#20469: Implement Ariki-Koike algebras
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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- | Reviewers:
koike | Work issues:
Authors: Travis Scrimshaw | Commit:
Report Upstream: N/A | 21a49dfec02d086a07e888ecef48529dffa302b6
Branch: | Stopgaps:
public/algebras/ariki_koike_algebras-20469|
Dependencies: |
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Comment (by tscrim):
Replying to [comment:3 andrew.mathas]:
> Thanks Travis. This looks interesting. I will have a play with it and
see if I can improve the multiplication with the tricks that I know about
these algebras -- I am not promising that this will be possible, only to
look!
Great, thanks. It is currently so recursive that it fails for `L2 * L1 *
L2` in `H(2,2)` (although `L1 * (L2)^2` works).
> - It would be better to use `L1`,`...`,`Ln` for the Jucys-Murphy
elements as this is what is commonly used in the literature and it's also
consistent with using `T1`,`T2`,`...` for the Hecke generators. Using
`l1`,`...`,`ln` looks strange to me.
Done.
> - I am not sure that it is worth the effort to define the Hu-Mathas as
the algebras that we defined are different to these when `q=1` and, more
importantly, the conversion between the two is easy in principle but messy
in practice.
This also leads to a question of what do we want to call the Hu-Mathas
variant? In the future, we can also add another basis to this for `q \neq
1`. We also should probably add the basis indexed by the colored
permutations (i.e., monomials in `T_0, ..., T_{n-1}`).
> - The default base ring should be the smallest ring that contains `q`,
`u0`,...,`ur`.
With `q` invertible, that is what it currently does (just in a slightly
funky way that groups the `u`'s together then the `q`'s).
> - At some stage I will polish the code that I have which implements the
Specht modules for these algebras. Given this it would probably be a good
idea to put this code in its own subdirectory so that the module code.
Similarly, my graded Specht module could go in here to.
I have moved this into a `hecke_algebras` folder to start. We can move the
rest of the Hecke algebras (e.g., Iwahori and Yokonuma) to this folder on
a followup ticket.
While adding the documentation, I also found some errors with the
multiplication that I have fixed.
--
Ticket URL: <http://trac.sagemath.org/ticket/20469#comment:5>
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