Kyoto Path Model for Affine Highest Weight Crystals¶
-
class
sage.combinat.crystals.kyoto_path_model.
KyotoPathModel
(crystals, weight, P)¶ Bases:
sage.combinat.crystals.tensor_product.TensorProductOfCrystals
The Kyoto path model for an affine highest weight crystal.
Note
Here we are using anti-Kashiwara notation and might differ from some of the literature.
Consider a Kac–Moody algebra
of affine Cartan type
, and we want to model the
-crystal
. First we consider the set of fundamental weights
of
and let
be the corresponding fundamental weights of the corresponding classical Lie algebra
. To model
, we start with a sequence of perfect
-crystals
of level
such that
where
is the canonical central element of
and
is the nonnegative weight lattice spanned by
.
Next we consider the crystal isomorphism
defined by
where
is the unique element in
such that
and
and
is the highest weight element in
. Iterating this, we obtain the following isomorphism:
We note by Lemma 10.6.2 in [HK02] that for any
there exists a finite
such that
Therefore we can model elements
as a
-crystal by considering an infinite list of elements
and defining the crystal structure by:
where
. To translate this into a finite list, we consider a finite sequence
and if
then we take the image as
. Similarly we remove
if we have
. Additionally if
then we consider this to be
.
We can then lift the
-crystal structure to a
-crystal structure by using a tensor product of the
affinization
of the of crystalsfor all
.
REFERENCES:
[HK02] Introduction to Quantum Groups and Crystal Bases. Jin Hong and Seok-Jin Kang. 2002. Volume 42. Graduate Studies in Mathematics. American Mathematical Society. INPUT:
B
– a single or list ofperfect crystal(s) of level
weight
– a weight in
EXAMPLES:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0]; mg [[[3]]] sage: mg.f_string([0,1,2,2]) [[[3]], [[3]], [[1]]] sage: x = mg.f_string([0,1,2]); x [[[2]], [[3]], [[1]]] sage: x.weight() Lambda[0]
An example of type
:
sage: B = crystals.KirillovReshetikhin(['A',5,2], 1,1) sage: La = RootSystem(['A',5,2]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0]; mg [[[-1]]] sage: mg.f_string([0,2,1,3]) [[[-3]], [[2]], [[-1]]] sage: mg.f_string([0,2,3,1]) [[[-3]], [[2]], [[-1]]]
An example of type
:
sage: B = crystals.KirillovReshetikhin(['D',3,2], 1,1) sage: La = RootSystem(['D',3,2]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0]; mg [[]] sage: mg.f_string([0,1,2,0]) [[[0]], [[1]], []]
An example using multiple crystals of the same level:
sage: B1 = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: B2 = crystals.KirillovReshetikhin(['A',2,1], 2,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel([B1, B2, B1], La[0]) sage: mg = C.module_generators[0]; mg [[[3]]] sage: mg.f_string([0,1,2,2]) [[[3]], [[1], [3]], [[3]]] sage: mg.f_string([0,1,2,2,2]) sage: mg.f_string([0,1,2,2,1,0]) [[[3]], [[2], [3]], [[1]], [[2]]] sage: mg.f_string([0,1,2,2,1,0,0,2]) [[[3]], [[1], [2]], [[1]], [[3]], [[1], [3]]]
By using the extended weight lattice, the Kyoto path model lifts the perfect crystals to their affinizations:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: P = RootSystem(['A',2,1]).weight_lattice(extended=True) sage: La = P.fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0]; mg [[[3]](0)] sage: x = mg.f_string([0,1,2]); x [[[2]](-1), [[3]](0), [[1]](0)] sage: x.weight() Lambda[0] - delta
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class
Element
(parent, *args, **kwds)¶ Bases:
sage.combinat.crystals.tensor_product.TensorProductOfRegularCrystalsElement
An element in the Kyoto path model.
-
e
(i)¶ Return the action of
on
self
.EXAMPLES:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0] sage: all(mg.e(i) is None for i in C.index_set()) True sage: mg.f(0).e(0) == mg True
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epsilon
(i)¶ Return
of
self
.EXAMPLES:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0] sage: [mg.epsilon(i) for i in C.index_set()] [0, 0, 0] sage: elt = mg.f(0) sage: [elt.epsilon(i) for i in C.index_set()] [1, 0, 0] sage: elt = mg.f_string([0,1,2]) sage: [elt.epsilon(i) for i in C.index_set()] [0, 0, 1] sage: elt = mg.f_string([0,1,2,2]) sage: [elt.epsilon(i) for i in C.index_set()] [0, 0, 2]
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f
(i)¶ Return the action of
on
self
.EXAMPLES:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0] sage: mg.f(2) sage: mg.f(0) [[[1]], [[2]]] sage: mg.f_string([0,1,2]) [[[2]], [[3]], [[1]]]
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phi
(i)¶ Return
of
self
.EXAMPLES:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0] sage: [mg.phi(i) for i in C.index_set()] [1, 0, 0] sage: elt = mg.f(0) sage: [elt.phi(i) for i in C.index_set()] [0, 1, 1] sage: elt = mg.f_string([0,1]) sage: [elt.phi(i) for i in C.index_set()] [0, 0, 2]
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truncate
(k=None)¶ Truncate
self
to have lengthk
and return as an element in a (finite) tensor product of crystals.INPUT:
k
– (optional) the length to truncate to; if not specified, then returns one more than the current non-ground-state elements (i.e. the current list inself
)
EXAMPLES:
sage: B1 = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: B2 = crystals.KirillovReshetikhin(['A',2,1], 2,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel([B1,B2,B1], La[0]) sage: mg = C.highest_weight_vector() sage: elt = mg.f_string([0,1,2,2,1,0]); elt [[[3]], [[2], [3]], [[1]], [[2]]] sage: t = elt.truncate(); t [[[3]], [[2], [3]], [[1]], [[2]]] sage: t.parent() is C.finite_tensor_product(4) True sage: elt.truncate(2) [[[3]], [[2], [3]]] sage: elt.truncate(10) [[[3]], [[2], [3]], [[1]], [[2]], [[1], [3]], [[2]], [[1]], [[2], [3]], [[1]], [[3]]]
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weight
()¶ Return the weight of
self
.EXAMPLES:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: P = RootSystem(['A',2,1]).weight_lattice(extended=True) sage: La = P.fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: mg = C.module_generators[0] sage: mg.weight() Lambda[0] sage: mg.f_string([0,1,2]).weight() Lambda[0] - delta
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-
KyotoPathModel.
finite_tensor_product
(k)¶ Return the finite tensor product of crystals of length
k
from truncatingself
.EXAMPLES:
sage: B1 = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: B2 = crystals.KirillovReshetikhin(['A',2,1], 2,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel([B1,B2,B1], La[0]) sage: C.finite_tensor_product(5) Full tensor product of the crystals [Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1), Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(2,1), Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1), Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1), Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(2,1)]
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KyotoPathModel.
weight_lattice_realization
()¶ Return the weight lattice realization used to express weights.
EXAMPLES:
sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1) sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights() sage: C = crystals.KyotoPathModel(B, La[0]) sage: C.weight_lattice_realization() Weight lattice of the Root system of type ['A', 2, 1] sage: P = RootSystem(['A',2,1]).weight_lattice(extended=True) sage: C = crystals.KyotoPathModel(B, P.fundamental_weight(0)) sage: C.weight_lattice_realization() Extended weight lattice of the Root system of type ['A', 2, 1]