/ticket/14261

[sagetrac-brant]

Attachment: trac_14261_iwahori_hecke.patch.gz

[tscrim]

comment:1

Hey Brant,

I've uploaded a new version of the patch which reworks the implementation to follow the WithRealizations API and deprecates the IwahoriHeckeAlgebraT class. If it wasn't ready for review, I think it now is.

Best,

Travis

[tscrim]

Changed author from Brant Jones to Brant Jones, Travis Scrimshaw

[tscrim]

comment:2

I should also state that I had to upload a new version of the patch because Brant's patch did not apply on 5.9 due to the changes in the doc. Also for got this.

For patchbot:

Apply: trac_14261-iwahori_hecke-ts.patch

[tscrim]

Description changed:

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+++ 
@@ -4,3 +4,9 @@
 
 See http://wiki.sagemath.org/HeckeAlgebras for some design discussion.
 
+---
+
+Apply:
+
+* [attachment: trac_14261-iwahori_hecke-ts.patch](https://github.com/sagemath/sage-prod/files/10657354/trac_14261-iwahori_hecke-ts.patch.gz)
+

[tscrim]

comment:3

New version which also adds bar involution on the T_basis and some tweaks to the documentation.

For patchbot:

Apply: trac_14261-iwahori_hecke-ts.patch

[AndrewMathas]

comment:4

A few comments:

• I think that it is a little dangerous implementing the Kazhdan-Lusztig C'-basis and calling it the C-basis. The documentation says clearly that this is the C'-basis but these names have been in use for 30+ years at this point so I think at the minimum you need to warn people about this. (This is even though I agree that KL got the names of these two bases "wrong":)

• As the code is essentially the same, it would be good to implement both KL C-basis and the KL C'-basis. This would solve the problem above but it also runs into the issue of what to call the C'-basis. Perhaps use C_basis and Cp_basis?

• Having just said that the code is 'the same' in terms of caching it would be better to implement the C'-basis using the observation that C_x' = \alpha(C_x), where \alpha is the Z-linear ring automorphism of the Hecke algebra which sends q^1/2} ^to -q^-1/2 and T_x to (-1)^\ell(x)q^-1/2\ell(x)T_x.

[tscrim]

comment:5

So which would you call the KL basis, C or C' (I believe calling [one of these] the KL is fairly standard)? I think the best name should be C_prime_basis since it is explicit, but we can have Cp and C_prime as shortcuts.

I'll also implement the ring automorphism \alpha and implement the C basis in the next version. Thanks for looking at this Andrew.

[nthiery]

comment:6

It's probably not necessary to put the _basis suffix. Elsewhere, we use Sym.s, Sym.p and so on and not Sym.s_basis.

Cheers,
Nicolas