#8649: to_tableau method broken for crystals of type B
------------------------------+---------------------------------------------
Reporter: aschilling | Owner: sage-combinat
Type: defect | Status: positive_review
Priority: major | Milestone: sage-4.4
Component: combinatorics | Keywords: crystals
Author: Anne Schilling | Upstream: N/A
Reviewer: Dan Bump | Merged:
Work_issues: |
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Comment(by aschilling):
Replying to [comment:6 nthiery]:
> Replying to [comment:5 aschilling]:
> > Thank you for the review. The mathematical reason why in type B the 0
is special,
> > is that it sits in a string of length 2:
> >
> > 1 -> 2 -> 0 -> -2 -> -1
> >
> > Hence a column of height 2 for example can become
> >
> > 2 0 0 0
> > 1 -2-> 1 -1-> 2 -2-> 0
> >
> > since in the tensor product always the rightmost unbracketed element
is changed.
> > For similar reasons, in the row case there cannot be two 0s.
> >
> > So this is really due to the string of length 2 which the other
crystal of letters
> > do not have. Nicolas, how else should the code be written to detect
this?
> > The paper by Kashiwara-Nakashima on tableaux for other types is also
pretty
> > case specific.
>
> In a perfect world, a crystal of letter would have a method which
> would take two letters a and b, and say whether b can be above a in a
> column. There would be a default implementation (returning a<b in the
> crystal of letters), and the type B crystal of letter would override
> this default implementation appropriately.
>
> If we are sure that no crystal of letter (or more generally crystal
> that we want to use as letter in a tableau) will ever have a letter
> called `0`, except for type `B`, or if we are willing to take the
> risk, then I guess we can be lazy, and just leave things as they are.
For the classical types ABCD, only type B has a 0. My plan was to
implement the exceptional types in a generic way at some point using
Littelmann paths or the alcove model and then possibly the
HighestWeightCrystal environment. The tableaux method is rather type
specific.
For type D, one can have columns of the form n-bar n n-bar n .... since
they are not comparable, so I think a<b for columns might not be the right
point of view. Rather not(a>=b) with the exception of a=b=0. But in any
case, one has to distinguish cases. Right now the tableaux method for
crystals is only supposed to work for type ABCD.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8649#comment:7>
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