Kyoto Path Model for Affine Highest Weight Crystals

class sage.combinat.crystals.kyoto_path_model.KyotoPathModel(crystals, weight, P)

Bases: TensorProductOfCrystals

The Kyoto path model for an affine highest weight crystal.

Note

Here we are using anti-Kashiwara notation and might differ from some of the literature.

Consider a Kac–Moody algebra \(\mathfrak{g}\) of affine Cartan type \(X\), and we want to model the \(U_q'(\mathfrak{g})\)-crystal \(B(\lambda)\). First we consider the set of fundamental weights \(\{\Lambda_i\}_{i \in I}\) of \(\mathfrak{g}\) and let \(\{\overline{\Lambda}_i\}_{i \in I_0}\) be the corresponding fundamental weights of the corresponding classical Lie algebra \(\mathfrak{g}_0\). To model \(B(\lambda)\), we start with a sequence of perfect \(U_q'(\mathfrak{g})\)-crystals \((B^{(i)})_i\) of level \(l\) such that

\[\lambda \in \overline{P}_l^+ = \left\{ \mu \in \overline{P}^+ \mid \langle c, \mu \rangle = l \right\}\]

where \(c\) is the canonical central element of \(U_q'(\mathfrak{g})\) and \(\overline{P}^+\) is the nonnegative weight lattice spanned by \(\{ \overline{\Lambda}_i \mid i \in I \}\).

Next we consider the crystal isomorphism \(\Phi_0 : B(\lambda_0) \to B^{(0)} \otimes B(\lambda_1)\) defined by \(u_{\lambda_0} \mapsto b^{(0)}_{\lambda_0} \otimes u_{\lambda_1}\) where \(b^{(0)}_{\lambda_0}\) is the unique element in \(B^{(0)}\) such that \(\varphi\left( b^{(0)}_{\lambda_0} \right) = \lambda_0\) and \(\lambda_1 = \varepsilon\left( b^{(0)}_{\lambda_0} \right)\) and \(u_{\mu}\) is the highest weight element in \(B(\mu)\). Iterating this, we obtain the following isomorphism:

\[\Phi_n : B(\lambda) \to B^{(0)} \otimes B^{(1)} \otimes \cdots \otimes B^{(N)} \otimes B(\lambda_{N+1}).\]

We note by Lemma 10.6.2 in [HK2002] that for any \(b \in B(\lambda)\) there exists a finite \(N\) such that

\[\Phi_N(b) = \left( \bigotimes_{k=0}^{N-1} b^{(k)} \right) \otimes u_{\lambda_N}.\]

Therefore we can model elements \(b \in B(\lambda)\) as a \(U_q'(\mathfrak{g})\)-crystal by considering an infinite list of elements \(b^{(k)} \in B^{(k)}\) and defining the crystal structure by:

\[\begin{split}\begin{aligned} \overline{\mathrm{wt}}(b) & = \lambda_N + \sum_{k=0}^{N-1} \overline{\mathrm{wt}}\left( b^{(k)} \right) \\ e_i(b) & = e_i\left( b^{\prime} \otimes b^{(N)} \right) \otimes u_{\lambda_N}, \\ f_i(b) & = f_i\left( b^{\prime} \otimes b^{(N)} \right) \otimes u_{\lambda_N}, \\ \varepsilon_i(b) & = \max\left( \varepsilon_i(b^{\prime}) - \varphi_i\left( b^{(N)} \right), 0 \right), \\ \varphi_i(b) & = \varphi_i(b^{\prime}) + \max\left( \varphi_i\left( b^{(N)} \right) - \varepsilon_i(b^{\prime}), 0 \right), \end{aligned}\end{split}\]

where \(b^{\prime} = b^{(0)} \otimes \cdots \otimes b^{(N-1)}\). To translate this into a finite list, we consider a finite sequence \(b^{(0)} \otimes \cdots \otimes b^{(N-1)} \otimes b^{(N)}_{\lambda_N}\) and if

\[f_i\left( b^{(0)} \otimes \cdots b^{(N-1)} \otimes b^{(N)}_{\lambda_N} \right) = b_0 \otimes \cdots \otimes b^{(N-1)} \otimes f_i\left( b^{(N)}_{\lambda_N} \right),\]

then we take the image as \(b^{(0)} \otimes \cdots \otimes f_i\left( b^{(N)}_{\lambda_N}\right) \otimes b^{(N+1)}_{\lambda_{N+1}}\). Similarly we remove \(b^{(N)}_{\lambda_{N}}\) if we have \(b_0 \otimes \cdots \otimes b^{(N-1)} \otimes b^{(N-1)}_{\lambda_{N-1}} \otimes b^{(N)}_{\lambda_N}\). Additionally if

\[e_i\left( b^{(0)} \otimes \cdots \otimes b^{(N-1)} \otimes b^{(N)}_{\lambda_N} \right) = b^{(0)} \otimes \cdots \otimes b^{(N-1)} \otimes e_i\left( b^{(N)}_{\lambda_N} \right),\]

then we consider this to be \(0\).

We can then lift the \(U_q'(\mathfrak{g})\)-crystal structure to a \(U_q(\mathfrak{g})\)-crystal structure by using a tensor product of the affinization of the of crystals \(B^{(i)}\) for all \(i\).

INPUT:

• B – a single or list of \(U_q^{\prime}\) perfect crystal(s) of level \(l\)

• weight – a weight in \(\overline{P}_l^+\)

EXAMPLES:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]; mg
[[[3]]]
sage: mg.f_string([0,1,2,2])
[[[3]], [[3]], [[1]]]
sage: x = mg.f_string([0,1,2]); x
[[[2]], [[3]], [[1]]]
sage: x.weight()
Lambda[0]

An example of type \(A_5^{(2)}\):

sage: B = crystals.KirillovReshetikhin(['A',5,2], 1,1)
sage: La = RootSystem(['A',5,2]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]; mg
[[[-1]]]
sage: mg.f_string([0,2,1,3])
[[[-3]], [[2]], [[-1]]]
sage: mg.f_string([0,2,3,1])
[[[-3]], [[2]], [[-1]]]

An example of type \(D_3^{(2)}\):

sage: B = crystals.KirillovReshetikhin(['D',3,2], 1,1)
sage: La = RootSystem(['D',3,2]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]; mg
[[]]
sage: mg.f_string([0,1,2,0])
[[[0]], [[1]], []]

An example using multiple crystals of the same level:

sage: B1 = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: B2 = crystals.KirillovReshetikhin(['A',2,1], 2,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel([B1, B2, B1], La[0])
sage: mg = C.module_generators[0]; mg
[[[3]]]
sage: mg.f_string([0,1,2,2])
[[[3]], [[1], [3]], [[3]]]
sage: mg.f_string([0,1,2,2,2])
sage: mg.f_string([0,1,2,2,1,0])
[[[3]], [[2], [3]], [[1]], [[2]]]
sage: mg.f_string([0,1,2,2,1,0,0,2])
[[[3]], [[1], [2]], [[1]], [[3]], [[1], [3]]]

By using the extended weight lattice, the Kyoto path model lifts the perfect crystals to their affinizations:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: P = RootSystem(['A',2,1]).weight_lattice(extended=True)
sage: La = P.fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]; mg
[[[3]](0)]
sage: x = mg.f_string([0,1,2]); x
[[[2]](-1), [[3]](0), [[1]](0)]
sage: x.weight()
Lambda[0] - delta

class Element

Bases: TensorProductOfRegularCrystalsElement

An element in the Kyoto path model.

e(i)

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

EXAMPLES:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]
sage: all(mg.e(i) is None for i in C.index_set())
True
sage: mg.f(0).e(0) == mg
True

epsilon(i)

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]
sage: [mg.epsilon(i) for i in C.index_set()]
[0, 0, 0]
sage: elt = mg.f(0)
sage: [elt.epsilon(i) for i in C.index_set()]
[1, 0, 0]
sage: elt = mg.f_string([0,1,2])
sage: [elt.epsilon(i) for i in C.index_set()]
[0, 0, 1]
sage: elt = mg.f_string([0,1,2,2])
sage: [elt.epsilon(i) for i in C.index_set()]
[0, 0, 2]

f(i)

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

EXAMPLES:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]
sage: mg.f(2)
sage: mg.f(0)
[[[1]], [[2]]]
sage: mg.f_string([0,1,2])
[[[2]], [[3]], [[1]]]

phi(i)

Return \(\varphi_i\) of self.

EXAMPLES:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]
sage: [mg.phi(i) for i in C.index_set()]
[1, 0, 0]
sage: elt = mg.f(0)
sage: [elt.phi(i) for i in C.index_set()]
[0, 1, 1]
sage: elt = mg.f_string([0,1])
sage: [elt.phi(i) for i in C.index_set()]
[0, 0, 2]

truncate(k=None)

Truncate self to have length k and return as an element in a (finite) tensor product of crystals.

INPUT:

• k – (optional) the length to truncate to; if not specified, then returns one more than the current non-ground-state elements (i.e. the current list in self)

EXAMPLES:

sage: B1 = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: B2 = crystals.KirillovReshetikhin(['A',2,1], 2,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel([B1,B2,B1], La[0])
sage: mg = C.highest_weight_vector()
sage: elt = mg.f_string([0,1,2,2,1,0]); elt
[[[3]], [[2], [3]], [[1]], [[2]]]
sage: t = elt.truncate(); t
[[[3]], [[2], [3]], [[1]], [[2]]]
sage: t.parent() is C.finite_tensor_product(4)
True
sage: elt.truncate(2)
[[[3]], [[2], [3]]]
sage: elt.truncate(10)
[[[3]], [[2], [3]], [[1]], [[2]], [[1], [3]],
 [[2]], [[1]], [[2], [3]], [[1]], [[3]]]

weight()

Return the weight of self.

EXAMPLES:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: P = RootSystem(['A',2,1]).weight_lattice(extended=True)
sage: La = P.fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: mg = C.module_generators[0]
sage: mg.weight()
Lambda[0]
sage: mg.f_string([0,1,2]).weight()
Lambda[0] - delta

finite_tensor_product(k)

Return the finite tensor product of crystals of length k from truncating self.

EXAMPLES:

sage: B1 = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: B2 = crystals.KirillovReshetikhin(['A',2,1], 2,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel([B1,B2,B1], La[0])
sage: C.finite_tensor_product(5)
Full tensor product of the crystals
 [Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1),
  Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(2,1),
  Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1),
  Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1),
  Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(2,1)]

weight_lattice_realization()

Return the weight lattice realization used to express weights.

EXAMPLES:

sage: B = crystals.KirillovReshetikhin(['A',2,1], 1,1)
sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()
sage: C = crystals.KyotoPathModel(B, La[0])
sage: C.weight_lattice_realization()
Weight lattice of the Root system of type ['A', 2, 1]

sage: P = RootSystem(['A',2,1]).weight_lattice(extended=True)
sage: C = crystals.KyotoPathModel(B, P.fundamental_weight(0))
sage: C.weight_lattice_realization()
Extended weight lattice of the Root system of type ['A', 2, 1]