#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-5.12
Component: combinatorics | Resolution:
Keywords: Iwahori Hecke | Merged in:
algebra | Reviewers: Andrew Mathas, Brant
Authors: Brant Jones, | Jones, Travis Scrimshaw
Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: #13735 #14014 |
#14678 #14516 |
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Comment (by andrew.mathas):
Replying to [comment:35 tscrim]:
> Thank you for reviewing this ticket Andrew.
Well, I haven't actually finished a review yet! Thanks for getting back to
me.
> I'm slightly hesitant about option 3 (implementing a generic Hecke
algebra), and it's probably due to my lack of expertise, but does it
always work for specializing to roots of unity? Or do problems just arise
with the representation theory? If it's okay, then I'm for option 3.
Actually, that made me have another (scarier) thought, does anyone know if
this works for fields of positive characteristic? If not or we don't know,
I think we should put that in the documentation somewhere that we assume
the base ring has characteristic 0.
The KL bases are defined over `Z[q^{1/2},q^{-1/2}]` so you can specialise
them to ANY ring provided these square roost exist. You can't, however,
compute these bases over any ring becaue the bar involution will not be
defined in general. Therefore, one really SHOULD compute them generically
and then specialise.
To be explicit, the group ring `RW` of the Coxeter group has a well-
defined KL basis for any ring `R`, however, you can't "see" the KL bases
of `RW` inside the group ring. The only way to compute the KL bases or
`RW` is to do it inside the generic Hecke algebra and then specialise.
Notice also that even though the KL bases are defined for `RW` the bar
involution is NOT defined on `RW`, so non-generic Hecke algebras should
not have a method for the bar involution either.
The KL bases also work in the root of unity cases...it is more that no one
has found a good use for them yet.
> This is correct [about `one_basis`]; it is suppose to return the index
of basis element for the Hecke's algebra's (multiplicative) identity. It's
something needed for `CombinatorialFreeModule` (at least, I'm pretty
sure).
I don't believe this. Combinaorial modules can be indexed anything so it
seems unlikely that they would require something in the index set to be a
multiplicative identity. In the code `one_basis` is used as a shortcut to
get to things like `T(1)`.
> I'm not really opposed to removing it, but I'm thinking it should be in
there for consistency (with perhaps modified behavior).
Is `is_field` a default method for an algebra? Most algebras are not
fields so this struck me as being a strange question to ask
mathematically. If you accept this then you should agree that it is a
strange method for an algebra. (But perhaps it is just me who is
strange!:)
> I don't like removing the base ring in the representation (the
`Univariate Laurent Polynomial Ring in v over Rational Field` part) since
it helps distinguish the Hecke algebras. The `in the X basis` part is a
standard paradigm from other `WithRealization` objects such as symmetric
functions.
I agree that this could be useful but I always err on the side of
readabiltly and ease of use.
I will try and incorporate some or all of the above into the review patch
and get it back to you tomorrow (possibly optimistic). Going via the
generic algebra turns out to be a minor rearrangement of code (I think),
so it shoudn't be too hard - I hope!
--
Ticket URL: <http://trac.sagemath.org/ticket/14261#comment:36>
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