In this document we briefly explain the construction and implementation of the Kirillov–Reshetikhin crystals of [FourierEtAl2009].
Kirillov–Reshetikhin (KR) crystals are finite-dimensional affine crystals
corresponding to Kirillov–Reshektikhin modules. They were first conjectured to
exist in [HatayamaEtAl2001]. The proof of their existence for nonexceptional
types was given in [OkadoSchilling2008] and their combinatorial models were
constructed in [FourierEtAl2009]. Kirillov-Reshetikhin crystals
are indexed first by their type (like
,
, ...) with
underlying index set
and two integers
and
.
The integers
only needs to satisfy
, whereas
is a node of the
finite Dynkin diagram
.
Their construction relies on several cases which we discuss separately. In
all cases when removing the zero arrows, the crystal decomposes as a (direct
sum of) classical crystals which gives the crystal structure for the index
set . Then the zero arrows are added by either
exploiting a symmetry of the Dynkin diagram or by using embeddings of crystals.
The Dynkin diagram for affine type has a rotational symmetry mapping
where we view the indices modulo
:
sage: C = CartanType(['A',3,1])
sage: C.dynkin_diagram()
0
O-------+
| |
| |
O---O---O
1 2 3
A3~
The classical decomposition of is the
highest weight
crystal
or equivalently the crystal of tableaux labelled
by the rectangular partition
:
In Sage we can see this via:
sage: K = crystals.KirillovReshetikhin(['A',3,1],1,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['A', 3] and shape(s) [[1]]
sage: K.list()
[[[1]], [[2]], [[3]], [[4]]]
sage: K = crystals.KirillovReshetikhin(['A',3,1],2,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['A', 3] and shape(s) [[1, 1]]
One can change between the classical and affine crystal using the methods lift and retract:
sage: K = crystals.KirillovReshetikhin(['A',3,1],2,1)
sage: b = K(rows=[[1],[3]]); type(b)
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_A_with_category.element_class'>
sage: b.lift()
[[1], [3]]
sage: type(b.lift())
<class 'sage.combinat.crystals.tensor_product.CrystalOfTableaux_with_category.element_class'>
sage: b = crystals.Tableaux(['A',3], shape = [1,1])(rows=[[1],[3]])
sage: K.retract(b)
[[1], [3]]
sage: type(K.retract(b))
<class 'sage.combinat.crystals.kirillov_reshetikhin.KR_type_A_with_category.element_class'>
The -arrows are obtained using the analogue of
, called
the promotion operator
, on the level of crystals via:
In Sage this can be achieved as follows:
sage: K = crystals.KirillovReshetikhin(['A',3,1],2,1)
sage: b = K.module_generator(); b
[[1], [2]]
sage: b.f(0)
sage: b.e(0)
[[2], [4]]
sage: K.promotion()(b.lift())
[[2], [3]]
sage: K.promotion()(b.lift()).e(1)
[[1], [3]]
sage: K.promotion_inverse()(K.promotion()(b.lift()).e(1))
[[2], [4]]
KR crystals are level crystals, meaning that the weight of all
elements in these crystals is zero:
sage: K = crystals.KirillovReshetikhin(['A',3,1],2,1)
sage: b = K.module_generator(); b.weight()
-Lambda[0] + Lambda[2]
sage: b.weight().level()
0
The KR crystal of type
looks as follows:
In Sage this can be obtained via:
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: G = K.digraph()
sage: view(G, tightpage=True) # optional - dot2tex graphviz
The Dynkin diagrams for types ,
,
are invariant under interchanging nodes
and
:
sage: n = 5
sage: C = CartanType(['D',n,1]); C.dynkin_diagram()
0 O O 5
| |
| |
O---O---O---O
1 2 3 4
D5~
sage: C = CartanType(['B',n,1]); C.dynkin_diagram()
O 0
|
|
O---O---O---O=>=O
1 2 3 4 5
B5~
sage: C = CartanType(['A',2*n-1,2]); C.dynkin_diagram()
O 0
|
|
O---O---O---O=<=O
1 2 3 4 5
B5~*
The underlying classical algebras obtained when removing node are
type
, respectively. The classical
decomposition into a
crystal is a direct sum:
where is obtained from
(or equivalently a rectangular
partition of shape
) by removing vertical dominoes. This in fact
only holds in the ranges
for type
, and
for types
and
:
sage: K = crystals.KirillovReshetikhin(['D',6,1],4,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 6] and shape(s)
[[], [1, 1], [1, 1, 1, 1], [2, 2], [2, 2, 1, 1], [2, 2, 2, 2]]
For type and
, one needs to be aware that
is a spin weight and hence corresponds in the partition language to a
column of height
and width
:
sage: K = crystals.KirillovReshetikhin(['B',3,1],3,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[1/2, 1/2, 1/2]]
As for type , the Dynkin automorphism induces a promotion-type
operator
on the level of crystals. In this case in can however
happen that the automorphism changes between classical components:
sage: K = crystals.KirillovReshetikhin(['D',4,1],2,1)
sage: b = K.module_generator(); b
[[1], [2]]
sage: K.automorphism(b)
[[2], [-1]]
sage: b = K(rows=[[2],[-2]])
sage: K.automorphism(b)
[]
This operator is used to define the affine crystal operators:
The KR crystals of types
,
,
and
are, respectively:
The Dynkin diagram of type has a symmetry
:
sage: C = CartanType(['C',4,1]); C.dynkin_diagram()
O=>=O---O---O=<=O
0 1 2 3 4
C4~
The classical subalgebra when removing the 0 node is of type .
However, in this case the crystal is not constructed using
, but rather using a virtual crystal construction.
of type
is realized inside
of type
using:
where and
are the crystal operator in the
ambient crystal
:
sage: K = crystals.KirillovReshetikhin(['C',3,1],1,2); K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['B', 4, 1]^* with (r,s)=(1,2)
The classical decomposition for is given by:
where is obtained from
(or equivalently a rectangular
partition of shape
) by removing horizontal dominoes:
sage: K = crystals.KirillovReshetikhin(['C',3,1],2,4)
sage: K.classical_decomposition()
The crystal of tableaux of type ['C', 3] and shape(s) [[], [2], [4], [2, 2], [4, 2], [4, 4]]
The KR crystal of type
looks as follows:
The Dynkin diagrams of types and
look as follows:
sage: C = CartanType(['D',5,2]); C.dynkin_diagram()
O=<=O---O---O=>=O
0 1 2 3 4
C4~*
sage: C = CartanType(['A',8,2]); C.dynkin_diagram()
O=<=O---O---O=<=O
0 1 2 3 4
BC4~
The classical subdiagram is of type for type
and
of type
for type
. The classical decomposition for these
KR crystals for
for type
and
for type
is given by:
where is obtained from
(or equivalently a rectangular
partition of shape
) by removing single boxes:
sage: K = crystals.KirillovReshetikhin(['D',5,2],2,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 4] and shape(s) [[], [1], [2], [1, 1], [2, 1], [2, 2]]
sage: K = crystals.KirillovReshetikhin(['A',8,2],2,2)
sage: K.classical_decomposition()
The crystal of tableaux of type ['C', 4] and shape(s) [[], [1], [2], [1, 1], [2, 1], [2, 2]]
The KR crystals are constructed using an injective map into a KR crystal of type
where
sage: K = crystals.KirillovReshetikhin(['D',5,2],1,2); K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['C', 4, 1] with (r,s)=(1,4)
sage: K = crystals.KirillovReshetikhin(['A',8,2],1,2); K.ambient_crystal()
Kirillov-Reshetikhin crystal of type ['C', 4, 1] with (r,s)=(1,4)
The KR crystals of type
and
look as follows:
As you can see from the Dynkin diagram for type , mapping the
nodes
yields the same diagram, but with relabelled nodes. In
this case the classical subdiagram is of type
instead of
. One
can also construct the KR crystal
of type
based on
this classical decomposition. In this case the classical decomposition is
the sum over all weights obtained from
by removing
horizontal dominoes:
sage: C = CartanType(['A',6,2]).dual()
sage: Kdual = crystals.KirillovReshetikhin(C,2,2)
sage: Kdual.classical_decomposition()
The crystal of tableaux of type ['B', 3] and shape(s) [[], [2], [2, 2]]
Looking at the picture, one can see that this implementation is
isomorphic to the other implementation based on the decomposition
up to a relabeling of the arrows:
sage: C = CartanType(['A',4,2])
sage: K = crystals.KirillovReshetikhin(C,1,1)
sage: Kdual = crystals.KirillovReshetikhin(C.dual(),1,1)
sage: G = K.digraph()
sage: Gdual = Kdual.digraph()
sage: f = { 1:1, 0:2, 2:0 }
sage: for u,v,label in Gdual.edges():
....: Gdual.set_edge_label(u,v,f[label])
sage: G.is_isomorphic(Gdual, edge_labels = True, certify = True)
(True, {[[1]]: [[-2]], [[2]]: [[-1]], [[-2]]: [[1]], [[-1]]: [[2]], []: [[0]]})
The KR crystals for types
and
were
excluded from the above discussion. They are associated to the exceptional
node
and in this case the classical decomposition is irreducible:
In Sage:
sage: K = crystals.KirillovReshetikhin(['C',2,1],2,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['C', 2] and shape(s) [[1, 1]]
sage: K = crystals.KirillovReshetikhin(['D',3,2],2,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['B', 2] and shape(s) [[1/2, 1/2]]
The KR crystals and
of type
are also special. They decompose as:
sage: K = crystals.KirillovReshetikhin(['D',4,1],4,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[1/2, 1/2, 1/2, 1/2]]
sage: K = crystals.KirillovReshetikhin(['D',4,1],3,1)
sage: K.classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[1/2, 1/2, 1/2, -1/2]]
In [JonesEtAl2010] the KR crystals for
in type
were constructed exploiting again a Dynkin diagram automorphism, namely the
automorphism
of order 3 which maps
:
sage: C = CartanType(['E',6,1]); C.dynkin_diagram()
O 0
|
|
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6~
The crystals and
are irreducible as classical crystals:
sage: K = crystals.KirillovReshetikhin(['E',6,1],1,1)
sage: K.classical_decomposition()
Direct sum of the crystals Family (Finite dimensional highest weight crystal of type ['E', 6] and highest weight Lambda[1],)
sage: K = crystals.KirillovReshetikhin(['E',6,1],6,1)
sage: K.classical_decomposition()
Direct sum of the crystals Family (Finite dimensional highest weight crystal of type ['E', 6] and highest weight Lambda[6],)
whereas for the adjoint node we have the decomposition
sage: K = crystals.KirillovReshetikhin(['E',6,1],2,1)
sage: K.classical_decomposition()
Direct sum of the crystals Family (Finite dimensional highest weight crystal of type ['E', 6] and highest weight 0,
Finite dimensional highest weight crystal of type ['E', 6] and highest weight Lambda[2])
The promotion operator on the crystal corresponding to can be calculated explicitly:
sage: K = crystals.KirillovReshetikhin(['E',6,1],1,1)
sage: promotion = K.promotion()
sage: u = K.module_generator(); u
[(1,)]
sage: promotion(u.lift())
[(-1, 6)]
The crystal is already of dimension 27. The elements
of this
crystal are labelled by tuples which specify their nonzero
and
. For example,
indicates that
and all others are equal to zero:
sage: K = crystals.KirillovReshetikhin(['E',6,1],1,1)
sage: K.cardinality()
27
A single column KR crystal is for any
.
In [LNSSS14I] and [LNSSS14II], it was shown that single column KR
crystals can be constructed by projecting level 0 crystals of LS paths onto
the classical weight lattice. We first verify that we do get an isomorphic
crystal for in type
:
sage: K = crystals.KirillovReshetikhin(['E',6,1], 1,1)
sage: K2 = crystals.kirillov_reshetikhin.LSPaths(['E',6,1], 1,1)
sage: K.digraph().is_isomorphic(K2.digraph(), edge_labels=True)
True
Here is an example in and we calculate its
classical decomposition:
sage: K = crystals.kirillov_reshetikhin.LSPaths(['E',8,1], 8,1)
sage: K.cardinality()
249
sage: L = [x for x in K if x.is_highest_weight([1,2,3,4,5,6,7,8])]
sage: map(lambda x: x.weight(), L)
[-2*Lambda[0] + Lambda[8], 0]
An important notion for finite-dimensional affine crystals is perfectness.
The crucial property is that a crystal is perfect of level
if there
is a bijection between level
dominant weights and elements in
For a precise definition of perfect crystals see [HongKang2002] .
In [FourierEtAl2010] it was proven that for the nonexceptional types
is perfect as long as
is an integer. Here
except
for
in type
and
in type
.
Here we verify this using Sage for of type
:
sage: K = crystals.KirillovReshetikhin(['C',3,1],1,1)
sage: Lambda = K.weight_lattice_realization().fundamental_weights(); Lambda
Finite family {0: Lambda[0], 1: Lambda[1], 2: Lambda[2], 3: Lambda[3]}
sage: [w.level() for w in Lambda]
[1, 1, 1, 1]
sage: Bmin = [b for b in K if b.Phi().level() == 1 ]; Bmin
[[[1]], [[2]], [[3]], [[-3]], [[-2]], [[-1]]]
sage: [b.Phi() for b in Bmin]
[Lambda[1], Lambda[2], Lambda[3], Lambda[2], Lambda[1], Lambda[0]]
As you can see, both and
satisfy
.
Hence there is no bijection between the minimal elements in
and level 1 weights. Therefore,
of type
is not perfect. However,
of type
is a perfect crystal:
sage: K = crystals.KirillovReshetikhin(['C',3,1],1,2)
sage: Lambda = K.weight_lattice_realization().fundamental_weights()
sage: Bmin = [b for b in K if b.Phi().level() == 1 ]
sage: [b.Phi() for b in Bmin]
[Lambda[0], Lambda[3], Lambda[2], Lambda[1]]
Perfect crystals can be used to construct infinite-dimensional highest weight crystals and Demazure crystals using the Kyoto path model [KKMMNN1992]. We construct Example 10.6.5 in [HongKang2002]:
sage: K = crystals.KirillovReshetikhin(['A',1,1], 1,1)
sage: La = RootSystem(['A',1,1]).weight_lattice().fundamental_weights()
sage: B = crystals.KyotoPathModel(K, La[0])
sage: B.highest_weight_vector()
[[[2]]]
sage: K = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: B = crystals.KyotoPathModel(K, La[0])
sage: B.highest_weight_vector()
[[[3]]]
sage: K = crystals.KirillovReshetikhin(['C',2,1], 2,1)
sage: La = RootSystem(['C',2,1]).weight_lattice().fundamental_weights()
sage: B = crystals.KyotoPathModel(K, La[1])
sage: B.highest_weight_vector()
[[[2], [-2]]]
For tensor products of Kirillov-Reshehtikhin crystals, there also exists
the important notion of the energy function. It can be defined as the sum
of certain local energy functions and the -matrix. In Theorem 7.5 in
[SchillingTingley2011] it was shown that for perfect crystals of the
same level the energy
is the same as the affine grading (up to a
normalization). The affine grading is defined as the minimal number of
applications of
to
to reach a ground state path. Computationally,
this algorithm is a lot more efficient than the computation involving
the
-matrix and has been implemented in Sage:
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: T = crystals.TensorProduct(K,K,K)
sage: hw = [b for b in T if all(b.epsilon(i)==0 for i in [1,2])]
sage: for b in hw:
....: print b, b.energy_function()
[[[1]], [[1]], [[1]]] 0
[[[1]], [[2]], [[1]]] 2
[[[2]], [[1]], [[1]]] 1
[[[3]], [[2]], [[1]]] 3
The affine grading can be computed even for nonperfect crystals:
sage: K = crystals.KirillovReshetikhin(['C',4,1],1,2)
sage: K1 = crystals.KirillovReshetikhin(['C',4,1],1,1)
sage: T = crystals.TensorProduct(K,K1)
sage: hw = [b for b in T if all(b.epsilon(i)==0 for i in [1,2,3,4])]
sage: for b in hw:
....: print b, b.affine_grading()
....:
[[], [[1]]] 1
[[[1, 1]], [[1]]] 2
[[[1, 2]], [[1]]] 1
[[[1, -1]], [[1]]] 0
The one-dimensional configuration sum of a crystal is the graded sum
by energy of the weight of all elements
:
Here is an example of how you can compute the one-dimensional configuration sum in Sage:
sage: K = crystals.KirillovReshetikhin(['A',2,1],1,1)
sage: T = crystals.TensorProduct(K,K)
sage: T.one_dimensional_configuration_sum()
B[-2*Lambda[1] + 2*Lambda[2]] + (q+1)*B[-Lambda[1]]
+ (q+1)*B[Lambda[1] - Lambda[2]] + B[2*Lambda[1]]
+ B[-2*Lambda[2]] + (q+1)*B[Lambda[2]]