Let be a CartanType with index set , and be a realization of the type weight lattice.
A type crystal is a colored oriented graph equipped with a weight function from the nodes to some realization of the type weight lattice such that:
Each edge is colored with a label in .
For each , each node has:
Furthermore, when they exist,
This crystal actually models a representation of a Lie algebra if it satisfies some further local conditions due to Stembridge [St2003].
REFERENCES:
[St2003] J. Stembridge, A local characterization of simply-laced crystals, Trans. Amer. Math. Soc. 355 (2003), no. 12, 4807-4823.
EXAMPLES:
We construct the type crystal on letters (or in representation theoretic terms, the highest weight crystal of type corresponding to the highest weight ):
sage: C = CrystalOfLetters(['A',5]); C
The crystal of letters for type ['A', 5]
It has a single highest weight element:
sage: C.highest_weight_vectors()
[1]
A crystal is an enumerated set (see EnumeratedSets); and we can count and list its elements in the usual way:
sage: C.cardinality()
6
sage: C.list()
[1, 2, 3, 4, 5, 6]
as well as use it in for loops:
sage: [x for x in C]
[1, 2, 3, 4, 5, 6]
Here are some more elaborate crystals (see their respective documentations):
sage: Tens = TensorProductOfCrystals(C, C)
sage: Spin = CrystalOfSpins(['B', 3])
sage: Tab = CrystalOfTableaux(['A', 3], shape = [2,1,1])
sage: Fast = FastCrystal(['B', 2], shape = [3/2, 1/2])
sage: KR = KirillovReshetikhinCrystal(['A',2,1],1,1)
One can get (currently) crude plotting via:
sage: Tab.plot()
For rank two crystals, there is an alternative method of getting metapost pictures. For more information see C.metapost?
See also the categories Crystals, ClassicalCrystals, FiniteCrystals, HighestWeightCrystals.
Caveat: this crystal library, although relatively featureful for classical crystals, is still in an early development stage, and the syntax details may be subject to changes.
TODO:
Most of the above features (except Littelmann/alcove paths) are in MuPAD-Combinat (see lib/COMBINAT/crystals.mu), which could provide inspiration.