Affine Crystals

class sage.combinat.crystals.affine.AffineCrystalFromClassical(cartan_type, classical_crystal, category=None)

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

This abstract class can be used for affine crystals that are constructed from a classical crystal. The zero arrows can be implemented using different methods (for example using a Dynkin diagram automorphisms or virtual crystals).

This is a helper class, mostly used to implement Kirillov-Reshetikhin crystals (see: sage.combinat.crystals.kirillov_reshetikhin.KirillovReshetikhin()).

For general information about crystals see sage.combinat.crystals.

INPUT:

  • cartan_type - The Cartan type of the resulting affine crystal
  • classical_crystal - instance of a classical crystal.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: A.list()
[[[1]], [[2]], [[3]]]
sage: A.cartan_type()
['A', 2, 1]
sage: A.index_set()
(0, 1, 2)
sage: b=A(rows=[[1]])
sage: b.weight()
-Lambda[0] + Lambda[1]
sage: b.classical_weight()
(1, 0, 0)
sage: [x.s(0) for x in A.list()]
[[[3]], [[2]], [[1]]]
sage: [x.s(1) for x in A.list()]
[[[2]], [[1]], [[3]]]
Element

alias of AffineCrystalFromClassicalElement

lift(affine_elt)

Lifts an affine crystal element to the corresponding classical crystal element

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A.list()[0]
sage: A.lift(b)
[[1]]
sage: A.lift(b).parent()
The crystal of tableaux of type ['A', 2] and shape(s) [[1]]
list()

Returns the list of all crystal elements using the underlying classical crystal

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: A.list()
[[[1]], [[2]], [[3]]]
retract(classical_elt)

Transforms a classical crystal element to the corresponding affine crystal element

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: t=C(rows=[[1]])
sage: t.parent()
The crystal of tableaux of type ['A', 2] and shape(s) [[1]]
sage: A.retract(t)
[[1]]
sage: A.retract(t).parent() is A
True
class sage.combinat.crystals.affine.AffineCrystalFromClassicalAndPromotion(cartan_type, classical_crystal, p_automorphism, p_inverse_automorphism, dynkin_node)

Bases: sage.combinat.crystals.affine.AffineCrystalFromClassical

Crystals that are constructed from a classical crystal and a Dynkin diagram automorphism \(\sigma\). In type \(A_n\), the Dynkin diagram automorphism is \(i \to i+1 \pmod n+1\) and the corresponding map on the crystal is the promotion operation \(\mathrm{pr}\) on tableaux. The affine crystal operators are given by \(f_0= \mathrm{pr}^{-1} f_{\sigma(0)} \mathrm{pr}\).

INPUT:

  • cartan_type - The Cartan type of the resulting affine crystal
  • classical_crystal - instance of a classical crystal.
  • automorphism, inverse_automorphism - A function on the elements of the classical_crystal
  • dynkin_node - Integer specifying the classical node in the image of the zero node under the automorphism sigma.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: A.list()
[[[1]], [[2]], [[3]]]
sage: A.cartan_type()
['A', 2, 1]
sage: A.index_set()
(0, 1, 2)
sage: b=A(rows=[[1]])
sage: b.weight()
-Lambda[0] + Lambda[1]
sage: b.classical_weight()
(1, 0, 0)
sage: [x.s(0) for x in A.list()]
[[[3]], [[2]], [[1]]]
sage: [x.s(1) for x in A.list()]
[[[2]], [[1]], [[3]]]
Element

alias of AffineCrystalFromClassicalAndPromotionElement

automorphism(x)

Gives the analogue of the affine Dynkin diagram automorphism on the level of crystals

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A.list()[0]
sage: A.automorphism(b)
[[2]]
inverse_automorphism(x)

Gives the analogue of the inverse of the affine Dynkin diagram automorphism on the level of crystals

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A.list()[0]
sage: A.inverse_automorphism(b)
[[3]]
class sage.combinat.crystals.affine.AffineCrystalFromClassicalAndPromotionElement

Bases: sage.combinat.crystals.affine.AffineCrystalFromClassicalElement

Elements of crystals that are constructed from a classical crystal and a Dynkin diagram automorphism. In type A, the automorphism is the promotion operation on tableaux.

This class is not instantiated directly but rather __call__ed from AffineCrystalFromClassicalAndPromotion. The syntax of this is governed by the (classical) CrystalOfTableaux.

Since this class inherits from AffineClassicalFromClassicalElement, the methods that need to be implemented are e0, f0 and possibly epsilon0 and phi0 if more efficient algorithms exist.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A(rows=[[1]])
sage: b._repr_()
'[[1]]'
e0()

Implements \(e_0\) using the automorphism as \(e_0 = \operatorname{pr}^{-1} e_{dynkin_node} \operatorname{pr}\)

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A(rows=[[1]])
sage: b.e0()
[[3]]
epsilon0()

Implements \(epsilon_0\) using the automorphism.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: [x.epsilon0() for x in A.list()]
[1, 0, 0]
f0()

Implements \(f_0\) using the automorphism as \(f_0 = \operatorname{pr}^{-1} f_{dynkin_node} \operatorname{pr}\)

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A(rows=[[3]])
sage: b.f0()
[[1]]
phi0()

Implements \(phi_0\) using the automorphism.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: [x.phi0() for x in A.list()]
[0, 0, 1]
class sage.combinat.crystals.affine.AffineCrystalFromClassicalElement

Bases: sage.structure.element_wrapper.ElementWrapper

Elements of crystals that are constructed from a classical crystal. The elements inherit many of their methods from the classical crystal using lift and retract.

This class is not instantiated directly but rather __call__ed from AffineCrystalFromClassical. The syntax of this is governed by the (classical) CrystalOfTableaux.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A(rows=[[1]])
sage: b._repr_()
'[[1]]'
classical_weight()

Returns the classical weight corresponding to self.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A(rows=[[1]])
sage: b.classical_weight()
(1, 0, 0)
e(i)

Returns the action of \(e_i\) on self.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A(rows=[[1]])
sage: b.e(0)
[[3]]
sage: b.e(1)
e0()

Assumes that \(e_0\) is implemented separately.

epsilon(i)

Returns the maximal time the crystal operator \(e_i\) can be applied to self.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: [x.epsilon(0) for x in A.list()]
[1, 0, 0]
sage: [x.epsilon(1) for x in A.list()]
[0, 1, 0]
epsilon0()

Uses \(epsilon_0\) from the super class, but should be implemented if a faster implementation exists.

f(i)

Returns the action of \(f_i\) on self.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A(rows=[[3]])
sage: b.f(0)
[[1]]
sage: b.f(2)
f0()

Assumes that \(f_0\) is implemented separately.

lift()

Lifts an affine crystal element to the corresponding classical crystal element

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b=A.list()[0]
sage: b.lift()
[[1]]
sage: b.lift().parent()
The crystal of tableaux of type ['A', 2] and shape(s) [[1]]
phi(i)

Returns the maximal time the crystal operator \(f_i\) can be applied to self.

EXAMPLES:

sage: n=2
sage: C=CrystalOfTableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A=AffineCrystalFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: [x.phi(0) for x in A.list()]
[0, 0, 1]
sage: [x.phi(1) for x in A.list()]
[1, 0, 0]
phi0()

Uses \(phi_0\) from the super class, but should be implemented if a faster implementation exists.

pp()

Method for pretty printing

EXAMPLES:

sage: K = KirillovReshetikhinCrystal(['D',3,2],1,1)
sage: t=K(rows=[[1]])
sage: t.pp()
1

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