Weight lattice realizations - Combinatorics

Skip to content
Back to top
View this page
Edit this page

# Weight lattice realizations

class sage.combinat.root_system.weight_lattice_realizations.WeightLatticeRealizations(base, name=None)

  Bases: `Category_over_base_ring`

  The category of weight lattice realizations over a given base ring.

  A weight lattice realization \(L\) over a base ring \(R\) is a free module (or vector space if \(R\) is a field) endowed with an embedding of the root lattice of some root system. By restriction, this embedding defines an embedding of the root lattice of this root system, which makes \(L\) a root lattice realization.

  Typical weight lattice realizations over \(\ZZ\) include the weight lattice, and ambient lattice. Typical weight lattice realizations over \(\QQ\) include the weight space, and ambient space.

  To describe the embedding, a weight lattice realization must implement a method `fundamental_weight`(i) returning for each `i()` in the index set the image of the fundamental weight \(\Lambda_i\) under the embedding.

  In order to be a proper root lattice realization, a weight lattice realization should also implement the scalar product with the coroot lattice; on the other hand, the embedding of the simple roots is given for free.

  See also

  - `RootSystem`

  - `RootLatticeRealizations`

  - `WeightSpace`

  - `AmbientSpace`

  EXAMPLES:

  Here, we consider the root system of type \(A_7\), and embed the weight lattice element \(x = \Lambda_1 + 2 \Lambda_3\) in several root lattice realizations:

  Sage

      sage: R = RootSystem(["A",7])
      sage: Lambda = R.weight_lattice().fundamental_weights()
      sage: x = Lambda[2] + 2 * Lambda[5]

      sage: L = R.weight_space()
      sage: L(x)
      Lambda[2] + 2*Lambda[5]

      sage: L = R.ambient_lattice()
      sage: L(x)
      (3, 3, 2, 2, 2, 0, 0, 0)

  Python

      >>> from sage.all import *
      >>> R = RootSystem(["A",Integer(7)])
      >>> Lambda = R.weight_lattice().fundamental_weights()
      >>> x = Lambda[Integer(2)] + Integer(2) * Lambda[Integer(5)]

      >>> L = R.weight_space()
      >>> L(x)
      Lambda[2] + 2*Lambda[5]

      >>> L = R.ambient_lattice()
      >>> L(x)
      (3, 3, 2, 2, 2, 0, 0, 0)

  We embed the weight space element \(x = \Lambda_1 + 1/2 \Lambda_3\) in the ambient space:

  Sage

      sage: Lambda = R.weight_space().fundamental_weights()
      sage: x = Lambda[2] + 1/2 * Lambda[5]

      sage: L = R.ambient_space()
      sage: L(x)
      (3/2, 3/2, 1/2, 1/2, 1/2, 0, 0, 0)

  [Python]

      >>> from sage.all import *
      >>> Lambda = R.weight_space().fundamental_weights()
      >>> x = Lambda[Integer(2)] + Integer(1)/Integer(2) * Lambda[Integer(5)]

      >>> L = R.ambient_space()
      >>> L(x)
      (3/2, 3/2, 1/2, 1/2, 1/2, 0, 0, 0)

  Of course, one can’t embed the weight space in the ambient lattice:

  Sage

      sage: L = R.ambient_lattice()
      sage: L(x)
      Traceback (most recent call last):
      ...
      TypeError: do not know how to make x (= Lambda[2] + 1/2*Lambda[5])
      an element of self (=Ambient lattice of the Root system of type ['A', 7])

  Python

      >>> from sage.all import *
      >>> L = R.ambient_lattice()
      >>> L(x)
      Traceback (most recent call last):
      ...
      TypeError: do not know how to make x (= Lambda[2] + 1/2*Lambda[5])
      an element of self (=Ambient lattice of the Root system of type ['A', 7])

  If \(K_1\) is a subring of \(K_2\), then one could in theory have an embedding from the weight space over \(K_1\) to any weight lattice realization over \(K_2\); this is not implemented:

  Sage

      sage: K1 = QQ
      sage: K2 = QQ['q']
      sage: L = R.ambient_space(K2)

      sage: Lambda = R.weight_space(K2).fundamental_weights()
      sage: L(Lambda[1])
      (1, 0, 0, 0, 0, 0, 0, 0)

      sage: Lambda = R.weight_space(K1).fundamental_weights()
      sage: L(Lambda[1])
      Traceback (most recent call last):
      ...
      TypeError: do not know how to make x (= Lambda[1]) an element
      of self (=Ambient space of the Root system of type ['A', 7])

  [Python]

      >>> from sage.all import *
      >>> K1 = QQ
      >>> K2 = QQ['q']
      >>> L = R.ambient_space(K2)

      >>> Lambda = R.weight_space(K2).fundamental_weights()
      >>> L(Lambda[Integer(1)])
      (1, 0, 0, 0, 0, 0, 0, 0)

      >>> Lambda = R.weight_space(K1).fundamental_weights()
      >>> L(Lambda[Integer(1)])
      Traceback (most recent call last):
      ...
      TypeError: do not know how to make x (= Lambda[1]) an element
      of self (=Ambient space of the Root system of type ['A', 7])

  class ElementMethods

    Bases: `object`

    symmetric_form(la)

      Return the symmetric form of `self` with `la`.

      Return the pairing \(( | )\) on the weight lattice. See Chapter 6 in Kac, Infinite Dimensional Lie Algebras for more details.

      Warning

      For affine root systems, if you are not working in the extended weight lattice/space, this may return incorrect results.

      EXAMPLES:

      Sage

          sage: # needs sage.graphs
          sage: P = RootSystem(['C',2]).weight_lattice()
          sage: al = P.simple_roots()
          sage: al[1].symmetric_form(al[1])
          2
          sage: al[1].symmetric_form(al[2])
          -2
          sage: al[2].symmetric_form(al[1])
          -2
          sage: Q = RootSystem(['C',2]).root_lattice()
          sage: alQ = Q.simple_roots()
          sage: all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
          ....:     for i in P.index_set() for j in P.index_set())
          True

          sage: # needs sage.graphs
          sage: P = RootSystem(['C',2,1]).weight_lattice(extended=True)
          sage: al = P.simple_roots()
          sage: al[1].symmetric_form(al[1])
          2
          sage: al[1].symmetric_form(al[2])
          -2
          sage: al[1].symmetric_form(al[0])
          -2
          sage: al[0].symmetric_form(al[1])
          -2
          sage: Q = RootSystem(['C',2,1]).root_lattice()
          sage: alQ = Q.simple_roots()
          sage: all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
          ....:     for i in P.index_set() for j in P.index_set())
          True
          sage: La = P.basis()
          sage: [La['delta'].symmetric_form(al) for al in P.simple_roots()]
          [0, 0, 0]
          sage: [La[0].symmetric_form(al) for al in P.simple_roots()]
          [1, 0, 0]

          sage: P = RootSystem(['C',2,1]).weight_lattice()
          sage: Q = RootSystem(['C',2,1]).root_lattice()
          sage: al = P.simple_roots()                                             # needs sage.graphs
          sage: alQ = Q.simple_roots()                                            # needs sage.graphs
          sage: all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])  # needs sage.graphs
          ....:     for i in P.index_set() for j in P.index_set())
          True

      Python

          >>> from sage.all import *
          >>> # needs sage.graphs
          >>> P = RootSystem(['C',Integer(2)]).weight_lattice()
          >>> al = P.simple_roots()
          >>> al[Integer(1)].symmetric_form(al[Integer(1)])
          2
          >>> al[Integer(1)].symmetric_form(al[Integer(2)])
          -2
          >>> al[Integer(2)].symmetric_form(al[Integer(1)])
          -2
          >>> Q = RootSystem(['C',Integer(2)]).root_lattice()
          >>> alQ = Q.simple_roots()
          >>> all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
          ...     for i in P.index_set() for j in P.index_set())
          True

          >>> # needs sage.graphs
          >>> P = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice(extended=True)
          >>> al = P.simple_roots()
          >>> al[Integer(1)].symmetric_form(al[Integer(1)])
          2
          >>> al[Integer(1)].symmetric_form(al[Integer(2)])
          -2
          >>> al[Integer(1)].symmetric_form(al[Integer(0)])
          -2
          >>> al[Integer(0)].symmetric_form(al[Integer(1)])
          -2
          >>> Q = RootSystem(['C',Integer(2),Integer(1)]).root_lattice()
          >>> alQ = Q.simple_roots()
          >>> all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])
          ...     for i in P.index_set() for j in P.index_set())
          True
          >>> La = P.basis()
          >>> [La['delta'].symmetric_form(al) for al in P.simple_roots()]
          [0, 0, 0]
          >>> [La[Integer(0)].symmetric_form(al) for al in P.simple_roots()]
          [1, 0, 0]

          >>> P = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice()
          >>> Q = RootSystem(['C',Integer(2),Integer(1)]).root_lattice()
          >>> al = P.simple_roots()                                             # needs sage.graphs
          >>> alQ = Q.simple_roots()                                            # needs sage.graphs
          >>> all(al[i].symmetric_form(al[j]) == alQ[i].symmetric_form(alQ[j])  # needs sage.graphs
          ...     for i in P.index_set() for j in P.index_set())
          True

      The result of \((\Lambda_0 | \alpha_0)\) should be \(1\), however we get \(0\) because we are not working in the extended weight lattice:

      Sage

          sage: La = P.basis()
          sage: [La[0].symmetric_form(al) for al in P.simple_roots()]             # needs sage.graphs
          [0, 0, 0]

      [Python]

          >>> from sage.all import *
          >>> La = P.basis()
          >>> [La[Integer(0)].symmetric_form(al) for al in P.simple_roots()]             # needs sage.graphs
          [0, 0, 0]

    to_weight_space(base_ring=None)

      Map `self` to the weight space.

      Warning

      Implemented for finite Cartan type.

      EXAMPLES:

      Sage

          sage: b = CartanType(['B',2]).root_system().ambient_space().from_vector(vector([1,-2])); b
          (1, -2)
          sage: b.to_weight_space()
          3*Lambda[1] - 4*Lambda[2]
          sage: b = CartanType(['B',2]).root_system().ambient_space().from_vector(vector([1/2,0])); b
          (1/2, 0)
          sage: b.to_weight_space()
          1/2*Lambda[1]
          sage: b.to_weight_space(ZZ)
          Traceback (most recent call last):
          ...
          TypeError: no conversion of this rational to integer
          sage: b = CartanType(['G',2]).root_system().ambient_space().from_vector(vector([4,-5,1])); b
          (4, -5, 1)
          sage: b.to_weight_space()
          -6*Lambda[1] + 5*Lambda[2]

      Python

          >>> from sage.all import *
          >>> b = CartanType(['B',Integer(2)]).root_system().ambient_space().from_vector(vector([Integer(1),-Integer(2)])); b
          (1, -2)
          >>> b.to_weight_space()
          3*Lambda[1] - 4*Lambda[2]
          >>> b = CartanType(['B',Integer(2)]).root_system().ambient_space().from_vector(vector([Integer(1)/Integer(2),Integer(0)])); b
          (1/2, 0)
          >>> b.to_weight_space()
          1/2*Lambda[1]
          >>> b.to_weight_space(ZZ)
          Traceback (most recent call last):
          ...
          TypeError: no conversion of this rational to integer
          >>> b = CartanType(['G',Integer(2)]).root_system().ambient_space().from_vector(vector([Integer(4),-Integer(5),Integer(1)])); b
          (4, -5, 1)
          >>> b.to_weight_space()
          -6*Lambda[1] + 5*Lambda[2]

  class ParentMethods

    Bases: `object`

    dynkin_diagram_automorphism_of_alcove_morphism(f)

      Return the Dynkin diagram automorphism induced by an alcove morphism.

      INPUT:

      - `f` – a linear map from `self` to `self` which preserves alcoves

      This method returns the Dynkin diagram automorphism for the decomposition \(f = d w\) (see `reduced_word_of_alcove_morphism()`), as a dictionary mapping elements of the index set to itself.

      EXAMPLES:

      Sage

          sage: R = RootSystem(["A",2,1]).weight_lattice()
          sage: alpha = R.simple_roots()                                          # needs sage.graphs
          sage: Lambda = R.fundamental_weights()

      Python

          >>> from sage.all import *
          >>> R = RootSystem(["A",Integer(2),Integer(1)]).weight_lattice()
          >>> alpha = R.simple_roots()                                          # needs sage.graphs
          >>> Lambda = R.fundamental_weights()

      Translations by elements of the root lattice induce a trivial Dynkin diagram automorphism:

      Sage

          sage: # needs sage.graphs sage.libs.gap
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[0].translation)
          {0: 0, 1: 1, 2: 2}
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[1].translation)
          {0: 0, 1: 1, 2: 2}
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[2].translation)
          {0: 0, 1: 1, 2: 2}

      [Python]

          >>> from sage.all import *
          >>> # needs sage.graphs sage.libs.gap
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(0)].translation)
          {0: 0, 1: 1, 2: 2}
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(1)].translation)
          {0: 0, 1: 1, 2: 2}
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(2)].translation)
          {0: 0, 1: 1, 2: 2}

      This is no more the case for translations by general elements of the (classical) weight lattice at level 0:

      Sage

          sage: omega1 = Lambda[1] - Lambda[0]
          sage: omega2 = Lambda[2] - Lambda[0]

          sage: # needs sage.graphs sage.libs.gap
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
          {0: 1, 1: 2, 2: 0}
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
          {0: 2, 1: 0, 2: 1}

          sage: # needs sage.graphs sage.libs.gap
          sage: R = RootSystem(['C',2,1]).weight_lattice()
          sage: alpha = R.simple_roots()
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[1].translation)
          {0: 2, 1: 1, 2: 0}

          sage: # needs sage.graphs sage.libs.gap
          sage: R = RootSystem(['D',5,1]).weight_lattice()
          sage: Lambda = R.fundamental_weights()
          sage: omega1 = Lambda[1] - Lambda[0]
          sage: omega2 = Lambda[2] - 2*Lambda[0]
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
          {0: 1, 1: 0, 2: 2, 3: 3, 4: 5, 5: 4}
          sage: R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
          {0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5}

      Python

          >>> from sage.all import *
          >>> omega1 = Lambda[Integer(1)] - Lambda[Integer(0)]
          >>> omega2 = Lambda[Integer(2)] - Lambda[Integer(0)]

          >>> # needs sage.graphs sage.libs.gap
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
          {0: 1, 1: 2, 2: 0}
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
          {0: 2, 1: 0, 2: 1}

          >>> # needs sage.graphs sage.libs.gap
          >>> R = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice()
          >>> alpha = R.simple_roots()
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(alpha[Integer(1)].translation)
          {0: 2, 1: 1, 2: 0}

          >>> # needs sage.graphs sage.libs.gap
          >>> R = RootSystem(['D',Integer(5),Integer(1)]).weight_lattice()
          >>> Lambda = R.fundamental_weights()
          >>> omega1 = Lambda[Integer(1)] - Lambda[Integer(0)]
          >>> omega2 = Lambda[Integer(2)] - Integer(2)*Lambda[Integer(0)]
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega1.translation)
          {0: 1, 1: 0, 2: 2, 3: 3, 4: 5, 5: 4}
          >>> R.dynkin_diagram_automorphism_of_alcove_morphism(omega2.translation)
          {0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5}

      Algorithm: computes \(w\) of the decomposition, and see how \(f\circ w^{-1}\) permutes the simple roots.

    embed_at_level(x, level=1)

      Embed the classical weight \(x\) in the level `level` hyperplane.

      This is achieved by translating the straightforward embedding of \(x\) by \(c\Lambda_0\) for \(c\) some appropriate scalar.

      INPUT:

      - `x` – an element of the corresponding classical weight/ambient lattice

      - `level` – integer or element of the base ring (default: 1)

      EXAMPLES:

      Sage

          sage: # needs sage.graphs
          sage: L = RootSystem(["B",3,1]).weight_space()
          sage: L0 = L.classical()
          sage: alpha = L0.simple_roots()
          sage: omega = L0.fundamental_weights()
          sage: L.embed_at_level(omega[1], 1)
          Lambda[1]
          sage: L.embed_at_level(omega[2], 1)
          -Lambda[0] + Lambda[2]
          sage: L.embed_at_level(omega[3], 1)
          Lambda[3]
          sage: L.embed_at_level(alpha[1], 1)
          Lambda[0] + 2*Lambda[1] - Lambda[2]

      Python

          >>> from sage.all import *
          >>> # needs sage.graphs
          >>> L = RootSystem(["B",Integer(3),Integer(1)]).weight_space()
          >>> L0 = L.classical()
          >>> alpha = L0.simple_roots()
          >>> omega = L0.fundamental_weights()
          >>> L.embed_at_level(omega[Integer(1)], Integer(1))
          Lambda[1]
          >>> L.embed_at_level(omega[Integer(2)], Integer(1))
          -Lambda[0] + Lambda[2]
          >>> L.embed_at_level(omega[Integer(3)], Integer(1))
          Lambda[3]
          >>> L.embed_at_level(alpha[Integer(1)], Integer(1))
          Lambda[0] + 2*Lambda[1] - Lambda[2]

    fundamental_weight(i)

      Return the \(i\)-th fundamental weight.

      INPUT:

      - `i` – an element of the index set

      By a slight notational abuse, for an affine type this method should also accept `'delta'` as input, and return the image of \(\delta\) of the extended weight lattice in this realization.

      This should be overridden by any subclass, and typically be implemented as a cached method for efficiency.

      EXAMPLES:

      Sage

          sage: L = RootSystem(["A",3]).ambient_lattice()
          sage: L.fundamental_weight(1)
          (1, 0, 0, 0)

          sage: L = RootSystem(["A",3,1]).weight_lattice(extended=True)
          sage: L.fundamental_weight(1)
          Lambda[1]
          sage: L.fundamental_weight("delta")
          delta

      Python

          >>> from sage.all import *
          >>> L = RootSystem(["A",Integer(3)]).ambient_lattice()
          >>> L.fundamental_weight(Integer(1))
          (1, 0, 0, 0)

          >>> L = RootSystem(["A",Integer(3),Integer(1)]).weight_lattice(extended=True)
          >>> L.fundamental_weight(Integer(1))
          Lambda[1]
          >>> L.fundamental_weight("delta")
          delta

    fundamental_weights()

      Return the family \((\Lambda_i)_{i\in I}\) of the fundamental weights.

      EXAMPLES:

      Sage

          sage: e = RootSystem(['A',3]).ambient_lattice()
          sage: f = e.fundamental_weights()
          sage: [f[i] for i in [1,2,3]]
          [(1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0)]

      Python

          >>> from sage.all import *
          >>> e = RootSystem(['A',Integer(3)]).ambient_lattice()
          >>> f = e.fundamental_weights()
          >>> [f[i] for i in [Integer(1),Integer(2),Integer(3)]]
          [(1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0)]

    is_extended()

      Return whether this is a realization of the extended weight lattice.

      See also

      `sage.combinat.root_system.weight_space.WeightSpace`

      EXAMPLES:

      Sage

          sage: RootSystem(["A",3,1]).weight_lattice().is_extended()
          False
          sage: RootSystem(["A",3,1]).weight_lattice(extended=True).is_extended()
          True

      Python

          >>> from sage.all import *
          >>> RootSystem(["A",Integer(3),Integer(1)]).weight_lattice().is_extended()
          False
          >>> RootSystem(["A",Integer(3),Integer(1)]).weight_lattice(extended=True).is_extended()
          True

      This method is irrelevant for finite root systems, since the weight lattice need not be extended to ensure that the root lattice embeds faithfully:

      Sage

          sage: RootSystem(["A",3]).weight_lattice().is_extended()
          False

      [Python]

          >>> from sage.all import *
          >>> RootSystem(["A",Integer(3)]).weight_lattice().is_extended()
          False

    reduced_word_of_alcove_morphism(f)

      Return the reduced word of an alcove morphism.

      INPUT:

      - `f` – a linear map from `self` to `self` which preserves alcoves

      Let \(A\) be the fundamental alcove. This returns a reduced word \(i_1, \ldots, i_k\) such that the affine Weyl group element \(w = s_{i_1} \circ \cdots \circ s_{i_k}\) maps the alcove \(f(A)\) back to \(A\). In other words, the alcove walk \(i_1, \ldots, i_k\) brings the fundamental alcove to the corresponding translated alcove.

      Let us throw in a bit of context to explain the main use case. It is customary to realize the alcove picture in the coroot or coweight lattice \(R^\vee\). The extended affine Weyl group is then the group of linear maps on \(R^\vee\) which preserve the alcoves. By [Kac “Infinite-dimensional Lie algebra”, Proposition 6.5] the affine Weyl group is the semidirect product of the associated finite Weyl group and the group of translations in the coroot lattice (the extended affine Weyl group uses the coweight lattice instead). In other words, an element of the extended affine Weyl group admits a unique decomposition of the form:

      \[f = d w ,\]

      where \(w\) is in the Weyl group, and \(d\) is a function which maps the fundamental alcove to itself. As \(d\) permutes the walls of the fundamental alcove, it permutes accordingly the corresponding simple roots, which induces an automorphism of the Dynkin diagram.

      This method returns a reduced word for \(w\), whereas the method `dynkin_diagram_automorphism_of_alcove_morphism()` returns \(d\) as a permutation of the nodes of the Dynkin diagram.

      Nota bene: recall that the coroot (resp. coweight) lattice is implemented as the root (resp weight) lattice of the dual root system. Hence, this method is implemented for weight lattice realizations, but in practice is most of the time used on the dual side.

      EXAMPLES:

      We start with type \(A\) which is simply laced; hence we do not have to worry about the distinction between the weight and coweight lattice:

      Sage

          sage: R = RootSystem(["A",2,1]).weight_lattice()
          sage: alpha = R.simple_roots()                                          # needs sage.graphs
          sage: Lambda = R.fundamental_weights()

      Python

          >>> from sage.all import *
          >>> R = RootSystem(["A",Integer(2),Integer(1)]).weight_lattice()
          >>> alpha = R.simple_roots()                                          # needs sage.graphs
          >>> Lambda = R.fundamental_weights()

      We consider first translations by elements of the root lattice:

      Sage

          sage: R.reduced_word_of_alcove_morphism(alpha[0].translation)           # needs sage.graphs
          [1, 2, 1, 0]
          sage: R.reduced_word_of_alcove_morphism(alpha[1].translation)           # needs sage.graphs
          [0, 2, 0, 1]
          sage: R.reduced_word_of_alcove_morphism(alpha[2].translation)           # needs sage.graphs
          [0, 1, 0, 2]

      [Python]

          >>> from sage.all import *
          >>> R.reduced_word_of_alcove_morphism(alpha[Integer(0)].translation)           # needs sage.graphs
          [1, 2, 1, 0]
          >>> R.reduced_word_of_alcove_morphism(alpha[Integer(1)].translation)           # needs sage.graphs
          [0, 2, 0, 1]
          >>> R.reduced_word_of_alcove_morphism(alpha[Integer(2)].translation)           # needs sage.graphs
          [0, 1, 0, 2]

      We continue with translations by elements of the classical weight lattice, embedded at level \(0\):

      > sage: omega1 = Lambda[1] - Lambda[0] sage: omega2 = Lambda[2] - Lambda[0]

      > sage: R.reduced_word_of_alcove_morphism(omega1.translation) # needs sage.graphs [0, 2] sage: R.reduced_word_of_alcove_morphism(omega2.translation) # needs sage.graphs [0, 1]

      The following tests ensure that the code agrees with the tables in Kashiwara’s private notes on affine quantum algebras (2008).

    reduced_word_of_translation(t)

      Given an element of the root lattice, this returns a reduced word \(i_1, \ldots, i_k\) such that the Weyl group element \(s_{i_1} \circ \cdots \circ s_{i_k}\) implements the “translation” where \(x\) maps to \(x + level(x)*t\). In other words, the alcove walk \(i_1, \ldots, i_k\) brings the fundamental alcove to the corresponding translated alcove.

      Note

      There are some technical conditions for \(t\) to actually be a translation; those are not tested (TODO: detail).

      EXAMPLES:

      Sage

          sage: # needs sage.graphs
          sage: R = RootSystem(["A",2,1]).weight_lattice()
          sage: alpha = R.simple_roots()
          sage: R.reduced_word_of_translation(alpha[1])
          [0, 2, 0, 1]
          sage: R.reduced_word_of_translation(alpha[2])
          [0, 1, 0, 2]
          sage: R.reduced_word_of_translation(alpha[0])
          [1, 2, 1, 0]

          sage: R = RootSystem(['D',5,1]).weight_lattice()
          sage: Lambda = R.fundamental_weights()
          sage: omega1 = Lambda[1] - Lambda[0]
          sage: omega2 = Lambda[2] - 2*Lambda[0]
          sage: R.reduced_word_of_translation(omega1)                             # needs sage.graphs
          [0, 2, 3, 4, 5, 3, 2, 0]
          sage: R.reduced_word_of_translation(omega2)                             # needs sage.graphs
          [0, 2, 1, 3, 2, 4, 3, 5, 3, 2, 1, 4, 3, 2]

      Python

          >>> from sage.all import *
          >>> # needs sage.graphs
          >>> R = RootSystem(["A",Integer(2),Integer(1)]).weight_lattice()
          >>> alpha = R.simple_roots()
          >>> R.reduced_word_of_translation(alpha[Integer(1)])
          [0, 2, 0, 1]
          >>> R.reduced_word_of_translation(alpha[Integer(2)])
          [0, 1, 0, 2]
          >>> R.reduced_word_of_translation(alpha[Integer(0)])
          [1, 2, 1, 0]

          >>> R = RootSystem(['D',Integer(5),Integer(1)]).weight_lattice()
          >>> Lambda = R.fundamental_weights()
          >>> omega1 = Lambda[Integer(1)] - Lambda[Integer(0)]
          >>> omega2 = Lambda[Integer(2)] - Integer(2)*Lambda[Integer(0)]
          >>> R.reduced_word_of_translation(omega1)                             # needs sage.graphs
          [0, 2, 3, 4, 5, 3, 2, 0]
          >>> R.reduced_word_of_translation(omega2)                             # needs sage.graphs
          [0, 2, 1, 3, 2, 4, 3, 5, 3, 2, 1, 4, 3, 2]

      A non simply laced case:

      Sage

          sage: R = RootSystem(["C",2,1]).weight_lattice()
          sage: Lambda = R.fundamental_weights()
          sage: c = R.cartan_type().translation_factors(); c                      # needs sage.graphs
          Finite family {0: 1, 1: 2, 2: 1}
          sage: R.reduced_word_of_translation((Lambda[1]-Lambda[0]) * c[1])       # needs sage.graphs
          [0, 1, 2, 1]
          sage: R.reduced_word_of_translation((Lambda[2]-Lambda[0]) * c[2])       # needs sage.graphs
          [0, 1, 0]

      [Python]

          >>> from sage.all import *
          >>> R = RootSystem(["C",Integer(2),Integer(1)]).weight_lattice()
          >>> Lambda = R.fundamental_weights()
          >>> c = R.cartan_type().translation_factors(); c                      # needs sage.graphs
          Finite family {0: 1, 1: 2, 2: 1}
          >>> R.reduced_word_of_translation((Lambda[Integer(1)]-Lambda[Integer(0)]) * c[Integer(1)])       # needs sage.graphs
          [0, 1, 2, 1]
          >>> R.reduced_word_of_translation((Lambda[Integer(2)]-Lambda[Integer(0)]) * c[Integer(2)])       # needs sage.graphs
          [0, 1, 0]

      See also `_test_reduced_word_of_translation()`.

      Todo

      - Add a picture in the doc

      - Add a method which, given an element of the classical weight lattice, constructs the appropriate value for t

    rho()

      EXAMPLES:

      Sage

          sage: RootSystem(['A',3]).ambient_lattice().rho()
          (3, 2, 1, 0)

      Python

          >>> from sage.all import *
          >>> RootSystem(['A',Integer(3)]).ambient_lattice().rho()
          (3, 2, 1, 0)

    rho_classical()

      Return the embedding at level 0 of \(\rho\) of the classical lattice.

      EXAMPLES:

      Sage

          sage: RootSystem(['C',4,1]).weight_lattice().rho_classical()            # needs sage.graphs
          -4*Lambda[0] + Lambda[1] + Lambda[2] + Lambda[3] + Lambda[4]
          sage: L = RootSystem(['D',4,1]).weight_lattice()
          sage: L.rho_classical().scalar(L.null_coroot())                         # needs sage.graphs
          0

      Python

          >>> from sage.all import *
          >>> RootSystem(['C',Integer(4),Integer(1)]).weight_lattice().rho_classical()            # needs sage.graphs
          -4*Lambda[0] + Lambda[1] + Lambda[2] + Lambda[3] + Lambda[4]
          >>> L = RootSystem(['D',Integer(4),Integer(1)]).weight_lattice()
          >>> L.rho_classical().scalar(L.null_coroot())                         # needs sage.graphs
          0

      Warning

      In affine type BC dual, this does not live in the weight lattice:

      Sage

          sage: L = CartanType(["BC",2,2]).dual().root_system().weight_space()
          sage: L.rho_classical()                                             # needs sage.graphs
          -3/2*Lambda[0] + Lambda[1] + Lambda[2]
          sage: L = CartanType(["BC",2,2]).dual().root_system().weight_lattice()
          sage: L.rho_classical()                                             # needs sage.graphs
          Traceback (most recent call last):
          ...
          ValueError: 5 is not divisible by 2

      Python

          >>> from sage.all import *
          >>> L = CartanType(["BC",Integer(2),Integer(2)]).dual().root_system().weight_space()
          >>> L.rho_classical()                                             # needs sage.graphs
          -3/2*Lambda[0] + Lambda[1] + Lambda[2]
          >>> L = CartanType(["BC",Integer(2),Integer(2)]).dual().root_system().weight_lattice()
          >>> L.rho_classical()                                             # needs sage.graphs
          Traceback (most recent call last):
          ...
          ValueError: 5 is not divisible by 2

    signs_of_alcovewalk(walk)

      Let walk = \([i_1,\ldots,i_n]\) denote an alcove walk starting from the fundamental alcove \(y_0\), crossing at step 1 the wall \(i_1\), and so on.

      For each \(k\), set \(w_k = s_{i_1} \circ s_{i_k}\), and denote by \(y_k = w_k(y_0)\) the alcove reached after \(k\) steps. Then, \(y_k\) is obtained recursively from \(y_{k-1}\) by applying the following reflection:

      \[y_k = s_{w_{k-1} \alpha_{i_k}} y_{k-1}.\]

      The step is said positive if \(w_{k-1} \alpha_{i_k}\) is a negative root (considering \(w_{k-1}\) as element of the classical Weyl group and \(\alpha_{i_k}\) as a classical root) and negative otherwise. The algorithm implemented here use the equivalent property:

          .. MATH:: \langle w_{k-1}^{-1} \rho_0, \alpha^\vee_{i_k}\rangle > 0

      Where \(\rho_0\) is the sum of the classical fundamental weights embedded at level 0 in this space (see `rho_classical()`), and \(\alpha^\vee_{i_k}\) is the simple coroot associated to \(\alpha_{i_k}\).

      This function returns a list of the form \([+1,+1,-1,...]\), where the \(k\)-th entry denotes whether the \(k\)-th step was positive or negative.

      See equation 3.4, of Ram: Alcove walks …, arXiv math/0601343v1

      EXAMPLES:

      Sage

          sage: L = RootSystem(['C',2,1]).weight_lattice()
          sage: L.signs_of_alcovewalk([1,2,0,1,2,1,2,0,1,2])                      # needs sage.libs.gap
          [-1, -1, 1, -1, 1, 1, 1, 1, 1, 1]

          sage: L = RootSystem(['A',2,1]).weight_lattice()
          sage: L.signs_of_alcovewalk([0,1,2,1,2,0,1,2,0,1,2,0])                  # needs sage.libs.gap
          [1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1]

          sage: L = RootSystem(['B',2,1]).coweight_lattice()
          sage: L.signs_of_alcovewalk([0,1,2,0,1,2])                              # needs sage.libs.gap
          [1, -1, 1, -1, 1, 1]

      Python

          >>> from sage.all import *
          >>> L = RootSystem(['C',Integer(2),Integer(1)]).weight_lattice()
          >>> L.signs_of_alcovewalk([Integer(1),Integer(2),Integer(0),Integer(1),Integer(2),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2)])                      # needs sage.libs.gap
          [-1, -1, 1, -1, 1, 1, 1, 1, 1, 1]

          >>> L = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice()
          >>> L.signs_of_alcovewalk([Integer(0),Integer(1),Integer(2),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2),Integer(0)])                  # needs sage.libs.gap
          [1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1]

          >>> L = RootSystem(['B',Integer(2),Integer(1)]).coweight_lattice()
          >>> L.signs_of_alcovewalk([Integer(0),Integer(1),Integer(2),Integer(0),Integer(1),Integer(2)])                              # needs sage.libs.gap
          [1, -1, 1, -1, 1, 1]

      Warning

      This method currently does not work in the weight lattice for type BC dual because \(\rho_0\) does not live in this lattice (but an integral multiple of it would do the job as well).

    simple_root(i)

      Return the \(i\)-th simple root.

      This default implementation takes the \(i\)-th simple root in the weight lattice and embeds it in `self`.

      EXAMPLES:

      Since all the weight lattice realizations in Sage currently implement a `simple_root` method, we have to call this one by hand:

      Sage

          sage: from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
          sage: simple_root = WeightLatticeRealizations(QQ).parent_class.simple_root.f
          sage: L = RootSystem("A3").ambient_space()
          sage: simple_root(L, 1)                                                 # needs sage.graphs
          (1, -1, 0, 0)
          sage: simple_root(L, 2)                                                 # needs sage.graphs
          (0, 1, -1, 0)
          sage: simple_root(L, 3)                                                 # needs sage.graphs
          (1, 1, 2, 0)

      Python

          >>> from sage.all import *
          >>> from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
          >>> simple_root = WeightLatticeRealizations(QQ).parent_class.simple_root.f
          >>> L = RootSystem("A3").ambient_space()
          >>> simple_root(L, Integer(1))                                                 # needs sage.graphs
          (1, -1, 0, 0)
          >>> simple_root(L, Integer(2))                                                 # needs sage.graphs
          (0, 1, -1, 0)
          >>> simple_root(L, Integer(3))                                                 # needs sage.graphs
          (1, 1, 2, 0)

      Note that this last root differs from the one implemented in `L` by a multiple of the vector `(1,1,1,1)`:

      Sage

          sage: L.simple_roots()                                                  # needs sage.graphs
          Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}

      [Python]

          >>> from sage.all import *
          >>> L.simple_roots()                                                  # needs sage.graphs
          Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}

      This is a harmless artefact of the \(SL\) versus \(GL\) interpretation of type \(A\); see the thematic tutorial on Lie Methods and Related Combinatorics in Sage for details.

    weyl_dimension(highest_weight)

      Return the dimension of the highest weight representation of highest weight `highest_weight`.

      EXAMPLES:

      Sage

          sage: RootSystem(['A',3]).ambient_lattice().weyl_dimension([2,1,0,0])
          20
          sage: P = RootSystem(['C',2]).weight_lattice()
          sage: La = P.basis()
          sage: P.weyl_dimension(La[1]+La[2])                                     # needs sage.graphs
          16

          sage: type(RootSystem(['A',3]).ambient_lattice().weyl_dimension([2,1,0,0]))
          <class 'sage.rings.integer.Integer'>

      Python

          >>> from sage.all import *
          >>> RootSystem(['A',Integer(3)]).ambient_lattice().weyl_dimension([Integer(2),Integer(1),Integer(0),Integer(0)])
          20
          >>> P = RootSystem(['C',Integer(2)]).weight_lattice()
          >>> La = P.basis()
          >>> P.weyl_dimension(La[Integer(1)]+La[Integer(2)])                                     # needs sage.graphs
          16

          >>> type(RootSystem(['A',Integer(3)]).ambient_lattice().weyl_dimension([Integer(2),Integer(1),Integer(0),Integer(0)]))
          <class 'sage.rings.integer.Integer'>

  super_categories()

    EXAMPLES:

    Sage

        sage: from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
        sage: WeightLatticeRealizations(QQ).super_categories()
        [Category of root lattice realizations over Rational Field]

    Python

        >>> from sage.all import *
        >>> from sage.combinat.root_system.weight_lattice_realizations import WeightLatticeRealizations
        >>> WeightLatticeRealizations(QQ).super_categories()
        [Category of root lattice realizations over Rational Field]