#14234: Restructuring Diagram/Partition Algebras to match category structure
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Reporter: ghseeli | Owner:
Type: enhancement | AlexGhitza
Priority: minor | Status:
Component: algebra | positive_review
Keywords: Partition algebra, diagram | Milestone:
algebra, days38, days40 | sage-5.12
Authors: Stephen Doty, Aaron Lauve, | Resolution:
George H. Seelinger | Merged in:
Report Upstream: N/A | Reviewers: Travis
Branch: | Scrimshaw
Stopgaps: | Work issues:
| Dependencies: #10630
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Comment (by darij):
This is a very promising patch! Some errors I found:
{{{
An propogating ideal.
A propogating ideal of rank `k` is a non-unital algebra with basis
indexed
by the collection of ideal set partitions of `\{1, \ldots, k, -1,
\ldots,
-k\}`. We say a set partition is ideal if its has propogating number
is
less than `k`.
This algebra is thus a subalgebra of the partition algebra.
For more information, see :class:`PartitionAlgebra`
}}}
I don't like this. First of all, it should be "propagating", not
"propogating" (fortunately the method name is correct). Second, this
should be "The", not "A", propagating ideal. Otherwise it sounds like
**some** ideal generated by ideal partitions -- and there are many of
those.
Actually I see the "propogating" typo elsewhere too. Also, a "with with"
in the definition of {{{ideal_diagrams}}}.
Also, typo: "ommitted".
I'm not very happy with the use of floats (as in "2.5") for the
"intermediate" partition algebras. Is it safe to assume that, say,
{{{range(1,int(k+0.5))}}} always ends at k-0.5, or can it happen that
k-0.5 is a tad smaller than an integer due to a rounding error and k-0.5
no longer falls in the interval?
There is a more serious issue with the intermediate algebras, and it's
this (from your doctest):
{{{
sage: da.partition_diagrams(1.5)
[{{2, -2}, {1, -1}}, {{-1}, {2, -2}, {1}}]
}}}
If I am to follow the Halverson-Ram conventions, this should contain three
more partition diagrams, e. g., {{1, -1, 2, -2}}. Generally, they define a
(k+1/2)-partition diagram, for k being an integer, to be any
(k+1)-partition diagram in which k+1 and -k-1 lie in the same block (but
don't need to be alone there). These are in bijections with set partitions
of a fixed (2k+1)-element set. What you define, instead, bijects with set
partitions of a 2k-element set, which is boring since these are already
the k-partition diagrams.
Moreover, {{{da.partition_diagrams(2)}}} returns a combinatorial class
whereas {{{da.partition_diagrams(1.5)}}} returns a list. I am a bit
puzzled because this hardly intended behaviour is doctested for...
--
Ticket URL: <http://trac.sagemath.org/ticket/14234#comment:18>
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