#11305: Bijection between Rigged Configurations and Crystal Paths
------------------------------------+---------------------------------------
Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.1
Component: combinatorics | Resolution:
Keywords: Crystals, days30 | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Travis Scrimshaw | Merged in:
Dependencies: | Stopgaps:
------------------------------------+---------------------------------------
Comment (by aschilling):
Hi Travis,
The patch looks close to ready. However, there are still a couple of
things that need to be fixed:
- I get doctest failures
{{{
sage -t "devel/sage-
combinat/sage/combinat/rigged_configurations/kleber_tree.py"
**********************************************************************
File "/Applications/sage-5.0/devel/sage-
combinat/sage/combinat/rigged_configurations/kleber_tree.py", line 41:
sage: for x in KT: x
Expected:
Kleber tree node with weight [2, 1, 2] and upwards edge root [0, 0, 0]
Kleber tree node with weight [3, 0, 1] and upwards edge root [0, 1, 1]
Kleber tree node with weight [1, 1, 1] and upwards edge root [1, 1, 1]
Kleber tree node with weight [0, 0, 2] and upwards edge root [2, 2, 1]
Kleber tree node with weight [0, 2, 2] and upwards edge root [1, 0, 0]
Kleber tree node with weight [1, 0, 3] and upwards edge root [1, 1, 0]
Kleber tree node with weight [2, 0, 0] and upwards edge root [0, 1, 1]
Kleber tree node with weight [0, 0, 2] and upwards edge root [1, 1, 0]
Kleber tree node with weight [0, 1, 0] and upwards edge root [1, 1, 1]
Kleber tree node with weight [0, 1, 0] and upwards edge root [0, 0, 1]
Got:
Kleber tree node with weight [2, 1, 2] and upwards edge root [0, 0, 0]
Kleber tree node with weight [0, 2, 2] and upwards edge root [1, 0, 0]
Kleber tree node with weight [1, 0, 3] and upwards edge root [1, 1, 0]
Kleber tree node with weight [1, 1, 1] and upwards edge root [1, 1, 1]
Kleber tree node with weight [0, 0, 2] and upwards edge root [2, 2, 1]
Kleber tree node with weight [3, 0, 1] and upwards edge root [0, 1, 1]
Kleber tree node with weight [2, 0, 0] and upwards edge root [0, 1, 1]
Kleber tree node with weight [0, 0, 2] and upwards edge root [1, 1, 0]
Kleber tree node with weight [0, 1, 0] and upwards edge root [1, 1, 1]
Kleber tree node with weight [0, 1, 0] and upwards edge root [0, 0, 1]
**********************************************************************
File "/Applications/sage-5.0/devel/sage-
combinat/sage/combinat/rigged_configurations/kleber_tree.py", line 57:
sage: for x in KT: x
Expected:
Kleber tree node with weight [1, 1, 2, 0, 0, 0, 0] and upwards edge
root [0, 0, 0, 0, 0, 0, 0]
Kleber tree node with weight [0, 0, 3, 0, 0, 0, 0] and upwards edge
root [1, 1, 0, 0, 0, 0, 0]
Kleber tree node with weight [0, 1, 1, 1, 0, 0, 0] and upwards edge
root [1, 1, 1, 0, 0, 0, 0]
Kleber tree node with weight [1, 0, 1, 0, 1, 0, 0] and upwards edge
root [1, 2, 2, 1, 0, 0, 0]
Kleber tree node with weight [0, 0, 1, 0, 0, 1, 0] and upwards edge
root [2, 3, 3, 2, 1, 0, 0]
Kleber tree node with weight [2, 0, 1, 1, 0, 0, 0] and upwards edge
root [0, 1, 1, 0, 0, 0, 0]
Kleber tree node with weight [1, 0, 0, 2, 0, 0, 0] and upwards edge
root [0, 1, 1, 0, 0, 0, 0]
Kleber tree node with weight [0, 0, 0, 1, 1, 0, 0] and upwards edge
root [1, 1, 1, 0, 0, 0, 0]
Got:
Kleber tree node with weight [1, 1, 2, 0, 0, 0, 0] and upwards edge
root [0, 0, 0, 0, 0, 0, 0]
Kleber tree node with weight [2, 0, 1, 1, 0, 0, 0] and upwards edge
root [0, 1, 1, 0, 0, 0, 0]
Kleber tree node with weight [0, 1, 1, 1, 0, 0, 0] and upwards edge
root [1, 1, 1, 0, 0, 0, 0]
Kleber tree node with weight [1, 0, 1, 0, 1, 0, 0] and upwards edge
root [1, 2, 2, 1, 0, 0, 0]
Kleber tree node with weight [0, 0, 1, 0, 0, 1, 0] and upwards edge
root [2, 3, 3, 2, 1, 0, 0]
Kleber tree node with weight [0, 0, 3, 0, 0, 0, 0] and upwards edge
root [1, 1, 0, 0, 0, 0, 0]
Kleber tree node with weight [1, 0, 0, 2, 0, 0, 0] and upwards edge
root [0, 1, 1, 0, 0, 0, 0]
Kleber tree node with weight [0, 0, 0, 1, 1, 0, 0] and upwards edge
root [1, 1, 1, 0, 0, 0, 0]
**********************************************************************
1 items had failures:
2 of 10 in __main__.example_0
***Test Failed*** 2 failures.
For whitespace errors, see the file
/Users/anne/.sage//tmp/kleber_tree_51023.py
[9.1 s]
}}}
- If you type sage -coverage filename for all the files you added/changed,
it says that there are still quite a few methods without doctests. Please
bring this up to 100% coverage!
- The file sage/combinat/rigged_configurations/crystal_path_element.py has
a comment
{{{
TODO:: Make this equal the number of rigged configurations via
iteration.
A proper is_highest_weight() function without needing to change
the index set
}}}
Would this be easy to fix? If so, please include it in the patch.
- Why do you return "None" in
sage/combinat/rigged_configurations/crystal_path_element.py for the action
of e and f if i=0? Perhaps you should have an assertion and return an
error "Only classical crystal operators are implemented" or something like
this.
- The description at the start of CrystalPaths needs to be changed
{{{
class CrystalPaths(AbstractCrystalPaths):
r"""
A crystal path is a tensor product of
:class:`CrystalPathFactors<certain crystals>` which are a
generalization
of Kirillov Reshetikhin (KR) crystals of some affine Kac-Moody
algebra.
The reason they are not KR crystals is because the bijection from
rigged
configurations (RC) to crystal paths does not always generate a valid
tableaux. In particular, for type `D_n^{(1)}` the highest weight RC's
will
correspond to Kashiwara-Nakashima crystals that are highest weight KR
crystal `B^{r,s}` with some vertical dominos removed from the
rectangle.
The bijection constructs the full rectangle, but in general it is not
semi-standard.
For more information on KR crystals, see
:mod:`sage.combinat.crystals.kirillov_reshetikin`.
}}}
What you call CrystalPaths is not a generalization of KR crystals, but
rather a *different model* or *representation* for KR crystals. This
representation at the end should be isomorphic to the model for KR
crystals in terms of Kashiwara-Nakashima tableaux. Also your link
`sage.combinat.crystals.kirillov_reshetikin' is broken since there is an h
missing. I suggest to change this description to
{{{
class CrystalPaths(AbstractCrystalPaths):
r"""
A crystal path is a tensor product of
:class:`CrystalPathFactors<certain crystals>`.
CrystalPaths provide a different model for tensor products of
Kirillov-Reshetikhin
crystals to the model in terms of Kashiwara-Nakashima tableaux.
Through the bijection with rigged configurations, the tableaux that
are produced
in the CrystalPaths model for type `D_n^{(1)}` are all of rectangular
shapes
and do not necessarily obey the usual strict increase in columns and
weak increase
in rows. The relation between the two tableaux models is given by a
filling map.
For more information see [OSS2011]_ .
REFERENCES:
.. [OSS2011] Masato Okado, Reiho Sakamoto, Anne Schilling
*Affine crystal structure on rigged configurations of type
`D_n^{(1)}`*
J. Algebraic Combinatorics, to appear,
doi:10.1007/s10801-012-0383-z (arXiv:1109.3523 [math.QA])
For more information on KR crystals, see
:mod:`sage.combinat.crystals.kirillov_reshetikhin`.
}}}
A similar change should be made to
{{{
class CrystalPathFactors(CrystalOfWords):
r"""
This is a generalized Kirillov Reshetikhin crystal `B^{r,s}` since it
is a
tableaux with (exactly) `r` rows and `s` columns, but does not need to
satisfy any row or column restrictions (such as being semi-standard).
For more information, see :class:`CrystalPaths`.
}}}
- I think the names "CrystalPaths" and "CrystalPathFactors" are not
optimal. Perhaps "KirillovReshetkhinTableaux" instead of
"CrystalPathFactors" as in http://arxiv.org/abs/1109.3523 and
"TensorProductsOfKirillovReshetkhinTableaux" or something like this.
- Something is not yet right with your code for CrystalPaths for type D.
The number of classically highest weight elements in KR tensor products
and CrystalPaths should be the same, but they are not. See also my comment
above that the two are just different representations of the same crystal.
{{{
sage: CP = CrystalPaths(['D', 4, 1], [[2,1],[2,1]]); CP
Crystal paths of type ['D', 4, 1] and tableau shape(s) [[1, 1], [1,
1]]
sage: len([b for b in CP if b.is_highest_weight()])
7
sage: KR = KirillovReshetikhinCrystal(['D',4,1],2,1)
sage: T = TensorProductOfCrystals(KR,KR)
sage: len([b for b in T if b.is_highest_weight(index_set=[1,2,3,4])])
10
}}}
- Your description of Kleber trees
{{{
r"""
Kleber trees
This is a standard tree data structure implementation
where the nodes
store a weight and are connected to their children and
their parent.
However this is also an edge labeled tree and we store the
end labels in
the child nodes since this makes the manipulations of this
Kleber's tree
easier to implement.
Note that iteration is done by breath-first since this is
how the list is
created. To do depth-first, call depth_first_iter().
}}}
is very implementation oriented. I think a brief mathematical summary
would be appropriate. Say what Kleber trees are used for, how the nodes
are labelled etc..
- Can you fix the method _latex_ in KleberTree line 343 and beyond?
- At the beginning of
sage/combinat/rigged_configurations/rigged_configuration_element.py please
give a brief description of what a rigged configuration element is (like a
sequence of partitions together with labelings of the parts).
- Could you please comment on your changes to root_system/cartan_type.py
and dynkin_diagram.py. Why are you making these changes?
- Last, but not least: you need to export your file so that it has the
right header. The patch should also contain a summary of the changes at
the top (before the code). You can achieve this by hg qrefresh -e and then
writing the summary.
So much for now!
Anne
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11305#comment:3>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/sage-trac?hl=en.
