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 B^{r,s} are indexed first by their type (like A_n^{(1)}, B_n^{(1)}, ...) with underlying index set I = \{0,1,\ldots, n\} and two integers r and s. The integers s only needs to satisfy s >0, whereas r is a node of the finite Dynkin diagram r \in I \setminus \{0\}.
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 I_0 = \{ 1,2,\ldots, n\}. 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 A has a rotational symmetry mapping \sigma: i \mapsto i+1 where we view the indices modulo n+1:
sage: C = CartanType(['A',3,1])
sage: C.dynkin_diagram()
0
O-------+
| |
| |
O---O---O
1 2 3
A3~
The classical decomposition of B^{r,s} is the A_n highest weight crystal B(s\omega_r) or equivalently the crystal of tableaux labelled by the rectangular partition (s^r):
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 0-arrows are obtained using the analogue of \sigma, called the promotion operator \mathrm{pr}, 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 0 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 B^{1,1} of type A_2^{(1)} 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, pdflatex=True, tightpage=True) # optional - dot2tex graphviz
The Dynkin diagrams for types D_n^{(1)}, B_n^{(1)}, A_{2n-1}^{(2)} are invariant under interchanging nodes 0 and 1:
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 0 are type \mathfrak{g}_0 = D_n, B_n, C_n, respectively. The classical decomposition into a \mathfrak{g}_0 crystal is a direct sum:
where \lambda is obtained from s\omega_r (or equivalently a rectangular partition of shape (s^r)) by removing vertical dominoes. This in fact only holds in the ranges 1\le r\le n-2 for type D_n^{(1)}, and 1 \le r \le n for types B_n^{(1)} and A_{2n-1}^{(2)}:
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 B_n^{(1)} and r=n, one needs to be aware that \omega_n is a spin weight and hence corresponds in the partition language to a column of height n and width 1/2:
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 A_n^{(1)}, the Dynkin automorphism induces a promotion-type operator \sigma 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 \sigma is used to define the affine crystal operators:
The KR crystals B^{1,1} of types D_3^{(1)}, B_2^{(1)}, and A_5^{(2)} are, respectively:
The Dynkin diagram of type C_n^{(1)} has a symmetry \sigma(i) = n-i:
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 C_n.
However, in this case the crystal B^{r,s} is not constructed using \sigma, but rather using a virtual crystal construction. B^{r,s} of type C_n^{(1)} is realized inside \hat{V}^{r,s} of type A_{2n+1}^{(2)} using:
where \hat{e}_i and \hat{f}_i are the crystal operator in the ambient crystal \hat{V}^{r,s}:
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 1 \le r < n is given by:
where \lambda is obtained from s\omega_r (or equivalently a rectangular partition of shape (s^r)) 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 B^{1,1} of type C_2^{(1)} looks as follows:
The Dynkin diagrams of types D_{n+1}^{(2)} and A_{2n}^{(2)} 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 B_n for type D_{n+1}^{(2)} and of type C_n for type A_{2n}^{(2)}. The classical decomposition for these KR crystals for 1\le r < n for type D_{n+1}^{(2)} and 1 \le r \le n for type A_{2n}^{(2)} is given by:
where \lambda is obtained from s\omega_r (or equivalently a rectangular partition of shape (s^r)) 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 C_n^{(1)}
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 B^{1,1} of type D_3^{(2)} and A_4^{(2)} look as follows:
As you can see from the Dynkin diagram for type A_{2n}^{(2)}, mapping the nodes i\mapsto n-i yields the same diagram, but with relabelled nodes. In this case the classical subdiagram is of type B_n instead of C_n. One can also construct the KR crystal B^{r,s} of type A_{2n}^{(2)} based on this classical decomposition. In this case the classical decomposition is the sum over all weights obtained from s \omega_r 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 C_n 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, {[[-2]]: [[1]], [[-1]]: [[2]], [[1]]: [[-2]], []: [[0]], [[2]]: [[-1]]})
The KR crystals B^{n,s} for types C_n^{(1)} and D_{n+1}^{(2)} were excluded from the above discussion. They are associated to the exceptional node r=n 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 B^{n,s} and B^{n-1,s} of type D_n^{(1)} 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 B^{r,s} for r=1,2,6 in type E_6^{(1)} were constructed exploiting again a Dynkin diagram automorphism, namely the automorphism \sigma of order 3 which maps 0\mapsto 1 \mapsto 6 \mapsto 0:
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 B^{1,s} and B^{6,s} 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 r=2 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 \sigma 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 B^{1,1} is already of dimension 27. The elements b of this crystal are labelled by tuples which specify their nonzero \phi_i(b) and \epsilon_i(b). For example, [-6,2] indicates that \phi_2([-6,2]) = \epsilon_6([-6,2]) = 1 and all others are equal to zero:
sage: K = crystals.KirillovReshetikhin(['E',6,1],1,1)
sage: K.cardinality()
27
An important notion for finite-dimensional affine crystals is perfectness. The crucial property is that a crystal B is perfect of level \ell if there is a bijection between level \ell dominant weights and elements in
For a precise definition of perfect crystals see [HongKang2002] . In [FourierEtAl2010] it was proven that for the nonexceptional types B^{r,s} is perfect as long as s/c_r is an integer. Here c_r=1 except c_r=2 for 1 \le r < n in type C_n^{(1)} and r=n in type B_n^{(1)}.
Here we verify this using Sage for B^{1,1} of type C_3^{(1)}:
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 b=1 and b=-2 satisfy \varphi(b)=\Lambda_1. Hence there is no bijection between the minimal elements in B_{\mathrm{min}} and level 1 weights. Therefore, B^{1,1} of type C_3^{(1)} is not perfect. However, B^{1,2} of type C_n^{(1)} 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 R-matrix. In Theorem 7.5 in [SchillingTingley2011] it was shown that for perfect crystals of the same level the energy D(b) is the same as the affine grading (up to a normalization). The affine grading is defined as the minimal number of applications of e_0 to b to reach a ground state path. Computationally, this algorithm is a lot more efficient than the computation involving the R-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 B is the graded sum by energy of the weight of all elements b \in B:
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]]