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 finitedimensional 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]. KirillovReshetikhin 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+
 
 
OOO
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, tightpage=True) # optional  dot2tex graphviz, not tested (opens external window)
Types \(D_n^{(1)}\), \(B_n^{(1)}\), \(A_{2n1}^{(2)}\)¶
The Dynkin diagrams for types \(D_n^{(1)}\), \(B_n^{(1)}\), \(A_{2n1}^{(2)}\) are invariant under interchanging nodes \(0\) and \(1\):
sage: n = 5
sage: C = CartanType(['D',n,1]); C.dynkin_diagram()
0 O O 5
 
 
OOOO
1 2 3 4
D5~
sage: C = CartanType(['B',n,1]); C.dynkin_diagram()
O 0


OOOO=>=O
1 2 3 4 5
B5~
sage: C = CartanType(['A',2*n1,2]); C.dynkin_diagram()
O 0


OOOO=<=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 \(1\le r\le n2\) for type \(D_n^{(1)}\), and \(1 \le r \le n\) for types \(B_n^{(1)}\) and \(A_{2n1}^{(2)}\):
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 promotiontype 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:
Type \(C_n^{(1)}\)¶
The Dynkin diagram of type \(C_n^{(1)}\) has a symmetry \(\sigma(i) = ni\):
sage: C = CartanType(['C',4,1]); C.dynkin_diagram()
O=>=OOO=<=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()
KirillovReshetikhin 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:
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=<=OOO=>=O
0 1 2 3 4
C4~*
sage: C = CartanType(['A',8,2]); C.dynkin_diagram()
O=<=OOO=<=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()
KirillovReshetikhin crystal of type ['C', 4, 1] with (r,s)=(1,4)
sage: K = crystals.KirillovReshetikhin(['A',8,2],1,2); K.ambient_crystal()
KirillovReshetikhin 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 \(i\mapsto ni\) 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)
True
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:
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 \(B^{n1,s}\) 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]]
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


OOOOO
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
Single column KR crystals¶
A single column KR crystal is \(B^{r,1}\) for any \(r \in I_0\).
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 \(B^{1,1}\) in type \(E_6^{(1)}\):
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 \(E_8^{(1)}\) 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: [x.weight() for x in L]
[2*Lambda[0] + Lambda[8], 0]
Applications¶
An important notion for finitedimensional 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 infinitedimensional 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 onedimensional configuration sum¶
For tensor products of KirillovReshehtikhin 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("{} {}".format(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("{} {}".format(b, b.affine_grading()))
[[], [[1]]] 1
[[[1, 1]], [[1]]] 2
[[[1, 2]], [[1]]] 1
[[[1, 1]], [[1]]] 0
The onedimensional 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 onedimensional 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]]