Hi Nicolas,
Yes, I agree, using coxeter3 will almost certainly be much more efficient.
I had a brief look at this and thought it looked two hard since I have only
a fleeting interest in this:)
In terms of syntax, chevie has a *LeftCells* function that returns all of
the cells of the Coxeter group, together with the corresponding
mu-coefficients -- this, I think, is equivalent to the information that
coxeter3 returns as it sounds like coxeter3 returns the W-graph for the
cell representation.
In terms of the suggested syntax, I think that this is partly driven by
what coxeter3 actually does. I was thinking of something like:
sage: W=CoxeterGroup("A3"); w=W.an_element()
sage: W.left_cell( w ) # left cell containing w
sage: W.left_cells() # all left cells -- as chevie's LeftCells(W) does now
In particular, rather the computing the complete decomposition of the
Coxeter group into a disjoint union of cells I think that it would be
useful to be able to compute only the cell containing a particular element.
With the implementation in chevie this isn't really an option, but it might
be with coxeter3. The right cells can be obtained from the left cells just
by taking inverses. It is not clear to me if we need both.
As coxeter 3 and chevie both seem to return the mu-coefficients as well
perhaps this information should also be returned -- it seems silly to
compute it and then discard it. In fact, I was thinking about the KL cells
only because I thought that I would implement the cell representations of
the Hecke algebras as anther example of Iwahori-Hecke algebta
representations. I was planning to do this using the implementation of the
KL-bases inside sage but if the full W-Graph data was available then this
would be much more efficient.
Andrew
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