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Affine Finite Crystals

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.

Type A_n^{(1)}

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):

B^{r,s} \cong B(s\omega_r) \quad \text{as a } \{1,2,\ldots,n\}\text{-crystal}

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:

f_0 = \mathrm{pr}^{-1} \circ f_1 \circ \mathrm{pr}
e_0 = \mathrm{pr}^{-1} \circ e_1 \circ \mathrm{pr}

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:

../_images/KR_A.png

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

Types D_n^{(1)}, B_n^{(1)}, A_{2n-1}^{(2)}

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:

B^{r,s} \cong \bigoplus_\lambda B(\lambda) \quad \text{as a } \{1,2,\ldots,n\}\text{-crystal}

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:

f_0 = \sigma \circ f_1 \circ \sigma
e_0 = \sigma \circ e_1 \circ \sigma

The KR crystals B^{1,1} of types D_3^{(1)}, B_2^{(1)}, and A_5^{(2)} are, respectively:

../_images/KR_D.png ../_images/KR_B.png ../_images/KR_Atwisted.png

Type C_n^{(1)}

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:

e_0 = \hat{e}_0 \hat{e}_1 \quad \text{and} \quad e_i = \hat{e}_{i+1} \quad \text{for} \quad 1\le i\le n
f_0 = \hat{f}_0 \hat{f}_1 \quad \text{and} \quad f_i = \hat{f}_{i+1} \quad \text{for} \quad 1\le i\le n

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:

B^{r,s} \cong \bigoplus_\lambda B(\lambda) \quad \text{as a } \{1,2,\ldots,n\}\text{-crystal}

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:

../_images/KR_C.png

Types D_{n+1}^{(2)}, A_{2n}^{(2)}

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:

B^{r,s} \cong \bigoplus_\lambda B(\lambda) \quad \text{as a } \{1,2,\ldots,n\}\text{-crystal}

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)}

S : B^{r,s} \to B^{r,2s}_{C_n^{(1)}} \quad \text{such that } S(e_ib) = e_i^{m_i}S(b) \text{ and } S(f_ib) = f_i^{m_i}S(b)

where

(m_0,\ldots,m_n) = (1,2,\ldots,2,1) \text{ for type } D_{n+1}^{(2)} \quad \text{and} \quad (1,2,\ldots,2,2) \text{ for type } A_{2n}^{(2)}.
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:

../_images/KR_Dtwisted.png ../_images/KR_Atwisted1.png

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]]})
../_images/KR_Atwisted_dual.png

Exceptional nodes

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:

B^{n,s} \cong B(s\omega_n).

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]]
../_images/KR_C_exceptional.png ../_images/KR_Dtwisted_exceptional.png

The KR crystals B^{n,s} and B^{n-1,s} of type D_n^{(1)} are also special. They decompose as:

B^{n,s} \cong B(s\omega_n) \quad \text{ and } \quad B^{n-1,s} \cong B(s\omega_{n-1}).
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]]

Type E_6^{(1)}

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

B^{2,s} \cong \bigoplus_{k=0}^s B(k\omega_2)
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
../_images/KR_E6.png

Applications

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

B_{\mathrm{min}} = \{ b \in B \mid \mathrm{lev}(\varphi(b)) = \ell \}\;.

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]]]

Energy function and one-dimensional configuration sum

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:

X(B) = \sum_{b \in B} x^{\mathrm{weight}(b)} q^{D(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]]