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 = crystals.HighestWeight(La[2])
sage: T.module_generator()
[[(2, -1), (1,)]]
sage: T = crystals.HighestWeight(0*La[2])
sage: T.module_generator()
[]

sage: C=CartanType(['E',7])
sage: La=C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(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 = crystals.HighestWeight(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 = crystals.HighestWeight(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, model=None)

Return the highest weight crystal of highest weight dominant_weight of the given model.

INPUT:

EXAMPLES:

sage: La = RootSystem(['A',2]).weight_lattice().fundamental_weights()
sage: wt = La[1] + La[2]
sage: crystals.HighestWeight(wt)
The crystal of tableaux of type ['A', 2] and shape(s) [[2, 1]]

sage: La = RootSystem(['C',2]).weight_lattice().fundamental_weights()
sage: wt = 5*La[1] + La[2]
sage: crystals.HighestWeight(wt)
The crystal of tableaux of type ['C', 2] and shape(s) [[6, 1]]

Some type \(E\) examples:

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

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

An example with an affine type:

sage: C = CartanType(['C',2,1])
sage: La = C.root_system().weight_lattice().fundamental_weights()
sage: T = crystals.HighestWeight(La[1])
sage: sorted(T.subcrystal(max_depth=3), key=str)
[(-Lambda[0] + 3*Lambda[1] - Lambda[2] - delta,),
 (-Lambda[0] + Lambda[1] + Lambda[2] - delta,),
 (-Lambda[1] + 2*Lambda[2] - delta,),
 (2*Lambda[0] - Lambda[1],),
 (Lambda[0] + Lambda[1] - Lambda[2],),
 (Lambda[0] - Lambda[1] + Lambda[2],),
 (Lambda[1],)]

Using the various models:

sage: La = RootSystem(['F',4]).weight_lattice().fundamental_weights()
sage: wt = La[1] + La[4]
sage: crystals.HighestWeight(wt)
The crystal of LS paths of type ['F', 4] and weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='NakajimaMonomials')
Highest weight crystal of modified Nakajima monomials of
 Cartan type ['F', 4] and highest weight Lambda[1] + Lambda[4]
sage: crystals.HighestWeight(wt, model='AlcovePaths')
Highest weight crystal of alcove paths of type ['F', 4] and weight Lambda[1] + Lambda[4]

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