Elementary Crystals

Let \lambda be a weight. The crystals T_{\lambda}, R_{\lambda}, B_i, and C are important objects in the tensor category of crystals. For example, the crystal T_0 is the neutral object in this category; i.e., T_0 \otimes B \cong B \otimes T_0 \cong B for any crystal B. We list some other properties of these crystals:

  • The crystal T_{\lambda} \otimes B(\infty) is the crystal of the Verma module with highest weight \lambda, where \lambda is a dominant integral weight.
  • Let u_{\infty} be the highest weight vector of B(\infty) and \lambda be a dominant integral weight. There is an embedding of crystals B(\lambda)
\longrightarrow T_{\lambda} \otimes B(\infty) sending u_{\lambda} \mapsto
t_{\lambda} \otimes u_{\infty} which is not strict, but the embedding B(\lambda) \longrightarrow C \otimes T_{\lambda} \otimes B(\infty) by u_{\lambda} \mapsto c \otimes t_{\lambda} \otimes u_{\infty} is a strict embedding.
  • For any dominant integral weight \lambda, there is a surjective crystal morphism \Psi_{\lambda} \colon R_{\lambda} \otimes B(\infty) \longrightarrow
B(\lambda). More precisely, if B = \{r_{\lambda} \otimes b \in R_{\lambda}
\otimes B(\infty) : \Psi_{\lambda}(r_{\lambda} \otimes b) \neq 0 \}, then B \cong B(\lambda) as crystals.
  • For all Cartan types and all weights \lambda, we have R_{\lambda} \cong C
\otimes T_{\lambda} as crystals.
  • For each i, there is a strict crystal morphism \Psi_i \colon B(\infty)
\longrightarrow B_i \otimes B(\infty) defined by u_{\infty} \mapsto
b_i(0) \otimes u_{\infty}, where u_\infty is the highest weight vector of B(\infty).

For more information on B(\infty), see InfinityCrystalOfTableaux.

Note

As with TensorProductOfCrystals, we are using the opposite of Kashiwara’s convention.

AUTHORS:

  • Ben Salisbury: Initial version

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.
class sage.combinat.crystals.elementary_crystals.AbstractSingleCrystalElement

Bases: sage.structure.element.Element

Abstract base class for elements in crystals with a single element.

e(i)

Return e_i of self, which is None for all i.

INPUT:

  • i – An element of the index set

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)
f(i)

Return f_i of self, which is None for all i.

INPUT:

  • i – An element of the index set

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)
class sage.combinat.crystals.elementary_crystals.ComponentCrystal(cartan_type)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

The component crystal.

Defined in [Kashiwara93], the component crystal C = \{c\} is the single element crystal whose crystal structure is defined by

\mathrm{wt}(c) = 0, \quad
e_i c = f_i c = 0, \quad
\varepsilon_i(c) = \varphi_i(c) = 0.

Note C \cong B(0), where B(0) is the highest weight crystal of highest weight 0.

INPUT:

  • cartan_type – A Cartan type
class Element

Bases: sage.combinat.crystals.elementary_crystals.AbstractSingleCrystalElement

Element of a component crystal.

epsilon(i)

Return \varepsilon_i of self, which is 0 for all i.

INPUT:

  • i – An element of the index set

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]
phi(i)

Return \varphi_i of self, which is 0 for all i.

INPUT:

  • i – An element of the index set

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]
weight()

Return the weight of self, which is always 0.

EXAMPLES:

sage: C = crystals.elementary.Component("F4")
sage: c = C.highest_weight_vector()
sage: c.weight()
(0, 0, 0, 0)
ComponentCrystal.cardinality()

Return the cardinality of self, which is always 1.

EXAMPLES:

sage: C = crystals.elementary.Component("E6")
sage: c = C.highest_weight_vector()
sage: C.cardinality()
1
class sage.combinat.crystals.elementary_crystals.ElementaryCrystal(cartan_type, i)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

The elementary crystal B_i.

For i an element of the index set of type X, the crystal B_i of type X is the set

B_i = \{ b_i(m) : m \in \ZZ \},

where the crystal stucture is given by

\begin{aligned}
\mathrm{wt}\bigl(b_i(m)\bigr) &= m\alpha_i \\
\varphi_j\bigl(b_i(m)\bigr) &= \begin{cases}
    m & \text{ if } j=i, \\
    -\infty & \text{ if } j\neq i,
\end{cases} \\
\varepsilon_j\bigl(b_i(m)\bigr) &= \begin{cases}
    -m & \text{ if } j=i, \\
    -\infty & \text{ if } j\neq i,
\end{cases} \\
e_j b_i(m) &= \begin{cases}
    b_i(m+1) & \text{ if } j=i, \\
    0 & \text{ if } j\neq i,
\end{cases} \\
f_j b_i(m) &= \begin{cases}
    b_i(m-1) & \text{ if } j=i, \\
    0 & \text{ if } j\neq i.
\end{cases}
\end{aligned}

The Kashiwara embedding theorem asserts there is a unique strict crystal embedding of crystals

B(\infty) \hookrightarrow B_i \otimes B(\infty),

satisfying certain properties (see [Kashiwara93]). The above embedding may be iterated to obtain a new embedding

B(\infty) \hookrightarrow B_{i_N} \otimes B_{i_{N-1}}
\otimes \cdots \otimes B_{i_2} \otimes B_{i_1} \otimes B(\infty),

which is a foundational object in the study of polyhedral realizations of crystals (see, for example, [NZ97]).

class Element(parent, m)

Bases: sage.structure.element.Element

Element of a B_i crystal.

e(i)

Return the action of e_i on self.

INPUT:

  • i – An element of the index set

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)
epsilon(i)

Return \varepsilon_i of self.

INPUT:

  • i – An element of the index set

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]]
f(i)

Return the action of f_i on self.

INPUT:

  • i – An element of the index set

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)
phi(i)

Return \varphi_i of self.

INPUT:

  • i – An element of the index set

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]]
weight()

Return the weight of self.

EXAMPLES:

sage: B = crystals.elementary.Elementary(['C',14],12)
sage: B(-385).weight()
-385*alpha[12]
ElementaryCrystal.weight_lattice_realization()

Return a realization of the lattice containing the weights of self.

EXAMPLES:

sage: B = crystals.elementary.Elementary(['A',4, 1], 2)
sage: B.weight_lattice_realization()
Root lattice of the Root system of type ['A', 4, 1]
class sage.combinat.crystals.elementary_crystals.RCrystal(cartan_type, weight)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

The crystal R_{\lambda}.

For a fixed weight \lambda, the crystal R_{\lambda} = \{ r_{\lambda} \} is a single element crystal with the crystal structure defined by

\mathrm{wt}(r_{\lambda}) = \lambda, \quad
e_i r_{\lambda} = f_i r_{\lambda} = 0, \quad
\varepsilon_i(r_{\lambda}) = -\langle h_i, \lambda\rangle, \quad
\varphi_i(r_{\lambda}) = 0,

where \{h_i\} are the simple coroots.

Tensoring R_{\lambda} with a crystal B results in shifting the weights of the vertices in B by \lambda and may also cut a subset out of the original graph of B. That is, \mathrm{wt}(r_{\lambda} \otimes b) =
\mathrm{wt}(b) + \lambda, where b \in B, provided r_{\lambda} \otimes
b \neq 0. For example, the crystal graph of B(\lambda) is the same as the crystal graph of R_{\lambda} \otimes B(\infty) generated from the component r_{\lambda} \otimes u_{\infty}.

INPUT:

  • cartan_type – A Cartan type
  • weight – An element of the weight lattice of type cartan_type

EXAMPLES:

We check by tensoring R_{\lambda} with B(\infty) results in a component of B(\lambda):

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
class Element

Bases: sage.combinat.crystals.elementary_crystals.AbstractSingleCrystalElement

Element of a R_{\lambda} crystal.

epsilon(i)

Return \varepsilon_i of self.

We have \varepsilon_i(r_{\lambda}) = -\langle h_i, \lambda
\rangle for all i, where h_i is a simple coroot.

INPUT:

  • i – An element of the index set

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]
phi(i)

Return \varphi_i of self, which is 0 for all i.

INPUT:

  • i – An element of the index set

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]
weight()

Return the weight of self, which is always \lambda.

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)
RCrystal.cardinality()

Return the cardinality of self, which is always 1.

EXAMPLES:

sage: La = RootSystem(['C',12]).weight_lattice().fundamental_weights()
sage: R = crystals.elementary.R(['C',12],La[9])
sage: R.cardinality()
1
RCrystal.weight_lattice_realization()

Return a realization of the lattice containing the weights of self.

EXAMPLES:

sage: La = RootSystem(['C',12]).weight_lattice().fundamental_weights()
sage: R = crystals.elementary.R(['C',12], La[9])
sage: R.weight_lattice_realization()
Ambient space of the Root system of type ['C', 12]

sage: ct = CartanMatrix([[2, -4], [-5, 2]])
sage: La = RootSystem(ct).weight_lattice().fundamental_weights()
sage: R = crystals.elementary.R(ct, La[1])
sage: R.weight_lattice_realization()
Weight space over the Rational Field of the Root system of type
[ 2 -4]
[-5  2]
class sage.combinat.crystals.elementary_crystals.TCrystal(cartan_type, weight)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

The crystal T_{\lambda}.

Let \lambda be a weight. As defined in [Kashiwara93] the crystal T_{\lambda} = \{ t_{\lambda} \} is a single element crystal with the crystal structure defined by

\mathrm{wt}(t_\lambda) = \lambda, \quad
e_i t_{\lambda} = f_i t_{\lambda} = 0, \quad
\varepsilon_i(t_{\lambda}) = \varphi_i(t_{\lambda}) = -\infty.

The crystal T_{\lambda} shifts the weights of the vertices in a crystal B by \lambda when tensored with B, but leaves the graph structure of B unchanged. That is to say, for all b \in B, we have \mathrm{wt}(b
\otimes t_{\lambda}) = \mathrm{wt}(b) + \lambda.

INPUT:

  • cartan_type – A Cartan type
  • weight – An element of the weight lattice of type cartan_type

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
class Element

Bases: sage.combinat.crystals.elementary_crystals.AbstractSingleCrystalElement

Element of a T_{\lambda} crystal.

epsilon(i)

Return \varepsilon_i of self, which is -\infty for all i.

INPUT:

  • i – An element of the index set

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]
phi(i)

Return \varphi_i of self, which is -\infty for all i.

INPUT:

  • i – An element of the index set

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]
weight()

Return the weight of self, which is always \lambda.

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)
TCrystal.cardinality()

Return the cardinality of self, which is always 1.

EXAMPLES:

sage: La = RootSystem(['C',12]).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(['C',12], La[9])
sage: T.cardinality()
1
TCrystal.weight_lattice_realization()

Return a realization of the lattice containing the weights of self.

EXAMPLES:

sage: La = RootSystem(['C',12]).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(['C',12], La[9])
sage: T.weight_lattice_realization()
Ambient space of the Root system of type ['C', 12]

sage: ct = CartanMatrix([[2, -4], [-5, 2]])
sage: La = RootSystem(ct).weight_lattice().fundamental_weights()
sage: T = crystals.elementary.T(ct, La[1])
sage: T.weight_lattice_realization()
Weight space over the Rational Field of the Root system of type
[ 2 -4]
[-5  2]