> On Friday, 7 November 2014 03:56:16 UTC+11, Nicolas M. Thiery wrote:
>
>>
>> This might be the occasion to add a ``cartan type'' for G(r,1,n).
>>
>
> Implementing these algebras properly would be painful...however,
> implementing their action on a representation in terms of the natural
> generators is not, which I guess is what you are suggesting.
>
I've come across a similar situation with quantum group representations
(where implementing the actual algebra is difficult compared to
implementing the representations and the action of natural generators).
What I ended up decided was best was have methods corresponding to each
type of generator, and so if/when the algebra is implemented, the action
would call the corresponding method. I'm very open to a discussion about
this.
>
> I have realised that attaching the seminormal representations to the Hecke
> algebras creates a few additional headaches that are associated with the
> choice of base ring and the choice of parameters of the Iwahori-Hecke
> algebra. For example, the seminormal representations are naturally defined
> over Q(q), but the Hecke algebra defined over Z[q,q^-1] also acts on them
> since this algebra embeds in the algebra over Q(q). Technically, however,
> the seminormal representation is not a representation for the Hecke algebra
> defined over Z[q,q^-1].
>
> The question here is whether we should follow the strict mathematical
> definitions and only allow the Hecke algebras with exactly the same base
> ring to act or should we be more flexible? The former is easier in terms
> of coding whereas the latter is more useful.
>
I think it depends on how you implement the action; where you could
perhaps just do the action of a term generator by generator and then
convert the coefficient into Q(q) and scale the result. Although (I
believe) the coercion framework would naturally do a pushout construction
of the Hecke algera over Q(q) and then do the action. I guess what I'm
saying if you try to make this very mathematically strict, you will likely
have a fight with the coercion system and it will be different than many
other parts of Sage.
>
> Then there are other annoyances in that the seminormal representations for
> the algebras with different quadratic relations, for example
> $(T_r-q)(T_r+1)=0$, $(T_r-q)(T_r+q^{-1})=0$ or $(T_r-q)(T_r+v)=0$, are all
> different and, of course, there are several different flavours of
> seminormal representations. Any suggestions on what we should try and
> support here would be appreciated.
>
The biggest thing (to me) is any convention choices that are made are
well-documented. What about the relations used for the
IwahoriHeckeAlgebra_nonstandard or some other "idiosyncratic (but
convenient)" relations?
>
> Another thought that occurs to me, which I don't promise to follow through
> on, is that I could use the seminormal representation to define Specht
> modules for these algebras over an arbitrary ring. It is possible to do
> this using some specialisation tricks. I am not sure how efficient this
> would be but I suspect that it would be at least as good as my
> implementation of the Murphy basis in chevie
> <http://webusers.imj-prg.fr/~jean.michel/chevie/index.html> that I wrote
> for the Hecke algebras of type A.
>
> In thinking about this it seems to me that currently there is no framework
> in sage for dealing with module that has more than one "natural" basis. Is
> this right? Of course, it is possible to define several different
> CombinatoriaFreeModules and coercions between them.
>
As Anne said, combinat/descent_algebra.py, or the Iwahori-Hecke algebras.
What you want to look (grep) for is "WithRealizations".
> A final question: where should this code go? Currently the Iwahori-Hecke
> algebras are defined in sage.algebras and the seminormal matrix
> representations in sage.combinat. I think the best place might be to put
> all of this code in a new directory sage.algebras.iwahoriheckeagebras/
>
> I agree with Anne that iwahori_hecke_algebras is a better name for the
subfolder as I can actually read that without getting cross-eyed. :P
However I would say it depends on how many new files (python modules) you
think you'll create. If it's only one or two, I think you can just put it
in the algebras folder.
Best,
Travis
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