Highest Weight Crystals¶
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class
sage.categories.highest_weight_crystals.
HighestWeightCrystalHomset
(X, Y, category=None)¶ Bases:
sage.categories.crystals.CrystalHomset
The set of crystal morphisms from a highest weight crystal to another crystal.
See also
See
sage.categories.crystals.CrystalHomset
for more information.-
Element
¶ alias of
HighestWeightCrystalMorphism
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-
class
sage.categories.highest_weight_crystals.
HighestWeightCrystalMorphism
(parent, on_gens, cartan_type=None, virtualization=None, scaling_factors=None, gens=None, check=True)¶ Bases:
sage.categories.crystals.CrystalMorphismByGenerators
A virtual crystal morphism whose domain is a highest weight crystal.
INPUT:
parent
– a homseton_gens
– a function or list that determines the image of the generators (if given a list, then this uses the order of the generators of the domain) of the domain underself
cartan_type
– (optional) a Cartan type; the default is the Cartan type of the domainvirtualization
– (optional) a dictionary whose keys are in the index set of the domain and whose values are lists of entries in the index set of the codomainscaling_factors
– (optional) a dictionary whose keys are in the index set of the domain and whose values are scaling factors for the weight, \(\varepsilon\) and \(\varphi\)gens
– (optional) a list of generators to define the morphism; the default is to use the highest weight vectors of the crystalcheck
– (default:True
) check if the crystal morphism is valid
-
sage.categories.highest_weight_crystals.
HighestWeightCrystals
¶ The category of highest weight crystals.
A crystal is highest weight if it is acyclic; in particular, every connected component has a unique highest weight element, and that element generate the component.
EXAMPLES:
sage: C = HighestWeightCrystals() sage: C Category of highest weight crystals sage: C.super_categories() [Category of crystals] sage: C.example() Highest weight crystal of type A_3 of highest weight omega_1