Affine Crystals

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

Bases: UniqueRepresentation, 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: KirillovReshetikhinCrystal()).

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 = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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

cardinality()

Return the cardinality of self.

EXAMPLES:

sage: C = crystals.Tableaux(['A',3],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['A',3,1],C,pr,pr_inverse,1)
sage: A.cardinality() == C.cardinality()
True

lift(affine_elt)

Lift an affine crystal element to the corresponding classical crystal element.

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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]]

retract(classical_elt)

Transform a classical crystal element to the corresponding affine crystal element.

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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, category=None)

Bases: 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 – an integer specifying the classical node in the image of the zero node under the automorphism sigma

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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)

Give the analogue of the affine Dynkin diagram automorphism on the level of crystals.

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b = A.list()[0]
sage: A.automorphism(b)
[[2]]

inverse_automorphism(x)

Give the analogue of the inverse of the affine Dynkin diagram automorphism on the level of crystals.

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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: 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) crystal.

Since this class inherits from AffineCrystalFromClassicalElement, 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 = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b = A(rows=[[1]])
sage: b._repr_()
'[[1]]'

e0()

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

EXAMPLES:

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

epsilon0()

Implement \(epsilon_0\) using the automorphism.

EXAMPLES:

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

f0()

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

EXAMPLES:

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

phi0()

Implement \(phi_0\) using the automorphism.

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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: 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) crystal.

EXAMPLES:

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

classical_weight()

Return the classical weight corresponding to self.

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['A',n,1],C,pr,pr_inverse,1)
sage: b = A(rows=[[1]])
sage: b.classical_weight()
(1, 0, 0)

e(i)

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

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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)

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

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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 \(\varepsilon_0\) from the super class, but should be implemented if a faster implementation exists.

EXAMPLES:

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

f(i)

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

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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()

Lift an affine crystal element to the corresponding classical crystal element.

EXAMPLES:

sage: n = 2
sage: C = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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 = crystals.Tableaux(['A',n],shape=[1])
sage: pr = attrcall("promotion")
sage: pr_inverse = attrcall("promotion_inverse")
sage: A = crystals.AffineFromClassicalAndPromotion(['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 \(\varphi_0\) from the super class, but should be implemented if a faster implementation exists.

EXAMPLES:

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

pp()

Method for pretty printing.

EXAMPLES:

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