#15508: Implement Fock space
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.0
Component: algebra | Resolution:
Keywords: Fock space | Merged in:
quantum group representations | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 4548a5b5788962a11afdb40c2786d816cda07183
public/modules/fock_space | Stopgaps:
Dependencies: #15289 |
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Comment (by andrew.mathas):
Hi Travis,
Thanks for both working on this and for letting me know about it. I just
had a quick glance through it and it looks very nice, but there is a lot
of code to digest! The fact that the new improved git-world makes it
difficult to separate the current ticket from its dependencies makes this
harder of course. I will try and look through this properly in due course
as it is one of the things that I care about - and implementing this was
on my to-do list.
A few comments:
* I noticed that you have implemented Fayer's LLT algorithm for computing
the canonical bases of the combinatorial Fock spaces. I haven't looked at
all at the details of your code but in his paper Fayers' has a conjecture
for the degrees of the polynomials that came up (they should always be
less than the defect/weight), and he mentioned that if this conjecture
were true then there are some shortcuts which speed up his algorithm.
Fayers' conjecture is now known to be true, so I was wondering whether you
have used this in your code.
* Am I right in thinking that you have only implemented the Fock spaces
`F(\Lambda)` such that `F(\Lambda)\cong
F(\Lambda_{a_1})\otimes\dots\otimes F(\Lambda_{a_k})`, for dominant
weights `\Lambda=\Lambda_{a_1}+\dots+\Lambda_{a_k}`?
In addition to these "standard" Fock spaces there are some more general
Fock spaces for `U_q(\widehat{sl}_e)` defined by Uglov. I am asking this
more for information/clarification as Uglov's Fock spaces don't come up in
my work - they arise in the cateogorification of rational Cherednik
algebras.
* In terms of displaying the elements of the Fock space I think that
{{{
|[3, 1]> + q*|[2, 2]> + q^ 2*|[2, 1, 1]>
}}}
is too cumbersome. I would prefer something like
{{{
|3, 1> + q*|2, 2> + q^ 2*|2, 1, 1>
}}}
and something similar for higher levels. Other alternatives would be
{{{
F(3, 1) + q*F(2, 2) + q^ 2*F(2, 1, 1)
}}}
or
{{{
F([3, 1]) + q*F([2, 2]) + q^ 2*F([2, 1, 1])
}}}
which both have the distinct advantage that you can cut and paste the
output back into `sage` (for a suitably defined shortcut `F`).
Cheers, Andrew
--
Ticket URL: <http://trac.sagemath.org/ticket/15508#comment:2>
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