#14261: Iwahori-Hecke algebra with several bases
-------------------------------------+-------------------------------------
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 |
-------------------------------------+-------------------------------------
Comment (by tscrim):
Thank you for reviewing this ticket Andrew.
Replying to [comment:34 andrew.mathas]:
> I'm part way through writing a review patch. Mostly I have just been
expanding on the documentation (and hopefully improving it). In
particular, I have added Iwahori-Hecke algebras to the algebra chapter of
the reference manual, as it didn't seem to exist anywhere else.
Yes, this should be done.
> There is one significant issue that needs to be addressed: currently the
code accepts two parameters `q1` and `q2` and defines the algebra with
generators `{T_s|s\in S}` and relations `(T_s-q1)(T_s-q2)=0` together with
the braid relations. I am unsure how to define the bar involution for such
parameters.
> ...
> The underlying problem is that the bar involution really only makes
sense in the generic case and currently there is no checking for this. On
the other hand, the Kazhdan-Lusztig bases do make sense whenever the squre
root of the prameters lives in the base ring as we can compute the KL
bases over `Z[v,v^-1]` and then specialise `v^2` to `q`. In particular,
the KL bases make sense for the integral groups rings `ZW`.
>
> I can think of a couple of solutions:
> * Leave everything as it is.
> * Only allow the KL bases to be implemenetd in the generic case.
> * Implement a generic Hecke algebra behind the scene and compute the KL
basis here, together with transition matrices, and then specialise these
results into the non-generic Hecke algebras whenever the square roots of
the parameters are (testably) well-defined.
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?
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.
> Here are a few other less important issues that I have come across:
> * I am confused by the `one_basis` method of all of the bases: why does
this return the identity element of the corresponding Coxeter group?
Initially I thought that this would return the identity element of the
Hecke algebra with respet to the current basis (which is implemented
as`T.one()`). If we really need a shorthand for the identity element of
the group shouldn't the method be called something like group_identity?
This is correct; 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).
> * Similarly, the method `is_field` strikes me as being strange: the
Hecke algebra is a field if only if the Coxeter group has rank zero and
the base ring is a field. Rather than being a method of the Hecke algebra
this should be (and is) a method of the base ring of the Hecke algebra.
I'm not really opposed to removing it, but I'm thinking it should be in
there for consistency (with perhaps modified behavior).
> * I think that the names for the bases are all too long to be useful and
that something shorter like "X-basis of Iwahori-Hecke algebra" or
"X-basis of Iwahori-Hecke algebra of type Y" feasible? Compare these with:
> {{{
> Iwahori-Hecke algebra of type A3 in v^2,-1 over Univariate Laurent
Polynomial Ring in v over Rational Field in the C basis
> Iwahori-Hecke algebra of type A3 in v^2,-1 over Univariate Laurent
Polynomial Ring in v over Rational Field in the Kazhdan-Lusztig basis
> Iwahori-Hecke algebra of type A3 in v^2,-1 over Univariate Laurent
Polynomial Ring in v over Rational Field in the standard basis
> }}}
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.
--
Ticket URL: <http://trac.sagemath.org/ticket/14261#comment:35>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/groups/opt_out.
