Highest Weight Crystals

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

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 homset
  • on_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 under self
  • cartan_type – (optional) a Cartan type; the default is the Cartan type of the domain
  • virtualization – (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 codomain
  • scaling_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 crystal
  • check – (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