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 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 (but technically incorrect).
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.
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.
So, if you think of rep as a module, this could be:
>
> rep.action(a, r)
>
> Then we can optionally setup the following as syntactic sugar:
>
> a*r or r*a
>
>
I'll implement both. I have the second alternative working already,
although some necessary sanity checks relating to the questions above are
missing.
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/
Andrew
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