Highest weight crystals

class sage.combinat.crystals.highest_weight_crystals.FiniteDimensionalHighestWeightCrystal_TypeE(dominant_weight)

Bases: sage.combinat.crystals.tensor_product.TensorProductOfCrystals

Commonalities for all finite dimensional type E highest weight crystals

Subclasses should setup an attribute column_crystal in their __init__ method before calling the __init__ method of this class.

Element

alias of TensorProductOfRegularCrystalsElement

module_generator()

This yields the module generator (or highest weight element) of the classical crystal of given dominant weight in self.

EXAMPLES:

sage: C=CartanType(['E',6])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[2])
sage: T.module_generator()
[[(2, -1), (1,)]]
sage: T = HighestWeightCrystal(0*La[2])
sage: T.module_generator()
[]

sage: C=CartanType(['E',7])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: T.module_generator()
[[(-7, 1), (7,)]]
class sage.combinat.crystals.highest_weight_crystals.FiniteDimensionalHighestWeightCrystal_TypeE6(dominant_weight)

Bases: sage.combinat.crystals.highest_weight_crystals.FiniteDimensionalHighestWeightCrystal_TypeE

Class of finite dimensional highest weight crystals of type \(E_6\).

EXAMPLES:

sage: C=CartanType(['E',6])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[2]); T
Finite dimensional highest weight crystal of type ['E', 6] and highest weight Lambda[2]
sage: B1 = T.column_crystal[1]; B1
The crystal of letters for type ['E', 6]
sage: B6 = T.column_crystal[6]; B6
The crystal of letters for type ['E', 6] (dual)
sage: t = T(B6([-1]),B1([-1,3])); t
[(-1,), (-1, 3)]
sage: [t.epsilon(i) for i in T.index_set()]
[2, 0, 0, 0, 0, 0]
sage: [t.phi(i) for i in T.index_set()]
[0, 0, 1, 0, 0, 0]
sage: TestSuite(t).run()
class sage.combinat.crystals.highest_weight_crystals.FiniteDimensionalHighestWeightCrystal_TypeE7(dominant_weight)

Bases: sage.combinat.crystals.highest_weight_crystals.FiniteDimensionalHighestWeightCrystal_TypeE

Class of finite dimensional highest weight crystals of type \(E_7\).

EXAMPLES:

sage: C=CartanType(['E',7])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: T.cardinality()
133
sage: B7 = T.column_crystal[7]; B7
The crystal of letters for type ['E', 7]
sage: t = T(B7([-5, 6]), B7([-2, 3])); t
[(-5, 6), (-2, 3)]
sage: [t.epsilon(i) for i in T.index_set()]
[0, 1, 0, 0, 1, 0, 0]
sage: [t.phi(i) for i in T.index_set()]
[0, 0, 1, 0, 0, 1, 0]
sage: TestSuite(t).run()
sage.combinat.crystals.highest_weight_crystals.HighestWeightCrystal(dominant_weight)

Returns an implementation of the highest weight crystal of highest weight \(dominant_weight\).

This is currently only implemented for crystals of type \(E_6\) and \(E_7\).

TODO: implement highest weight crystals for classical types \(A_n\), \(B_n\), \(C_n\), \(D_n\) using tableaux.

EXAMPLES:

sage: C=CartanType(['E',6])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: T.cardinality()
27
sage: T = HighestWeightCrystal(La[6])
sage: T.cardinality()
27
sage: T = HighestWeightCrystal(La[2])
sage: T.cardinality()
78
sage: T = HighestWeightCrystal(La[4])
sage: T.cardinality()
2925
sage: T = HighestWeightCrystal(La[3])
sage: T.cardinality()
351
sage: T = HighestWeightCrystal(La[5])
sage: T.cardinality()
351

sage: C=CartanType(['E',7])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: T.cardinality()
133
sage: T = HighestWeightCrystal(La[2])
sage: T.cardinality()
912
sage: T = HighestWeightCrystal(La[3])
sage: T.cardinality()
8645
sage: T = HighestWeightCrystal(La[4])
sage: T.cardinality()
365750
sage: T = HighestWeightCrystal(La[5])
sage: T.cardinality()
27664
sage: T = HighestWeightCrystal(La[6])
sage: T.cardinality()
1539
sage: T = HighestWeightCrystal(La[7])
sage: T.cardinality()
56

sage: C = CartanType(['C',2,1])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = HighestWeightCrystal(La[1])
sage: [p for p in T.subcrystal(max_depth=3)]
[(Lambda[1],), (Lambda[0] - Lambda[1] + Lambda[2],), (-Lambda[0] + Lambda[1] + Lambda[2] - delta,),
(Lambda[0] + Lambda[1] - Lambda[2],), (-Lambda[0] + 3*Lambda[1] - Lambda[2] - delta,), (2*Lambda[0] - Lambda[1],),
(-Lambda[1] + 2*Lambda[2] - delta,)]

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