#20564: KleshchevPartitions
-------------------------------------+-------------------------------------
Reporter: andrew.mathas | Owner: andrew.mathas
Type: enhancement | Status: new
Priority: major | Milestone: sage-7.2
Component: combinatorics | Resolution:
Keywords: Kleshchev | Merged in:
partition tuples | Reviewers:
Authors: Andrew Mathas | Work issues:
Report Upstream: N/A | Commit:
Branch: | ac574c7f3156effe84e8f9247aeac877e23d52b3
u/andrew.mathas/kleshchevpartitions| Stopgaps:
Dependencies: |
-------------------------------------+-------------------------------------
Old description:
> Implement subclasses of `PartitionTuples` for Kleshchev partition tuples.
> These partition tuples index the simple modules for the cyclotomic Hecke
> algebras of type A and the crystal basis of corresponding integrable
> highest weight modules for the affine special linear groups.
>
> {{{#!python
> sage: KleshchevPartitions(5,[3,2,1],size=3)[:]
> [([], [], [2, 1]),
> ([1], [], [1, 1]),
> ([], [], [1, 1, 1]),
> ([], [1], [2]),
> ([], [], [3]),
> ([1], [], [2]),
> ([2], [], [1]),
> ([], [1], [1, 1]),
> ([], [1, 1], [1]),
> ([], [2], [1]),
> ([1], [2], []),
> ([], [3], []),
> ([1], [1], [1]),
> ([1, 1], [1], []),
> ([2], [1], []),
> ([3], [], [])
> ]
> sage: KleshchevPartitions(5,[3,2,1],3,direction="down")[:]
> [([], [1], [1, 1]),
> ([1], [], [1, 1]),
> ([], [], [1, 1, 1]),
> ([], [1, 1], [1]),
> ([], [1], [2]),
> ([1], [1], [1]),
> ([1, 1], [], [1]),
> ([2], [], [1]),
> ([], [1, 1, 1], []),
> ([], [1, 1], [1]),
> ([1], [1, 1], []),
> ([1, 1], [1], []),
> ([1], [2], []),
> ([2], [1], []),
> ([1, 1, 1], [], []),
> ([1, 1], [1], []),
> ([2, 1], [], []),
> ([3], [], [])
> ]
> }}}
>
> There is significant overlap between this patch and
> http://trac.sagemath.org/ticket/20469, which implements the corresponding
> crystals so these patches should be merged. Unfortunately, the labellings
> used for these sets of multipartitions in the two patches are different:
> {{{#!python
> sage: C = crystals.KleshchevPartitions(4, [3,2,1])
> sage: [p for p in C.subcrystal(max_depth=3) if sum(sum(l) for l in
> p.value.value) == 3]
> [([], [], [1, 1, 1]),
> ([1], [1, 1], []),
> ([1], [2], []),
> ([2, 1], [], []),
> ([], [1, 1], [1]),
> ([1, 1], [1], []),
> ([1, 1], [], [1]),
> ([], [1], [2]),
> ([1, 1, 1], [], []),
> ([2], [1], []),
> ([], [1], [1, 1]),
> ([1], [1], [1]),
> ([2], [], [1]),
> ([3], [], []),
> ([], [1, 1, 1], []),
> ([1], [], [1, 1])
> ]
> }}}
New description:
Implement subclasses of `PartitionTuples` for Kleshchev partition tuples.
These partition tuples index the simple modules for the cyclotomic Hecke
algebras of type A and the crystal basis of corresponding integrable
highest weight modules for the affine special linear groups.
{{{#!python
sage: KleshchevPartitions(5,[3,2,1],size=3)[:]
sage: sorted(KleshchevPartitions(5,[3,2,1],size=3)[:])
[([], [], [1, 1, 1]),
([], [], [2, 1]),
([], [], [3]),
([], [1], [1, 1]),
([], [1], [2]),
([], [1, 1], [1]),
([], [2], [1]),
([], [3], []),
([1], [], [1, 1]),
([1], [], [2]),
([1], [1], [1]),
([1], [2], []),
([1, 1], [1], []),
([2], [], [1]),
([2], [1], []),
([3], [], [])
]
sage: sorted(KleshchevPartitions(5,[3,2,1],3,direction="down")[:])
[([], [], [1, 1, 1]),
([], [1], [1, 1]),
([], [1], [2]),
([], [1, 1], [1]),
([], [1, 1], [1]),
([], [1, 1, 1], []),
([1], [], [1, 1]),
([1], [1], [1]),
([1], [1, 1], []),
([1], [2], []),
([1, 1], [], [1]),
([1, 1], [1], []),
([1, 1], [1], []),
([1, 1, 1], [], []),
([2], [], [1]),
([2], [1], []),
([2, 1], [], []),
([3], [], [])
]
}}}
There is significant overlap between this patch and part of #15584, which
implements the corresponding crystals so these patches should be merged.
The labellings used for these sets of multipartitions in #15584 matches
the `"down"` direction:
{{{#!python
sage: C = crystals.KleshchevPartitions(4, [3,2,1])
sage: sorted([p for p in C.subcrystal(max_depth=3) if sum(sum(l) for l in
p.value.value) == 3],
....: key=lambda x: [list(part) for part in x.value.value])
[([], [], [1, 1, 1]),
([], [1], [1, 1]),
([], [1], [2]),
([], [1, 1], [1]),
([], [1, 1, 1], []),
([1], [], [1, 1]),
([1], [1], [1]),
([1], [1, 1], []),
([1], [2], []),
([1, 1], [], [1]),
([1, 1], [1], []),
([1, 1, 1], [], []),
([2], [], [1]),
([2], [1], []),
([2, 1], [], []),
([3], [], [])]
}}}
--
Comment (by tscrim):
There are also duplicates in the `"down"` iteration. For this, I think we
should just use the crystal iteration from #15584, which seems to match.
Do you agree?
--
Ticket URL: <http://trac.sagemath.org/ticket/20564#comment:9>
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