Let be a weight. The crystals
,
,
,
and
are important objects in the tensor category of crystals.
For example, the crystal
is the neutral object in this category; i.e.,
for any crystal
. We list
some other properties of these crystals:
For more information on , see
InfinityCrystalOfTableaux.
Note
As with TensorProductOfCrystals, we are using the opposite of Kashiwara’s convention.
AUTHORS:
REFERENCES:
[Kashiwara93] | (1, 2, 3) M. Kashiwara. The Crystal Base and Littelmann’s Refined Demazure Character Formula. Duke Math. J. 71 (3), pp. 839–858, 1993. |
[NZ97] | T. Nakashima and A. Zelevinsky. Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras. Adv. Math. 131, pp. 253–278, 1997. |
Bases: sage.structure.element.Element
Abstract base class for elements in crystals with a single element.
Return of self, which is None for all
.
INPUT:
EXAMPLES:
sage: ct = CartanType(['A',2])
sage: la = RootSystem(ct).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(ct,la[1])
sage: t = T.highest_weight_vector()
sage: t.e(1)
sage: t.e(2)
Return of self, which is None for all
.
INPUT:
EXAMPLES:
sage: ct = CartanType(['A',2])
sage: la = RootSystem(ct).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(ct,la[1])
sage: t = T.highest_weight_vector()
sage: t.f(1)
sage: t.f(2)
Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation
The component crystal.
Defined in [Kashiwara93], the component crystal is the single
element crystal whose crystal structure is defined by
Note , where
is the highest weight crystal of highest
weight
.
INPUT:
Bases: sage.combinat.crystals.elementary_crystals.AbstractSingleCrystalElement
Element of a component crystal.
Return of self, which is
for all
.
INPUT:
EXAMPLES:
sage: C = crystals.elementary.Component("C5")
sage: c = C.highest_weight_vector()
sage: [c.epsilon(i) for i in C.index_set()]
[0, 0, 0, 0, 0]
Return of self, which is
for all
.
INPUT:
EXAMPLES:
sage: C = crystals.elementary.Component("C5")
sage: c = C.highest_weight_vector()
sage: [c.phi(i) for i in C.index_set()]
[0, 0, 0, 0, 0]
Return the weight of self, which is always .
EXAMPLES:
sage: C = crystals.elementary.Component("F4")
sage: c = C.highest_weight_vector()
sage: c.weight()
(0, 0, 0, 0)
Return the cardinality of self, which is always .
EXAMPLES:
sage: C = crystals.elementary.Component("E6")
sage: c = C.highest_weight_vector()
sage: C.cardinality()
1
Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation
The elementary crystal .
For an element of the index set of type
, the crystal
of type
is the set
where the crystal stucture is given by
The Kashiwara embedding theorem asserts there is a unique strict crystal embedding of crystals
satisfying certain properties (see [Kashiwara93]). The above embedding may be iterated to obtain a new embedding
which is a foundational object in the study of polyhedral realizations of crystals (see, for example, [NZ97]).
Bases: sage.structure.element.Element
Element of a crystal.
Return the action of on self.
INPUT:
EXAMPLES:
sage: B = crystals.elementary.Elementary(['E',7],1)
sage: B(3).e(1)
4
sage: B(172).e_string([1]*171)
343
sage: B(0).e(2)
Return of self.
INPUT:
EXAMPLES:
sage: B = crystals.elementary.Elementary(['F',4],3)
sage: [[B(j).epsilon(i) for i in B.index_set()] for j in range(5)]
[[-inf, -inf, 0, -inf],
[-inf, -inf, -1, -inf],
[-inf, -inf, -2, -inf],
[-inf, -inf, -3, -inf],
[-inf, -inf, -4, -inf]]
Return the action of on self.
INPUT:
EXAMPLES:
sage: B = crystals.elementary.Elementary(['E',7],1)
sage: B(3).f(1)
2
sage: B(172).f_string([1]*171)
1
sage: B(0).e(2)
Return of self.
INPUT:
EXAMPLES:
sage: B = crystals.elementary.Elementary(['E',8,1],4)
sage: [[B(m).phi(j) for j in B.index_set()] for m in range(44,49)]
[[-inf, -inf, -inf, -inf, 44, -inf, -inf, -inf, -inf],
[-inf, -inf, -inf, -inf, 45, -inf, -inf, -inf, -inf],
[-inf, -inf, -inf, -inf, 46, -inf, -inf, -inf, -inf],
[-inf, -inf, -inf, -inf, 47, -inf, -inf, -inf, -inf],
[-inf, -inf, -inf, -inf, 48, -inf, -inf, -inf, -inf]]
Return the weight of self.
EXAMPLES:
sage: B = crystals.elementary.Elementary(['C',14],12)
sage: B(-385).weight()
-385*alpha[12]
Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation
The crystal .
For a fixed weight , the crystal
is a single element crystal with the crystal structure defined by
where are the simple coroots.
Tensoring with a crystal
results in shifting the weights
of the vertices in
by
and may also cut a subset out of the
original graph of
. That is,
, where
, provided
. For example, the crystal graph of
is the same as
the crystal graph of
generated from the
component
.
INPUT:
EXAMPLES:
We check by tensoring with
results in a
component of
:
sage: B = crystals.infinity.Tableaux("A2")
sage: R = crystals.elementary.R("A2", B.Lambda()[1]+B.Lambda()[2])
sage: T = crystals.TensorProduct(R, B)
sage: mg = T(R.highest_weight_vector(), B.highest_weight_vector())
sage: S = T.subcrystal(generators=[mg])
sage: for x in S: x.weight()
(2, 1, 0)
(2, 0, 1)
(1, 2, 0)
(1, 1, 1)
(1, 1, 1)
(1, 0, 2)
(0, 2, 1)
(0, 1, 2)
sage: C = crystals.Tableaux("A2", shape=[2,1])
sage: for x in C: x.weight()
(2, 1, 0)
(1, 2, 0)
(1, 1, 1)
(1, 0, 2)
(0, 1, 2)
(2, 0, 1)
(1, 1, 1)
(0, 2, 1)
sage: GT = T.digraph(subset=S)
sage: GC = C.digraph()
sage: GT.is_isomorphic(GC, edge_labels=True)
True
Bases: sage.combinat.crystals.elementary_crystals.AbstractSingleCrystalElement
Element of a crystal.
Return of self.
We have for all
, where
is a simple coroot.
INPUT:
EXAMPLES:
sage: la = RootSystem(['A',2]).weight_lattice().fundamental_weights()
sage: R = crystals.elementary.R("A2",la[1])
sage: r = R.highest_weight_vector()
sage: [r.epsilon(i) for i in R.index_set()]
[-1, 0]
Return of self, which is
for all
.
INPUT:
EXAMPLES:
sage: la = RootSystem("C5").weight_lattice().fundamental_weights()
sage: R = crystals.elementary.R("C5",la[4]+la[5])
sage: r = R.highest_weight_vector()
sage: [r.phi(i) for i in R.index_set()]
[0, 0, 0, 0, 0]
Return the weight of self, which is always .
EXAMPLES:
sage: ct = CartanType(['C',5])
sage: la = RootSystem(ct).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(ct,la[4]+la[5]-la[1]-la[2])
sage: t = T.highest_weight_vector()
sage: t.weight()
(0, 1, 2, 2, 1)
Return the cardinality of self, which is always .
EXAMPLES:
sage: La = RootSystem(['C',12]).weight_lattice().fundamental_weights()
sage: R = crystals.elementary.R(['C',12],La[9])
sage: R.cardinality()
1
Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation
The crystal .
Let be a weight. As defined in [Kashiwara93] the crystal
is a single element crystal with the
crystal structure defined by
The crystal shifts the weights of the vertices in a crystal
by
when tensored with
, but leaves the graph structure of
unchanged. That is to say, for all
, we have
.
INPUT:
EXAMPLES:
sage: ct = CartanType(['A',2])
sage: C = crystals.Tableaux(ct, shape=[1])
sage: for x in C: x.weight()
(1, 0, 0)
(0, 1, 0)
(0, 0, 1)
sage: La = RootSystem(ct).ambient_space().fundamental_weights()
sage: TLa = crystals.elementary.T(ct, 3*(La[1] + La[2]))
sage: TP = crystals.TensorProduct(TLa, C)
sage: for x in TP: x.weight()
(7, 3, 0)
(6, 4, 0)
(6, 3, 1)
sage: G = C.digraph()
sage: H = TP.digraph()
sage: G.is_isomorphic(H,edge_labels=True)
True
Bases: sage.combinat.crystals.elementary_crystals.AbstractSingleCrystalElement
Element of a crystal.
Return of self, which is
for all
.
INPUT:
EXAMPLES:
sage: ct = CartanType(['C',5])
sage: la = RootSystem(ct).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(ct,la[4]+la[5]-la[1]-la[2])
sage: t = T.highest_weight_vector()
sage: [t.epsilon(i) for i in T.index_set()]
[-inf, -inf, -inf, -inf, -inf]
Return of self, which is
for all
.
INPUT:
EXAMPLES:
sage: ct = CartanType(['C',5])
sage: la = RootSystem(ct).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(ct,la[4]+la[5]-la[1]-la[2])
sage: t = T.highest_weight_vector()
sage: [t.phi(i) for i in T.index_set()]
[-inf, -inf, -inf, -inf, -inf]
Return the weight of self, which is always .
EXAMPLES:
sage: ct = CartanType(['C',5])
sage: la = RootSystem(ct).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(ct,la[4]+la[5]-la[1]-la[2])
sage: t = T.highest_weight_vector()
sage: t.weight()
(0, 1, 2, 2, 1)
Return the cardinality of self, which is always .
EXAMPLES:
sage: La = RootSystem(['C',12]).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(['C',12], La[9])
sage: T.cardinality()
1