#20469: Implement Ariki-Koike algebras
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: new
Priority: major | Milestone: sage-7.2
Component: algebra | Resolution:
Keywords: hecke algebra, | Merged in:
complex reflection group, ariki- |
koike |
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/algebras/ariki_koike_algebras-20469|
e9b6a594ae1e089b51b37081f35743402447a960
Dependencies: | Stopgaps:
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Comment (by andrew.mathas):
Hi Travis,
I have been through your code and fixed the recursion issue. It was
actually an infinite recursion loop. The problem was that as the terms in
the products `....T_i L_k...` were being put into standard form "letter-
by-letter" (so changing the previous expression to `\sum ... L_m T_j...`)
you sometimes ended up going around in circles by pushing a `T_i` past an
`L_k` that then created a large power of some `L_m` that, when reduced,
got you back to the previous situation. I have rewritten the
`product_on_basis` code to avoid this, so I'm afraid that I've replaced
this section of your code. he product code is now less recursive with
terms largely being rewritten into standard form "in place".
On top of this I proved a formula for the expansion of `L_k^m` tat,
embarrassingly, I later found in one of my papers. This was my first guess
for improving the multiplication issues but once I'd made this change I
discovered the recursion loop. The other main change is that I changed
`L_k` to `q**{1-k}*L_k` because this renormalisation is what is normally
used in the literature as it works better with the combinatorics.
I have made a start at adding the documentation. As a result of e
rescaling of the `L_k`'s quite a lot of doc-tests are currently failing,
being off by the corresponding powers of `q` - I am happy to fix these but
I left them in at this stage so that you can compare if you wish. I am
also happy to fill out the documentation as I know this area quite well.
Other issues that we could think about are:
- whether to allow the two parameter version: `(T_i-q1)(T_i+q2)=0`.
robably quite painful as all of the product formulas will change
- whether to implement the degenerate algebras. This might be easy as it
could be done as a derived class with slightly different `_product_LTwTv`,
`_product_Tw_L`, and `_Li_power` methods)
- I think having a slightly shorter `_repr_` string would be a good idea:
`Ariki-Koike algebra of rank 5 and parameters q,u0,u2,u3` is enough I
think
- implementing other bases? This is probably best left for another
ticket...especially as I think that the realisations code requires that
the same indexing sets be used
Any way, I think that the code now works. Please have a look and let me
know what you think. Happy to be a reviewer or coauthor on the ticket as
you think best.
Andrew
--
Ticket URL: <https://trac.sagemath.org/ticket/20469#comment:8>
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