# 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. 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].

AUTHORS:

• Travis Scrimshaw (2012-01-03): Initial version
• Travis Scrimshaw (2012-11-14): Added bijection to KR crystals
class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxBn(cartan_type, r, s)

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()

Element

alias of KRTableauxSpinElement

from_kirillov_reshetikhin_crystal(krc)

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]]

class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxDTwistedSpin(cartan_type, r, s)

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

Element

alias of KRTableauxSpinElement

class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxRectangle(cartan_type, r, s)

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:

• $$A_n^{(1)}$$ for all $$1 \leq r \leq n$$,
• $$C_n^{(1)}$$ when $$r = n$$.

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

from_kirillov_reshetikhin_crystal(krc)

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]]

class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxSpin(cartan_type, r, s)

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()

Element

alias of KRTableauxSpinElement

class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxSpinElement(parent, list, **options)

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).

e(i)

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)

epsilon(i)

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

f(i)

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]]

left_split()

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

phi(i)

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

to_array(rows=True)

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:

• rows – (Default: True) Set to True if the resulting array is by row, otherwise it is by column.

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]]

class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxTypeBox(cartan_type, r, s)

Kirillov-Reshetikhin tableaux $$B^{r,s}$$ of type:

• $$A_{2n}^{(2)}$$ for all $$r \leq n$$,
• $$D_{n+1}^{(2)}$$ for all $$r < n$$.

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

class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxTypeHorizonal(cartan_type, r, s)

Kirillov-Reshetikhin tableaux $$B^{r,s}$$ of type:

• $$C_n^{(1)}$$ for $$1 \leq r < n$$,
• $$A_{2n}^{(2)\dagger}$$ for $$1 \leq r \leq n$$.

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()

from_kirillov_reshetikhin_crystal(krc)

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]]

class sage.combinat.rigged_configurations.kr_tableaux.KRTableauxTypeVertical(cartan_type, r, s)

Kirillov-Reshetkihn tableaux $$B^{r,s}$$ of type:

• $$D_n^{(1)}$$ for all $$1 \leq r < n-1$$,
• $$B_n^{(1)}$$ for all $$1 \leq r < n$$,
• $$A_{2n-1}^{(2)}$$ for all $$1 \leq r \leq n$$.

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

from_kirillov_reshetikhin_crystal(krc)

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]]

class sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableaux(cartan_type, r, s)

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:

• type $$D_n^{(1)}$$ when $$r \leq n-2$$,
• type $$B_n^{(1)}$$ when $$r < n$$,
• type $$A_{2n-1}^{(2)}$$ for all $$r$$,

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:

• type $$C_n^{(1)}$$ when $$r < n$$,
• type $$A_{2n}^{(2)\dagger}$$ for all $$r$$,

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:

• type $$A_{2n}^{(2)}$$ for all $$r$$,
• type $$D_{n+1}^{(2)}$$ when $$r < n$$,

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:

• cartan_type – the Cartan type
• r – the Dynkin diagram index (typically the number of rows)
• s – the number of columns

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

Element
classical_decomposition()

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]]

from_kirillov_reshetikhin_crystal(krc)

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]]

kirillov_reshetikhin_crystal()

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)

module_generator(i=None, **options)

Return the specified module generator.

INPUT:

• i – The index of the module generator

We can also get a module generator by using one of the following optional arguments:

• shape – The associated shape
• column_shape – The shape given as columns (a column of length $$k$$ correspond to a classical weight $$\omega_k$$)
• weight – The weight
• classical_weight – The classical weight

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]]

r()

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

s()

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

class sage.combinat.rigged_configurations.kr_tableaux.KirillovReshetikhinTableauxElement(parent, list, **options)

A Kirillov-Reshetikhin tableau.

classical_weight()

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)

e(i)

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]]

epsilon(i)

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

f(i)

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]]

left_split()

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))

lusztig_involution()

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]]

phi(i)

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

pp()

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

right_split()

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))

to_array(rows=True)

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:

• rows – (Default: True) Set to True if the resulting array is by row, otherwise it is by column.

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]]

to_classical_highest_weight(index_set=None)

Return the classical highest weight element corresponding to self.

INPUT:

• index_set – (Default: None) Return the highest weight with respect to the index set. If None is passed in, then this uses the classical index set.

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]]

to_kirillov_reshetikhin_crystal()

Construct a KashiwaraNakashimaTableaux() element from self.

We construct the Kirillov-Reshetikhin crystal element as follows:

1. Determine the shape $$\lambda$$ of the KR crystal from the weight.
2. Determine a path $$e_{i_1} e_{i_2} \cdots e_{i_k}$$ to the highest weight.
3. Apply $$f_{i_k} \cdots f_{i_2} f_{i_1}$$ to a highest weight KR crystal of shape $$\lambda$$.

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

to_tableau()

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'>

weight()

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]


#### Previous topic

Tensor Product of Kirillov-Reshetikhin Tableaux Elements

Kleber Trees