Bases: sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets
Direct sum of crystals.
Given a list of crystals of the same Cartan type,
one can form the direct sum
.
INPUT:
- crystals – a list of crystals of the same Cartan type
- keepkey – a boolean
The option keepkey is by default set to False, assuming
that the crystals are all distinct. In this case the elements of
the direct sum are just represented by the elements in the
crystals . If the crystals are not all distinct, one should
set the keepkey option to True. In this case, the
elements of the direct sum are represented as tuples
where
.
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: C1 = crystals.Tableaux(['A',2],shape=[1,1])
sage: B = crystals.DirectSum([C,C1])
sage: B.list()
[1, 2, 3, [[1], [2]], [[1], [3]], [[2], [3]]]
sage: [b.f(1) for b in B]
[2, None, None, None, [[2], [3]], None]
sage: B.module_generators
[1, [[1], [2]]]
sage: B = crystals.DirectSum([C,C], keepkey=True)
sage: B.list()
[(0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3)]
sage: B.module_generators
[(0, 1), (1, 1)]
sage: b = B( tuple([0,C(1)]) )
sage: b
(0, 1)
sage: b.weight()
(1, 0, 0)
The following is required, because DirectSumOfCrystals takes the same arguments as DisjointUnionEnumeratedSets (which see for details).
TESTS:
sage: C = crystals.Letters(['A',2])
sage: B = crystals.DirectSum([C,C], keepkey=True)
sage: B
Direct sum of the crystals Family (The crystal of letters for type ['A', 2], The crystal of letters for type ['A', 2])
sage: TestSuite(C).run()
Bases: sage.structure.element_wrapper.ElementWrapper
A class for elements of direct sums of crystals
Returns the action of on self.
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: B = crystals.DirectSum([C,C], keepkey=True)
sage: [[b, b.e(2)] for b in B]
[[(0, 1), None], [(0, 2), None], [(0, 3), (0, 2)], [(1, 1), None], [(1, 2), None], [(1, 3), (1, 2)]]
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: B = crystals.DirectSum([C,C], keepkey=True)
sage: b = B( tuple([0,C(2)]) )
sage: b.epsilon(2)
0
Returns the action of on self.
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: B = crystals.DirectSum([C,C], keepkey=True)
sage: [[b,b.f(1)] for b in B]
[[(0, 1), (0, 2)], [(0, 2), None], [(0, 3), None], [(1, 1), (1, 2)], [(1, 2), None], [(1, 3), None]]
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: B = crystals.DirectSum([C,C], keepkey=True)
sage: b = B( tuple([0,C(2)]) )
sage: b.phi(2)
1
Returns the weight of self.
EXAMPLES:
sage: C = crystals.Letters(['A',2])
sage: B = crystals.DirectSum([C,C], keepkey=True)
sage: b = B( tuple([0,C(2)]) )
sage: b
(0, 2)
sage: b.weight()
(0, 1, 0)