Affinization Crystals

class sage.combinat.crystals.affinization.AffinizationOfCrystal(B)

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

An affiniziation of a crystal.

Let \mathfrak{g} be a Kac-Moody algebra of affine type. The affinization of a finite U_q^{\prime}(\mathfrak{g})-crystal B is the (infinite) U_q(\mathfrak{g})-crystal with underlying set:

B^{\mathrm{aff}} = \{ b(m) \mid b \in B, m \in \ZZ \}

and crystal structure determined by:

\begin{aligned}
    e_i(b(m)) & =
    \begin{cases}
      (e_0 b)(m+1) & i = 0, \\
      (e_i b)(m)   & i \neq 0,
    \end{cases} \\
    f_i(b(m)) &=
    \begin{cases}
      (f_0 b)(m-1) & i = 0, \\
      (f_i b)(m)   & i \neq 0,
    \end{cases} \\
    \mathrm{wt}(b(m)) &= \mathrm{wt}(b) + m \delta.
\end{aligned}

EXAMPLES:

We first construct a Kirillov-Reshetikhin crystal and then take it’s corresponding affinization:

sage: K = crystals.KirillovReshetikhin(['A',2,1], 2, 2)
sage: A = K.affinization()

Next we construct an affinization crystal from a tensor product of KR crystals:

sage: KT = crystals.TensorProductOfKirillovReshetikhinTableaux(['C',2,1], [[1,2],[2,1]])
sage: A = crystals.AffinizationOf(KT)

REFERENCES:

class Element(parent, b, m)

Bases: sage.structure.element.Element

An element in an affinization crystal.

e(i)

Return the action of e_i on self.

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization()
sage: mg = A.module_generators[0]
sage: mg.e(0)
[[1, 2], [2, 3]](1)
sage: mg.e(1)
sage: mg.e(0).e(1)
[[1, 1], [2, 3]](1)
epsilon(i)

Return \varepsilon_i of self.

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization()
sage: mg = A.module_generators[0]
sage: mg.epsilon(0)
2
sage: mg.epsilon(1)
0
f(i)

Return the action of f_i on self.

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization()
sage: mg = A.module_generators[0]
sage: mg.f(2)
[[1, 1], [2, 3]](0)
sage: mg.f(2).f(2).f(0)
sage: mg.f_string([2,1,1])
sage: mg.f_string([2,1])
[[1, 2], [2, 3]](0)
sage: mg.f_string([2,1,0])
[[1, 1], [2, 2]](-1)
phi(i)

Return \varphi_i of self.

INPUT:

  • i – an element of the index set

EXAMPLES:

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization()
sage: mg = A.module_generators[0]
sage: mg.phi(0)
0
sage: mg.phi(2)
2
weight()

Return the weight of self.

The weight \mathrm{wt} of an element is:

\mathrm{wt}\bigl( b(m) \bigr) = \mathrm{wt}(b) + m \delta,

where \delta is the null root.

EXAMPLES:

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization()
sage: mg = A.module_generators[0]
sage: mg.weight()
-2*Lambda[0] + 2*Lambda[2]
sage: mg.e(0).weight()
-Lambda[1] + Lambda[2] + delta
sage: mg.e(0).e(0).weight()
2*Lambda[0] - 2*Lambda[1] + 2*delta
AffinizationOfCrystal.digraph(subset=None, index_set=None)

Return the DiGraph associated with self. See digraph() for more information.

EXAMPLES:

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization()
sage: S = A.subcrystal(max_depth=3)
sage: G = A.digraph(subset=S)
AffinizationOfCrystal.weight_lattice_realization()

Return the weight lattice realization of self.

EXAMPLES:

sage: A = crystals.KirillovReshetikhin(['A',2,1], 1, 1).affinization()
sage: A.weight_lattice_realization()
Extended weight lattice of the Root system of type ['A', 2, 1]