#14261: Iwahori-Hecke algebra with several bases
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Reporter: brant | Owner:
sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.10
Component: combinatorics | Resolution:
Keywords: Iwahori Hecke algebra | Work issues:
Report Upstream: N/A | Reviewers: Andrew
Mathas?, Dan Bump?
Authors: Brant Jones, Travis Scrimshaw | Merged in:
Dependencies: | Stopgaps:
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Comment (by andrew.mathas):
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 the problem above but it also
runs into the problem 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 he 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.
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14261#comment:4>
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