Kirillov-Reshetikhin tableaux are rectangular tableaux with \(r\) rows and \(s\) columns that naturally arise under the bijection between rigged configurations and tableaux [RigConBijection]. They are in bijection with the elements of the Kirillov-Reshetikhin crystal \(B^{r,s}\) under the (inverse) filling map [OSS13] [SchScr]. They do not have to satisfy the semistandard row or column restrictions. These tensor products are the result from the bijection from rigged configurations [RigConBijection].
For more information, see KirillovReshetikhinTableaux and TensorProductOfKirillovReshetikhinTableaux.
AUTHORS:
REFERENCES:
| [OSS13] | Masato Okado, Reiho Sakamoto, and Anne Schilling. Affine crystal structure on rigged configurations of type \(D_n^{(1)}\). J. Algebraic Combinatorics, 37 (2013). 571-599. Arxiv 1109.3523. |
Bases: sage.combinat.rigged_configurations.kr_tableaux.KRTableauxTypeHorizonal
Kirillov-Reshetkhin tableaux \(B^{n,s}\) of type \(B_n^{(1)}\).
TESTS:
sage: KRT = crystals.KirillovReshetikhin(['B', 2, 1], 2, 3, model='KR')
sage: TestSuite(KRT).run()
alias of KRTableauxSpinElement
Construct an element of self from the Kirillov-Reshetikhin crystal element krc.
EXAMPLES:
sage: KR = crystals.KirillovReshetikhin(['B',3,1], 3, 3, model='KR')
sage: C = crystals.KirillovReshetikhin(['B',3,1], 3, 3, model='KN')
sage: krc = C.module_generators[1].f_string([3,2,3,1,3,3]); krc
[++-, [[2], [0], [-3]]]
sage: KR.from_kirillov_reshetikhin_crystal(krc)
[[1, 1, 2], [2, 2, -3], [-3, -3, -1]]
Bases: sage.combinat.rigged_configurations.kr_tableaux.KRTableauxRectangle
Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type \(D_n^{(2)}\) with \(r = n\).
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 2], 1, 1, model='KR')
sage: KRT.cardinality()
8
sage: KRC = crystals.KirillovReshetikhin(['D', 4, 2], 1, 1, model='KN')
sage: KRT.cardinality() == KRC.cardinality()
True
alias of KRTableauxSpinElement
Bases: sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableaux
Kirillov-Reshetkhin tableaux \(B^{r,s}\) whose module generator is a single \(r \times s\) rectangle.
These are Kirillov-Reshetkhin tableaux \(B^{r,s}\) of type:
TESTS:
sage: KRT = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
sage: TestSuite(KRT).run()
sage: KRT = crystals.KirillovReshetikhin(['C', 3, 1], 3, 2, model='KR')
sage: TestSuite(KRT).run() # long time
Construct a KirillovReshetikhinTableauxElement.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
sage: C = crystals.KirillovReshetikhin(['A',4,1], 2, 1, model='KN')
sage: krc = C(4,3); krc
[[3], [4]]
sage: KRT.from_kirillov_reshetikhin_crystal(krc)
[[3], [4]]
Bases: sage.combinat.rigged_configurations.kr_tableaux.KRTableauxRectangle
Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type \(D_n^{(1)}\) with \(r = n, n-1\).
TESTS:
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 3, 2, model='KR')
sage: TestSuite(KRT).run()
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 4, 2, model='KR')
sage: TestSuite(KRT).run()
alias of KRTableauxSpinElement
Bases: sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableauxElement
Kirillov-Reshetikhin tableau for spinors.
Here we are in the embedding \(B(\Lambda_n) \hookrightarrow B(2 \Lambda_n)\), so \(e_i\) and \(f_i\) act by \(e_i^2\) and \(f_i^2\) respectively for all \(i \neq 0\). We do this so our columns are full width (as opposed to half width and/or uses a \(\pm\) representation).
Calculate the action of \(e_i\) on self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
sage: KRT(-1, -4, 3, 2).e(1)
[[1], [3], [-4], [-2]]
sage: KRT(-1, -4, 3, 2).e(3)
Compute \(\varepsilon_i\) of self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
sage: KRT(-1, -4, 3, 2).epsilon(1)
1
sage: KRT(-1, -4, 3, 2).epsilon(3)
0
Calculate the action of \(f_i\) on self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
sage: KRT(-1, -4, 3, 2).f(1)
sage: KRT(-1, -4, 3, 2).f(3)
[[2], [4], [-3], [-1]]
Return the image of self under the left column splitting map.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 4, 3, model='KR')
sage: elt = KRT(-3,-4,2,1,-3,-4,2,1,-2,-4,3,1); elt.pp()
1 1 1
2 2 3
-4 -4 -4
-3 -3 -2
sage: elt.left_split().pp()
1 (X) 1 1
2 2 3
-4 -4 -4
-3 -3 -2
Compute \(\varphi_i\) of self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 1, model='KR')
sage: KRT(-1, -4, 3, 2).phi(1)
0
sage: KRT(-1, -4, 3, 2).phi(3)
1
Return a 2-dimensional array representation of this Kirillov-Reshetikhin element.
If the output is in rows, then it outputs the top row first (in the English convention) from left to right.
For example: if the reading word is \([2, 1, 4, 3]\), so as a \(2 \times 2\) tableau:
1 3
2 4
we output [[1, 3], [2, 4]].
If the output is in columns, then it outputs the leftmost column first with the bottom element first. In other words this parses the reading word into its columns.
Continuing with the previous example, the output would be [[2, 1], [4, 3]].
INPUT:
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 4, 3, model='KR')
sage: elt = KRT(-3,-4,2,1,-3,-4,2,1,-2,-4,3,1)
sage: elt.to_array()
[[1, 1, 1], [2, 2, 3], [-4, -4, -4], [-3, -3, -2]]
sage: elt.to_array(False)
[[-3, -4, 2, 1], [-3, -4, 2, 1], [-2, -4, 3, 1]]
Bases: sage.combinat.rigged_configurations.kr_tableaux.KRTableauxTypeVertical
Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type:
TESTS:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 2], 2, 2, model='KR')
sage: TestSuite(KRT).run()
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 2], 2, 2, model='KR')
sage: TestSuite(KRT).run() # long time
Bases: sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableaux
Kirillov-Reshetikhin tableaux \(B^{r,s}\) of type:
TESTS:
sage: KRT = crystals.KirillovReshetikhin(['C', 3, 1], 2, 2, model='KR')
sage: TestSuite(KRT).run() # long time
sage: KRT = crystals.KirillovReshetikhin(CartanType(['A', 4, 2]).dual(), 2, 2, model='KR')
sage: TestSuite(KRT).run()
Construct an element of self from the Kirillov-Reshetikhin crystal element krc.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['C',4,1], 2, 3, model='KR')
sage: C = crystals.KirillovReshetikhin(['C',4,1], 2, 3, model='KN')
sage: krc = C(4,3); krc
[[3], [4]]
sage: KRT.from_kirillov_reshetikhin_crystal(krc)
[[3, -2, 1], [4, -1, 2]]
Bases: sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableaux
Kirillov-Reshetkihn tableaux \(B^{r,s}\) of type:
TESTS:
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 1, 1, model='KR')
sage: TestSuite(KRT).run()
sage: KRT = crystals.KirillovReshetikhin(['B', 3, 1], 2, 2, model='KR')
sage: TestSuite(KRT).run() # long time
sage: KRT = crystals.KirillovReshetikhin(['A', 5, 2], 2, 2, model='KR')
sage: TestSuite(KRT).run() # long time
Construct an element of self from the Kirillov-Reshetikhin crystal element krc.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
sage: C = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KN')
sage: krc = C(4,3); krc
[[3], [4]]
sage: KRT.from_kirillov_reshetikhin_crystal(krc)
[[3, -2, 1], [4, -1, 2]]
Bases: sage.combinat.crystals.tensor_product.CrystalOfWords
Kirillov-Reshetikhin tableaux.
Kirillov-Reshetikhin tableaux are rectangular tableaux with \(r\) rows and \(s\) columns that naturally arise under the bijection between rigged configurations and tableaux [RigConBijection]. They are in bijection with the elements of the Kirillov-Reshetikhin crystal \(B^{r,s}\) under the (inverse) filling map.
Whenever \(B^{r,s} \cong B(s\Lambda_r)\) as a classical crystal (which is the case for \(B^{r,s}\) in type \(A_n^{(1)}\), \(B^{n,s}\) in type \(C_n^{(1)}\) and \(D_{n+1}^{(2)}\), \(B^{n,s}\) and \(B^{n-1,s}\) in type \(D_n^{(1)}\)) then the filling map is trivial.
For \(B^{r,s}\) in:
the filling map is defined in [OSS2011].
For the spinor cases in type \(D_n^{(1)}\), the crystal \(B^{k,s}\) where \(k = n-1, n\), is isomorphic as a classical crystal to \(B(s\Lambda_k)\), and here we consider the Kirillov-Reshetikhin tableaux as living in \(B(2s \Lambda_k)\) under the natural doubling map. In this case, the crystal operators \(e_i\) and \(f_i\) act as \(e_i^2\) and \(f_i^2\) respectively. See [BijectionDn].
For the spinor case in type \(B_n^{(1)}\), the crystal \(B^{n,s}\), we consider the images under the natural doubling map into \(B^{n,2s}\). The classical components of this crystal are now given by removing \(2 \times 2\) boxes. The filling map is the same as below (see the non-spin type \(C_n^{(1)}\)).
For \(B^{r,s}\) in:
the filling map is given as follows. Suppose we are considering the (classically) highest weight element in the classical component \(B(\lambda)\). Then we fill it in with the horizontal dominoes \([\bar{\imath}, i]\) in the \(i\)-th row from the top (in English notation) and reordering the columns so that they are increasing. Recall from above that \(B^{n,s} \cong B(s\Lambda_n)\) in type \(C^{(1)}_n\).
For \(B^{r,s}\) in:
the filling map is the same as given in [OSS2011] except for the rightmost column which is given by the column \([1, 2, \ldots, k, \emptyset, \ldots \emptyset]\) where \(k = (r+x-1)/2\) in Step 3 of [OSS2011].
For the spinor case in type \(D_{n+1}^{(2)}\), the crystal \(B^{n,s}\), we define the filling map in the same way as in type \(D_n^{(1)}\).
Note
The filling map and classical decompositions in non-spinor cases can be classified by how the special node \(0\) connects with the corresponding classical diagram.
The classical crystal stucture is given by the usual Kashiwara-Nakashima tableaux rules. That is to embed this into \(B(\Lambda_1)^{\otimes n s}\) by using the reading word and then applying the classical crystal operator. The affine crystal stucture is given by converting to the corresponding KR crystal element, performing the affine crystal operator, and pulling back to a KR tableau.
For more information about the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin tableaux, see TensorProductOfKirillovReshetikhinTableaux.
Note
The tableaux for all non-simply-laced types are provably correct if the bijection with rigged configurations holds. Therefore this is currently only proven for \(B^{r,1}\) or \(B^{1,s}\) and in general for types \(A_n^{(1)}\) and \(D_n^{(1)}\).
INPUT:
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
sage: elt = KRT(4, 3); elt
[[3], [4]]
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 2, 1, model='KR')
sage: elt = KRT(-1, 1); elt
[[1], [-1]]
We can create highest weight crystals from a given shape or weight:
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
sage: KRT.module_generator(shape=[1,1])
[[1, 1], [2, -1]]
sage: KRT.module_generator(column_shape=[2])
[[1, 1], [2, -1]]
sage: WS = RootSystem(['D',4,1]).weight_space()
sage: KRT.module_generator(weight=WS.sum_of_terms([[0,-2],[2,1]]))
[[1, 1], [2, -1]]
sage: WSC = RootSystem(['D',4]).weight_space()
sage: KRT.module_generator(classical_weight=WSC.fundamental_weight(2))
[[1, 1], [2, -1]]
We can go between KashiwaraNakashimaTableaux() and KirillovReshetikhinTableaux elements:
sage: KRCrys = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KN')
sage: KRTab = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
sage: elt = KRCrys(3, 2); elt
[[2], [3]]
sage: k = KRTab(elt); k
[[2, 1], [3, -1]]
sage: KRCrys(k)
[[2], [3]]
We check that the classical weights in the classical decompositions agree in a few different type:
sage: KRCrys = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KN')
sage: KRTab = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
True
sage: KRCrys = crystals.KirillovReshetikhin(['B', 3, 1], 2, 2, model='KN')
sage: KRTab = crystals.KirillovReshetikhin(['B', 3, 1], 2, 2, model='KR')
sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
True
sage: KRCrys = crystals.KirillovReshetikhin(['C', 3, 1], 2, 2, model='KN')
sage: KRTab = crystals.KirillovReshetikhin(['C', 3, 1], 2, 2, model='KR')
sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
True
sage: KRCrys = crystals.KirillovReshetikhin(['D', 4, 2], 2, 2, model='KN')
sage: KRTab = crystals.KirillovReshetikhin(['D', 4, 2], 2, 2, model='KR')
sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
True
sage: KRCrys = crystals.KirillovReshetikhin(['A', 4, 2], 2, 2, model='KN')
sage: KRTab = crystals.KirillovReshetikhin(['A', 4, 2], 2, 2, model='KR')
sage: all(t.classical_weight() == KRCrys(t).classical_weight() for t in KRTab)
True
alias of KirillovReshetikhinTableauxElement
Return the classical crystal decomposition of self.
EXAMPLES:
sage: crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR').classical_decomposition()
The crystal of tableaux of type ['D', 4] and shape(s) [[], [1, 1], [2, 2]]
Construct an element of self from the Kirillov-Reshetikhin crystal element krc.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
sage: C = crystals.KirillovReshetikhin(['A',4,1], 2, 1, model='KN')
sage: krc = C(4,3); krc
[[3], [4]]
sage: KRT.from_kirillov_reshetikhin_crystal(krc)
[[3], [4]]
Return the corresponding KR crystal in the Kashiwara-Nakashima model.
EXAMPLES:
sage: crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR').kirillov_reshetikhin_crystal()
Kirillov-Reshetikhin crystal of type ['A', 4, 1] with (r,s)=(2,1)
Return the specified module generator.
INPUT:
We can also get a module generator by using one of the following optional arguments:
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D', 4, 1], 2, 2, model='KR')
sage: KRT.module_generator(1)
[[1, 1], [2, -1]]
sage: KRT.module_generator(shape=[1,1])
[[1, 1], [2, -1]]
sage: KRT.module_generator(column_shape=[2])
[[1, 1], [2, -1]]
sage: WS = RootSystem(['D',4,1]).weight_space()
sage: KRT.module_generator(weight=WS.sum_of_terms([[0,-2],[2,1]]))
[[1, 1], [2, -1]]
sage: WSC = RootSystem(['D',4]).weight_space()
sage: KRT.module_generator(classical_weight=WSC.fundamental_weight(2))
[[1, 1], [2, -1]]
sage: KRT = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
sage: KRT.module_generator()
[[1, 1], [2, 2]]
Return the value \(r\) for this tableaux class which corresponds to the number of rows.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
sage: KRT.r()
2
Return the value \(s\) for this tableaux class which corresponds to the number of columns.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 1, model='KR')
sage: KRT.s()
1
Return the tensor product of self with crystals.
If crystals is a list of (a tensor product of) KR tableaux, this returns a TensorProductOfKirillovReshetikhinTableaux.
EXAMPLES:
sage: K = crystals.KirillovReshetikhin(['A', 3, 1], 2, 2, model='KR')
sage: TP = crystals.TensorProductOfKirillovReshetikhinTableaux(['A', 3, 1], [[1,3],[3,1]])
sage: K.tensor(TP, K)
Tensor product of Kirillov-Reshetikhin tableaux of type ['A', 3, 1]
and factor(s) ((2, 2), (1, 3), (3, 1), (2, 2))
sage: C = crystals.KirillovReshetikhin(['A',3,1], 3, 1, model='KN')
sage: K.tensor(K, C)
Full tensor product of the crystals
[Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (2, 2),
Kirillov-Reshetikhin tableaux of type ['A', 3, 1] and shape (2, 2),
Kirillov-Reshetikhin crystal of type ['A', 3, 1] with (r,s)=(3,1)]
Bases: sage.combinat.crystals.tensor_product.TensorProductOfRegularCrystalsElement
A Kirillov-Reshetikhin tableau.
For more information, see KirillovReshetikhinTableaux and TensorProductOfKirillovReshetikhinTableaux.
Return the classical weight of self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: elt = KRT(3,2,-1,1); elt
[[2, 1], [3, -1]]
sage: elt.classical_weight()
(0, 1, 1, 0)
Perform the action of \(e_i\) on self.
Todo
Implement a direct action of \(e_0\) without moving to KR crystals.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: KRT.module_generators[0].e(0)
[[-2, 1], [-1, -1]]
Compute \(\epsilon_i\) of self.
Todo
Implement a direct action of \(\epsilon_0\) without moving to KR crystals.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: KRT.module_generators[0].epsilon(0)
2
Perform the action of \(f_i\) on self.
Todo
Implement a direct action of \(f_0\) without moving to KR crystals.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: KRT.module_generators[0].f(0)
[[1, 1], [2, -1]]
Return the image of self under the left column splitting map.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
sage: mg = KRT.module_generators[1]; mg.pp()
1 -2 1
2 -1 2
sage: ls = mg.left_split(); ls.pp()
1 (X) -2 1
2 -1 2
sage: ls.parent()
Tensor product of Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and factor(s) ((2, 1), (2, 2))
Return the result of the classical Lusztig involution on self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
sage: mg = KRT.module_generators[1]
sage: mg.lusztig_involution()
[[-2, -2, 1], [-1, -1, 2]]
sage: elt = mg.f_string([2,1,3,2]); elt
[[3, -2, 1], [4, -1, 2]]
sage: elt.lusztig_involution()
[[-4, -2, 1], [-3, -1, 2]]
Compute \(\phi_i\) of self.
Todo
Compute \(\phi_0\) without moving to KR crystals.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: KRT.module_generators[0].phi(0)
2
Pretty print self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 2, model='KR')
sage: elt = KRT(2, 1, 4, 3); elt
[[1, 3], [2, 4]]
sage: elt.pp()
1 3
2 4
Return the image of self under the right column splitting map.
Let \(\ast\) denote the Lusztig involution, and \(\mathrm{ls}\) as the left splitting map. The right splitting map is defined as \(\mathrm{rs} := \ast \circ \mathrm{ls} \circ \ast\).
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 3, model='KR')
sage: mg = KRT.module_generators[1]; mg.pp()
1 -2 1
2 -1 2
sage: ls = mg.right_split(); ls.pp()
-2 1 (X) 1
-1 2 2
sage: ls.parent()
Tensor product of Kirillov-Reshetikhin tableaux of type ['D', 4, 1] and factor(s) ((2, 2), (2, 1))
Return a 2-dimensional array representation of this Kirillov-Reshetikhin element.
If the output is in rows, then it outputs the top row first (in the English convention) from left to right.
For example: if the reading word is \([2, 1, 4, 3]\), so as a \(2 \times 2\) tableau:
1 3
2 4
we output [[1, 3], [2, 4]].
If the output is in columns, then it outputs the leftmost column first with the bottom element first. In other words this parses the reading word into its columns.
Continuing with the previous example, the output would be [[2, 1], [4, 3]].
INPUT:
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 2, model='KR')
sage: elt = KRT(2, 1, 4, 3)
sage: elt.to_array()
[[1, 3], [2, 4]]
sage: elt.to_array(False)
[[2, 1], [4, 3]]
Return the classical highest weight element corresponding to self.
INPUT:
OUTPUT:
A pair [H, f_str] where H is the highest weight element and f_str is a list of \(a_i\) of \(f_{a_i}\) needed to reach H.
EXAMPLES:
sage: KRTab = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: elt = KRTab(3,2,-1,1); elt
[[2, 1], [3, -1]]
sage: elt.to_classical_highest_weight()
[[[1, 1], [2, -1]], [1, 2]]
Construct a KashiwaraNakashimaTableaux() element from self.
We construct the Kirillov-Reshetikhin crystal element as follows:
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: elt = KRT(3,2,-1,1); elt
[[2, 1], [3, -1]]
sage: elt.to_kirillov_reshetikhin_crystal()
[[2], [3]]
TESTS:
Spinor tests:
sage: KRT = crystals.KirillovReshetikhin(['D',4,1], 4, 3, model='KR')
sage: KRC = crystals.KirillovReshetikhin(['D',4,1], 4, 3, model='KN')
sage: elt = KRT(-3,-4,2,1,-3,-4,2,1,-2,-4,3,1); elt
[[1, 1, 1], [2, 2, 3], [-4, -4, -4], [-3, -3, -2]]
sage: ret = elt.to_kirillov_reshetikhin_crystal(); ret
[++--, [[1], [3], [-4], [-3]]]
sage: test = KRT(ret); test
[[1, 1, 1], [2, 2, 3], [-4, -4, -4], [-3, -3, -2]]
sage: test == elt
True
Return a Tableau object of self.
EXAMPLES:
sage: KRT = crystals.KirillovReshetikhin(['A', 4, 1], 2, 2, model='KR')
sage: elt = KRT(2, 1, 4, 3); elt
[[1, 3], [2, 4]]
sage: t = elt.to_tableau(); t
[[1, 3], [2, 4]]
sage: type(t)
<class 'sage.combinat.tableau.Tableaux_all_with_category.element_class'>
Return the weight of self.
EXAMPLES:
sage: KR = crystals.KirillovReshetikhin(['D',4,1], 2, 2, model='KR')
sage: KR.module_generators[1].weight()
-2*Lambda[0] + Lambda[2]