Crystals of letters

class sage.combinat.crystals.letters.BKKLetter[source]

Bases: Letter

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['A', [2, 1]])
sage: c = C(-2)
sage: c.e(-2)
-3
sage: c = C(1)
sage: c.e(0)
-1
sage: c = C(2)
sage: c.e(1)
1
sage: c.e(-2)

>>> from sage.all import *
>>> C = crystals.Letters(['A', [Integer(2), Integer(1)]])
>>> c = C(-Integer(2))
>>> c.e(-Integer(2))
-3
>>> c = C(Integer(1))
>>> c.e(Integer(0))
-1
>>> c = C(Integer(2))
>>> c.e(Integer(1))
1
>>> c.e(-Integer(2))
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['A', [2, 1]])
sage: c = C.an_element()
sage: c.f(-2)
-2
sage: c = C(-1)
sage: c.f(0)
1
sage: c = C(1)
sage: c.f(1)
2
sage: c.f(-2)

>>> from sage.all import *
>>> C = crystals.Letters(['A', [Integer(2), Integer(1)]])
>>> c = C.an_element()
>>> c.f(-Integer(2))
-2
>>> c = C(-Integer(1))
>>> c.f(Integer(0))
1
>>> c = C(Integer(1))
>>> c.f(Integer(1))
2
>>> c.f(-Integer(2))
weight()[source]

Return weight of self.

EXAMPLES:

sage: C = crystals.Letters(['A', [2, 1]])
sage: c = C(-1)
sage: c.weight()
(0, 0, 1, 0, 0)
sage: c = C(2)
sage: c.weight()
(0, 0, 0, 0, 1)

>>> from sage.all import *
>>> C = crystals.Letters(['A', [Integer(2), Integer(1)]])
>>> c = C(-Integer(1))
>>> c.weight()
(0, 0, 1, 0, 0)
>>> c = C(Integer(2))
>>> c.weight()
(0, 0, 0, 0, 1)
class sage.combinat.crystals.letters.ClassicalCrystalOfLetters(cartan_type, element_class, element_print_style=None, dual=None)[source]

Bases: UniqueRepresentation, Parent

A generic class for classical crystals of letters.

All classical crystals of letters should be instances of this class or of subclasses. To define a new crystal of letters, one only needs to implement a class for the elements (which subclasses Letter), with appropriate \(e_i\) and \(f_i\) operations. If the module generator is not \(1\), one also needs to define the subclass ClassicalCrystalOfLetters for the crystal itself.

The basic assumption is that crystals of letters are small, but used intensively as building blocks. Therefore, we explicitly build in memory the list of all elements, the crystal graph and its transitive closure, so as to make the following operations constant time: list, cmp, (todo: phi, epsilon, e, and f with caching)

list()[source]

Return a list of the elements of self.

EXAMPLES:

sage: C = crystals.Letters(['A',5])
sage: C.list()
[1, 2, 3, 4, 5, 6]

>>> from sage.all import *
>>> C = crystals.Letters(['A',Integer(5)])
>>> C.list()
[1, 2, 3, 4, 5, 6]
lt_elements(x, y)[source]

Return True if and only if there is a path from x to y in the crystal graph, when x is not equal to y.

Because the crystal graph is classical, it is a directed acyclic graph which can be interpreted as a poset. This function implements the comparison function of this poset.

EXAMPLES:

sage: C = crystals.Letters(['A', 5])
sage: x = C(1)
sage: y = C(2)
sage: C.lt_elements(x,y)
True
sage: C.lt_elements(y,x)
False
sage: C.lt_elements(x,x)
False
sage: C = crystals.Letters(['D', 4])
sage: C.lt_elements(C(4),C(-4))
False
sage: C.lt_elements(C(-4),C(4))
False

>>> from sage.all import *
>>> C = crystals.Letters(['A', Integer(5)])
>>> x = C(Integer(1))
>>> y = C(Integer(2))
>>> C.lt_elements(x,y)
True
>>> C.lt_elements(y,x)
False
>>> C.lt_elements(x,x)
False
>>> C = crystals.Letters(['D', Integer(4)])
>>> C.lt_elements(C(Integer(4)),C(-Integer(4)))
False
>>> C.lt_elements(C(-Integer(4)),C(Integer(4)))
False
class sage.combinat.crystals.letters.ClassicalCrystalOfLettersWrapped(cartan_type)[source]

Bases: ClassicalCrystalOfLetters

Crystal of letters by wrapping another crystal.

This is used for a crystal of letters of type \(E_8\) and \(F_4\).

This class follows the same output as the other crystal of letters, where \(b\) is represented by the “letter” with \(\varphi_i(b)\) (resp., \(\varepsilon_i\)) number of \(i\)’s (resp., \(-i\)’s or \(\bar{i}\)’s). However, this uses an auxiliary crystal to construct these letters to avoid hardcoding the crystal elements and the corresponding edges; in particular, the 248 nodes of \(E_8\).

class sage.combinat.crystals.letters.CrystalOfBKKLetters(ct, dual)[source]

Bases: ClassicalCrystalOfLetters

Crystal of letters for Benkart-Kang-Kashiwara supercrystals.

This implements the \(\mathfrak{gl}(m|n)\) crystal of Benkart, Kang and Kashiwara [BKK2000].

EXAMPLES:

sage: C = crystals.Letters(['A', [1, 1]]); C
The crystal of letters for type ['A', [1, 1]]

sage: C = crystals.Letters(['A', [2,4]], dual=True); C
The crystal of letters for type ['A', [2, 4]] (dual)

>>> from sage.all import *
>>> C = crystals.Letters(['A', [Integer(1), Integer(1)]]); C
The crystal of letters for type ['A', [1, 1]]

>>> C = crystals.Letters(['A', [Integer(2),Integer(4)]], dual=True); C
The crystal of letters for type ['A', [2, 4]] (dual)
Element[source]

alias of BKKLetter

sage.combinat.crystals.letters.CrystalOfLetters(cartan_type, element_print_style=None, dual=None)[source]

Return the crystal of letters of the given type.

For classical types, this is a combinatorial model for the crystal with highest weight \(\Lambda_1\) (the first fundamental weight).

Any irreducible classical crystal appears as the irreducible component of the tensor product of several copies of this crystal (plus possibly one copy of the spin crystal, see CrystalOfSpins). See [KN1994]. Elements of this irreducible component have a fixed shape, and can be fit inside a tableau shape. Otherwise said, any irreducible classical crystal is isomorphic to a crystal of tableaux with cells filled by elements of the crystal of letters (possibly tensored with the crystal of spins).

We also have the crystal of fundamental representation of the general linear Lie superalgebra, which are used as letters inside of tableaux following [BKK2000]. Similarly, all of these crystals appear as a subcrystal of a sufficiently large tensor power of this crystal.

INPUT:

  • T – a Cartan type

EXAMPLES:

sage: C = crystals.Letters(['A',5])
sage: C.list()
[1, 2, 3, 4, 5, 6]
sage: C.cartan_type()
['A', 5]

>>> from sage.all import *
>>> C = crystals.Letters(['A',Integer(5)])
>>> C.list()
[1, 2, 3, 4, 5, 6]
>>> C.cartan_type()
['A', 5]

For type \(E_6\), one can also specify how elements are printed. This option is usually set to None and the default representation is used. If one chooses the option ‘compact’, the elements are printed in the more compact convention with 27 letters +abcdefghijklmnopqrstuvwxyz and the 27 letters -ABCDEFGHIJKLMNOPQRSTUVWXYZ for the dual crystal.

EXAMPLES:

sage: C = crystals.Letters(['E',6], element_print_style = 'compact')
sage: C
The crystal of letters for type ['E', 6]
sage: C.list()
[+, a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z]
sage: C = crystals.Letters(['E',6], element_print_style = 'compact', dual = True)
sage: C
The crystal of letters for type ['E', 6] (dual)
sage: C.list()
[-, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z]

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)], element_print_style = 'compact')
>>> C
The crystal of letters for type ['E', 6]
>>> C.list()
[+, a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z]
>>> C = crystals.Letters(['E',Integer(6)], element_print_style = 'compact', dual = True)
>>> C
The crystal of letters for type ['E', 6] (dual)
>>> C.list()
[-, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z]
class sage.combinat.crystals.letters.CrystalOfQueerLetters(ct)[source]

Bases: ClassicalCrystalOfLetters

Queer crystal of letters elements.

The index set is of the form \(\{-n, \ldots, -1, 1, \ldots, n\}\). For \(1 < i \leq n\), the operators \(e_{-i}\) and \(f_{-i}\) are defined as

\[f_{-i} = s_{w^{-1}_i} f_{-1} s_{w_i}, \quad e_{-i} = s_{w^{-1}_i} e_{-1} s_{w_i},\]

where \(w_i = s_2 \cdots s_i s_1 \cdots s_{i-1}\) and \(s_i\) is the reflection along the \(i\)-string in the crystal. See [GJK+2014].

Element[source]

alias of QueerLetter_element

index_set()[source]

Return index set of self.

EXAMPLES:

sage: Q = crystals.Letters(['Q',3])
sage: Q.index_set()
(1, 2, -2, -1)

>>> from sage.all import *
>>> Q = crystals.Letters(['Q',Integer(3)])
>>> Q.index_set()
(1, 2, -2, -1)
class sage.combinat.crystals.letters.Crystal_of_letters_type_A_element[source]

Bases: Letter

Type \(A\) crystal of letters elements.

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['A',4])
sage: [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1), (3, 2, 2), (4, 3, 3), (5, 4, 4)]

>>> from sage.all import *
>>> C = crystals.Letters(['A',Integer(4)])
>>> [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1), (3, 2, 2), (4, 3, 3), (5, 4, 4)]
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['A',4])
sage: [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != 0]
[(2, 1), (3, 2), (4, 3), (5, 4)]

>>> from sage.all import *
>>> C = crystals.Letters(['A',Integer(4)])
>>> [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != Integer(0)]
[(2, 1), (3, 2), (4, 3), (5, 4)]
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['A',4])
sage: [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2), (2, 2, 3), (3, 3, 4), (4, 4, 5)]

>>> from sage.all import *
>>> C = crystals.Letters(['A',Integer(4)])
>>> [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2), (2, 2, 3), (3, 3, 4), (4, 4, 5)]
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['A',4])
sage: [(c,i) for i in C.index_set() for c in C if c.phi(i) != 0]
[(1, 1), (2, 2), (3, 3), (4, 4)]

>>> from sage.all import *
>>> C = crystals.Letters(['A',Integer(4)])
>>> [(c,i) for i in C.index_set() for c in C if c.phi(i) != Integer(0)]
[(1, 1), (2, 2), (3, 3), (4, 4)]
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['A',3])]
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['A',Integer(3)])]
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_B_element[source]

Bases: Letter

Type \(B\) crystal of letters elements.

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['B',4])
sage: [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3),
 (4, 3, 3),
 (-3, 3, -4),
 (0, 4, 4),
 (-4, 4, 0)]

>>> from sage.all import *
>>> C = crystals.Letters(['B',Integer(4)])
>>> [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3),
 (4, 3, 3),
 (-3, 3, -4),
 (0, 4, 4),
 (-4, 4, 0)]
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['B',3])
sage: [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != 0]
[(2, 1), (-1, 1), (3, 2), (-2, 2), (0, 3), (-3, 3)]

>>> from sage.all import *
>>> C = crystals.Letters(['B',Integer(3)])
>>> [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != Integer(0)]
[(2, 1), (-1, 1), (3, 2), (-2, 2), (0, 3), (-3, 3)]
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['B',4])
sage: [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2),
 (-2, 1, -1),
 (2, 2, 3),
 (-3, 2, -2),
 (3, 3, 4),
 (-4, 3, -3),
 (4, 4, 0),
 (0, 4, -4)]

>>> from sage.all import *
>>> C = crystals.Letters(['B',Integer(4)])
>>> [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2),
 (-2, 1, -1),
 (2, 2, 3),
 (-3, 2, -2),
 (3, 3, 4),
 (-4, 3, -3),
 (4, 4, 0),
 (0, 4, -4)]
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['B',3])
sage: [(c,i) for i in C.index_set() for c in C if c.phi(i) != 0]
[(1, 1), (-2, 1), (2, 2), (-3, 2), (3, 3), (0, 3)]

>>> from sage.all import *
>>> C = crystals.Letters(['B',Integer(3)])
>>> [(c,i) for i in C.index_set() for c in C if c.phi(i) != Integer(0)]
[(1, 1), (-2, 1), (2, 2), (-3, 2), (3, 3), (0, 3)]
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['B',3])]
[(1, 0, 0),
 (0, 1, 0),
 (0, 0, 1),
 (0, 0, 0),
 (0, 0, -1),
 (0, -1, 0),
 (-1, 0, 0)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['B',Integer(3)])]
[(1, 0, 0),
 (0, 1, 0),
 (0, 0, 1),
 (0, 0, 0),
 (0, 0, -1),
 (0, -1, 0),
 (-1, 0, 0)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_C_element[source]

Bases: Letter

Type \(C\) crystal of letters elements.

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['C',4])
sage: [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3),
 (4, 3, 3),
 (-3, 3, -4),
 (-4, 4, 4)]

>>> from sage.all import *
>>> C = crystals.Letters(['C',Integer(4)])
>>> [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3),
 (4, 3, 3),
 (-3, 3, -4),
 (-4, 4, 4)]
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['C',3])
sage: [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != 0]
[(2, 1), (-1, 1), (3, 2), (-2, 2), (-3, 3)]

>>> from sage.all import *
>>> C = crystals.Letters(['C',Integer(3)])
>>> [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != Integer(0)]
[(2, 1), (-1, 1), (3, 2), (-2, 2), (-3, 3)]
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['C',4])
sage: [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2), (-2, 1, -1), (2, 2, 3),
 (-3, 2, -2), (3, 3, 4), (-4, 3, -3), (4, 4, -4)]

>>> from sage.all import *
>>> C = crystals.Letters(['C',Integer(4)])
>>> [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2), (-2, 1, -1), (2, 2, 3),
 (-3, 2, -2), (3, 3, 4), (-4, 3, -3), (4, 4, -4)]
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['C',3])
sage: [(c,i) for i in C.index_set() for c in C if c.phi(i) != 0]
[(1, 1), (-2, 1), (2, 2), (-3, 2), (3, 3)]

>>> from sage.all import *
>>> C = crystals.Letters(['C',Integer(3)])
>>> [(c,i) for i in C.index_set() for c in C if c.phi(i) != Integer(0)]
[(1, 1), (-2, 1), (2, 2), (-3, 2), (3, 3)]
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['C',3])]
[(1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, -1), (0, -1, 0), (-1, 0, 0)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['C',Integer(3)])]
[(1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, -1), (0, -1, 0), (-1, 0, 0)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_D_element[source]

Bases: Letter

Type \(D\) crystal of letters elements.

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['D',5])
sage: [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3),
 (4, 3, 3),
 (-3, 3, -4),
 (5, 4, 4),
 (-4, 4, -5),
 (-5, 5, 4),
 (-4, 5, 5)]

>>> from sage.all import *
>>> C = crystals.Letters(['D',Integer(5)])
>>> [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3),
 (4, 3, 3),
 (-3, 3, -4),
 (5, 4, 4),
 (-4, 4, -5),
 (-5, 5, 4),
 (-4, 5, 5)]
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['D',4])
sage: [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != 0]
[(2, 1), (-1, 1), (3, 2), (-2, 2), (4, 3), (-3, 3), (-4, 4), (-3, 4)]

>>> from sage.all import *
>>> C = crystals.Letters(['D',Integer(4)])
>>> [(c,i) for i in C.index_set() for c in C if c.epsilon(i) != Integer(0)]
[(2, 1), (-1, 1), (3, 2), (-2, 2), (4, 3), (-3, 3), (-4, 4), (-3, 4)]
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['D',5])
sage: [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2),
 (-2, 1, -1),
 (2, 2, 3),
 (-3, 2, -2),
 (3, 3, 4),
 (-4, 3, -3),
 (4, 4, 5),
 (-5, 4, -4),
 (4, 5, -5),
 (5, 5, -4)]

>>> from sage.all import *
>>> C = crystals.Letters(['D',Integer(5)])
>>> [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2),
 (-2, 1, -1),
 (2, 2, 3),
 (-3, 2, -2),
 (3, 3, 4),
 (-4, 3, -3),
 (4, 4, 5),
 (-5, 4, -4),
 (4, 5, -5),
 (5, 5, -4)]
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['D',4])
sage: [(c,i) for i in C.index_set() for c in C if c.phi(i) != 0]
[(1, 1), (-2, 1), (2, 2), (-3, 2), (3, 3), (-4, 3), (3, 4), (4, 4)]

>>> from sage.all import *
>>> C = crystals.Letters(['D',Integer(4)])
>>> [(c,i) for i in C.index_set() for c in C if c.phi(i) != Integer(0)]
[(1, 1), (-2, 1), (2, 2), (-3, 2), (3, 3), (-4, 3), (3, 4), (4, 4)]
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['D',4])]
[(1, 0, 0, 0),
 (0, 1, 0, 0),
 (0, 0, 1, 0),
 (0, 0, 0, 1),
 (0, 0, 0, -1),
 (0, 0, -1, 0),
 (0, -1, 0, 0),
 (-1, 0, 0, 0)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['D',Integer(4)])]
[(1, 0, 0, 0),
 (0, 1, 0, 0),
 (0, 0, 1, 0),
 (0, 0, 0, 1),
 (0, 0, 0, -1),
 (0, 0, -1, 0),
 (0, -1, 0, 0),
 (-1, 0, 0, 0)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_E6_element[source]

Bases: LetterTuple

Type \(E_6\) crystal of letters elements. This crystal corresponds to the highest weight crystal \(B(\Lambda_1)\).

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['E',6])
sage: C((-1,3)).e(1)
(1,)
sage: C((-2,-3,4)).e(2)
(-3, 2)
sage: C((1,)).e(1)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)])
>>> C((-Integer(1),Integer(3))).e(Integer(1))
(1,)
>>> C((-Integer(2),-Integer(3),Integer(4))).e(Integer(2))
(-3, 2)
>>> C((Integer(1),)).e(Integer(1))
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['E',6])
sage: C((1,)).f(1)
(-1, 3)
sage: C((-6,)).f(1)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)])
>>> C((Integer(1),)).f(Integer(1))
(-1, 3)
>>> C((-Integer(6),)).f(Integer(1))
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['E',6])]
[(0, 0, 0, 0, 0, -2/3, -2/3, 2/3),
 (-1/2, 1/2, 1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, 1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, 1/2, -1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (-1/2, -1/2, -1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, 1/2, 1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (-1/2, -1/2, 1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (-1/2, 1/2, -1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, -1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (0, 0, 0, 0, 1, 1/3, 1/3, -1/3),
 (1/2, 1/2, 1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (-1/2, -1/2, 1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (-1/2, 1/2, -1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, -1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (0, 0, 0, 1, 0, 1/3, 1/3, -1/3),
 (-1/2, 1/2, 1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, 1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (0, 0, 1, 0, 0, 1/3, 1/3, -1/3),
 (1/2, 1/2, -1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (0, 1, 0, 0, 0, 1/3, 1/3, -1/3),
 (1, 0, 0, 0, 0, 1/3, 1/3, -1/3),
 (0, -1, 0, 0, 0, 1/3, 1/3, -1/3),
 (0, 0, -1, 0, 0, 1/3, 1/3, -1/3),
 (0, 0, 0, -1, 0, 1/3, 1/3, -1/3),
 (0, 0, 0, 0, -1, 1/3, 1/3, -1/3),
 (-1/2, -1/2, -1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (-1, 0, 0, 0, 0, 1/3, 1/3, -1/3)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['E',Integer(6)])]
[(0, 0, 0, 0, 0, -2/3, -2/3, 2/3),
 (-1/2, 1/2, 1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, 1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, 1/2, -1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (-1/2, -1/2, -1/2, 1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, 1/2, 1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (-1/2, -1/2, 1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (-1/2, 1/2, -1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, -1/2, -1/2, 1/2, -1/6, -1/6, 1/6),
 (0, 0, 0, 0, 1, 1/3, 1/3, -1/3),
 (1/2, 1/2, 1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (-1/2, -1/2, 1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (-1/2, 1/2, -1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, -1/2, 1/2, -1/2, -1/6, -1/6, 1/6),
 (0, 0, 0, 1, 0, 1/3, 1/3, -1/3),
 (-1/2, 1/2, 1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (1/2, -1/2, 1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (0, 0, 1, 0, 0, 1/3, 1/3, -1/3),
 (1/2, 1/2, -1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (0, 1, 0, 0, 0, 1/3, 1/3, -1/3),
 (1, 0, 0, 0, 0, 1/3, 1/3, -1/3),
 (0, -1, 0, 0, 0, 1/3, 1/3, -1/3),
 (0, 0, -1, 0, 0, 1/3, 1/3, -1/3),
 (0, 0, 0, -1, 0, 1/3, 1/3, -1/3),
 (0, 0, 0, 0, -1, 1/3, 1/3, -1/3),
 (-1/2, -1/2, -1/2, -1/2, -1/2, -1/6, -1/6, 1/6),
 (-1, 0, 0, 0, 0, 1/3, 1/3, -1/3)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_E6_element_dual[source]

Bases: LetterTuple

Type \(E_6\) crystal of letters elements. This crystal corresponds to the highest weight crystal \(B(\Lambda_6)\). This crystal is dual to \(B(\Lambda_1)\) of type \(E_6\).

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['E',6], dual = True)
sage: C((-1,)).e(1)
(1, -3)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)], dual = True)
>>> C((-Integer(1),)).e(Integer(1))
(1, -3)
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['E',6], dual = True)
sage: C((6,)).f(6)
(5, -6)
sage: C((6,)).f(1)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)], dual = True)
>>> C((Integer(6),)).f(Integer(6))
(5, -6)
>>> C((Integer(6),)).f(Integer(1))
lift()[source]

Lift an element of self to the crystal of letters crystals.Letters(['E',6]) by taking its inverse weight.

EXAMPLES:

sage: C = crystals.Letters(['E',6], dual = True)
sage: b = C.module_generators[0]
sage: b.lift()
(-6,)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)], dual = True)
>>> b = C.module_generators[Integer(0)]
>>> b.lift()
(-6,)
retract(p)[source]

Retract element p, which is an element in crystals.Letters(['E',6]) to an element in crystals.Letters(['E',6], dual=True) by taking its inverse weight.

EXAMPLES:

sage: C = crystals.Letters(['E',6])
sage: Cd = crystals.Letters(['E',6], dual = True)
sage: b = Cd.module_generators[0]
sage: p = C((-1,3))
sage: b.retract(p)
(1, -3)
sage: b.retract(None)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)])
>>> Cd = crystals.Letters(['E',Integer(6)], dual = True)
>>> b = Cd.module_generators[Integer(0)]
>>> p = C((-Integer(1),Integer(3)))
>>> b.retract(p)
(1, -3)
>>> b.retract(None)
weight()[source]

Return the weight of self.

EXAMPLES:

sage: C = crystals.Letters(['E',6], dual = True)
sage: b=C.module_generators[0]
sage: b.weight()
(0, 0, 0, 0, 1, -1/3, -1/3, 1/3)
sage: [v.weight() for v in C]
[(0, 0, 0, 0, 1, -1/3, -1/3, 1/3),
(0, 0, 0, 1, 0, -1/3, -1/3, 1/3),
(0, 0, 1, 0, 0, -1/3, -1/3, 1/3),
(0, 1, 0, 0, 0, -1/3, -1/3, 1/3),
(-1, 0, 0, 0, 0, -1/3, -1/3, 1/3),
(1, 0, 0, 0, 0, -1/3, -1/3, 1/3),
(1/2, 1/2, 1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(0, -1, 0, 0, 0, -1/3, -1/3, 1/3),
(-1/2, -1/2, 1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(0, 0, -1, 0, 0, -1/3, -1/3, 1/3),
(-1/2, 1/2, -1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, -1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(0, 0, 0, -1, 0, -1/3, -1/3, 1/3),
(-1/2, 1/2, 1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, 1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(1/2, 1/2, -1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(-1/2, -1/2, -1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(0, 0, 0, 0, -1, -1/3, -1/3, 1/3),
(-1/2, 1/2, 1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, 1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, 1/2, -1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(-1/2, -1/2, -1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, 1/2, 1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(-1/2, -1/2, 1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(-1/2, 1/2, -1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, -1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(0, 0, 0, 0, 0, 2/3, 2/3, -2/3)]

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)], dual = True)
>>> b=C.module_generators[Integer(0)]
>>> b.weight()
(0, 0, 0, 0, 1, -1/3, -1/3, 1/3)
>>> [v.weight() for v in C]
[(0, 0, 0, 0, 1, -1/3, -1/3, 1/3),
(0, 0, 0, 1, 0, -1/3, -1/3, 1/3),
(0, 0, 1, 0, 0, -1/3, -1/3, 1/3),
(0, 1, 0, 0, 0, -1/3, -1/3, 1/3),
(-1, 0, 0, 0, 0, -1/3, -1/3, 1/3),
(1, 0, 0, 0, 0, -1/3, -1/3, 1/3),
(1/2, 1/2, 1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(0, -1, 0, 0, 0, -1/3, -1/3, 1/3),
(-1/2, -1/2, 1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(0, 0, -1, 0, 0, -1/3, -1/3, 1/3),
(-1/2, 1/2, -1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, -1/2, 1/2, 1/2, 1/6, 1/6, -1/6),
(0, 0, 0, -1, 0, -1/3, -1/3, 1/3),
(-1/2, 1/2, 1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, 1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(1/2, 1/2, -1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(-1/2, -1/2, -1/2, -1/2, 1/2, 1/6, 1/6, -1/6),
(0, 0, 0, 0, -1, -1/3, -1/3, 1/3),
(-1/2, 1/2, 1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, 1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, 1/2, -1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(-1/2, -1/2, -1/2, 1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, 1/2, 1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(-1/2, -1/2, 1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(-1/2, 1/2, -1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(1/2, -1/2, -1/2, -1/2, -1/2, 1/6, 1/6, -1/6),
(0, 0, 0, 0, 0, 2/3, 2/3, -2/3)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_E7_element[source]

Bases: LetterTuple

Type \(E_7\) crystal of letters elements. This crystal corresponds to the highest weight crystal \(B(\Lambda_7)\).

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['E',7])
sage: C((7,)).e(7)
sage: C((-7,6)).e(7)
(7,)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(7)])
>>> C((Integer(7),)).e(Integer(7))
>>> C((-Integer(7),Integer(6))).e(Integer(7))
(7,)
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['E',7])
sage: C((-7,)).f(7)
sage: C((7,)).f(7)
(-7, 6)

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(7)])
>>> C((-Integer(7),)).f(Integer(7))
>>> C((Integer(7),)).f(Integer(7))
(-7, 6)
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['E',7])]
[(0, 0, 0, 0, 0, 1, -1/2, 1/2), (0, 0, 0, 0, 1, 0, -1/2, 1/2), (0, 0, 0,
1, 0, 0, -1/2, 1/2), (0, 0, 1, 0, 0, 0, -1/2, 1/2), (0, 1, 0, 0, 0, 0,
-1/2, 1/2), (-1, 0, 0, 0, 0, 0, -1/2, 1/2), (1, 0, 0, 0, 0, 0, -1/2,
1/2), (1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 0, 0), (0, -1, 0, 0, 0, 0, -1/2,
1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, 1/2, 0, 0), (0, 0, -1, 0, 0, 0, -1/2,
1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, 1/2, 0, 0), (1/2, -1/2, -1/2, 1/2,
1/2, 1/2, 0, 0), (0, 0, 0, -1, 0, 0, -1/2, 1/2), (-1/2, 1/2, 1/2, -1/2,
1/2, 1/2, 0, 0), (1/2, -1/2, 1/2, -1/2, 1/2, 1/2, 0, 0), (1/2, 1/2,
-1/2, -1/2, 1/2, 1/2, 0, 0), (-1/2, -1/2, -1/2, -1/2, 1/2, 1/2, 0, 0),
(0, 0, 0, 0, -1, 0, -1/2, 1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, 1/2, 0, 0),
(1/2, -1/2, 1/2, 1/2, -1/2, 1/2, 0, 0), (1/2, 1/2, -1/2, 1/2, -1/2, 1/2,
0, 0), (-1/2, -1/2, -1/2, 1/2, -1/2, 1/2, 0, 0), (1/2, 1/2, 1/2, -1/2,
-1/2, 1/2, 0, 0), (-1/2, -1/2, 1/2, -1/2, -1/2, 1/2, 0, 0), (-1/2, 1/2,
-1/2, -1/2, -1/2, 1/2, 0, 0), (1/2, -1/2, -1/2, -1/2, -1/2, 1/2, 0, 0),
(0, 0, 0, 0, 0, 1, 1/2, -1/2), (0, 0, 0, 0, 0, -1, -1/2, 1/2), (-1/2,
1/2, 1/2, 1/2, 1/2, -1/2, 0, 0), (1/2, -1/2, 1/2, 1/2, 1/2, -1/2, 0, 0),
(1/2, 1/2, -1/2, 1/2, 1/2, -1/2, 0, 0), (-1/2, -1/2, -1/2, 1/2, 1/2,
-1/2, 0, 0), (1/2, 1/2, 1/2, -1/2, 1/2, -1/2, 0, 0), (-1/2, -1/2, 1/2,
-1/2, 1/2, -1/2, 0, 0), (-1/2, 1/2, -1/2, -1/2, 1/2, -1/2, 0, 0), (1/2,
-1/2, -1/2, -1/2, 1/2, -1/2, 0, 0), (0, 0, 0, 0, 1, 0, 1/2, -1/2), (1/2,
1/2, 1/2, 1/2, -1/2, -1/2, 0, 0), (-1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 0,
0), (-1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 0, 0), (1/2, -1/2, -1/2, 1/2,
-1/2, -1/2, 0, 0), (0, 0, 0, 1, 0, 0, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2,
-1/2, -1/2, 0, 0), (1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 0, 0), (0, 0, 1,
0, 0, 0, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 0, 0), (0, 1, 0,
0, 0, 0, 1/2, -1/2), (1, 0, 0, 0, 0, 0, 1/2, -1/2), (0, -1, 0, 0, 0, 0,
1/2, -1/2), (0, 0, -1, 0, 0, 0, 1/2, -1/2), (0, 0, 0, -1, 0, 0, 1/2,
-1/2), (0, 0, 0, 0, -1, 0, 1/2, -1/2), (0, 0, 0, 0, 0, -1, 1/2, -1/2),
(-1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 0, 0), (-1, 0, 0, 0, 0, 0, 1/2,
-1/2)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['E',Integer(7)])]
[(0, 0, 0, 0, 0, 1, -1/2, 1/2), (0, 0, 0, 0, 1, 0, -1/2, 1/2), (0, 0, 0,
1, 0, 0, -1/2, 1/2), (0, 0, 1, 0, 0, 0, -1/2, 1/2), (0, 1, 0, 0, 0, 0,
-1/2, 1/2), (-1, 0, 0, 0, 0, 0, -1/2, 1/2), (1, 0, 0, 0, 0, 0, -1/2,
1/2), (1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 0, 0), (0, -1, 0, 0, 0, 0, -1/2,
1/2), (-1/2, -1/2, 1/2, 1/2, 1/2, 1/2, 0, 0), (0, 0, -1, 0, 0, 0, -1/2,
1/2), (-1/2, 1/2, -1/2, 1/2, 1/2, 1/2, 0, 0), (1/2, -1/2, -1/2, 1/2,
1/2, 1/2, 0, 0), (0, 0, 0, -1, 0, 0, -1/2, 1/2), (-1/2, 1/2, 1/2, -1/2,
1/2, 1/2, 0, 0), (1/2, -1/2, 1/2, -1/2, 1/2, 1/2, 0, 0), (1/2, 1/2,
-1/2, -1/2, 1/2, 1/2, 0, 0), (-1/2, -1/2, -1/2, -1/2, 1/2, 1/2, 0, 0),
(0, 0, 0, 0, -1, 0, -1/2, 1/2), (-1/2, 1/2, 1/2, 1/2, -1/2, 1/2, 0, 0),
(1/2, -1/2, 1/2, 1/2, -1/2, 1/2, 0, 0), (1/2, 1/2, -1/2, 1/2, -1/2, 1/2,
0, 0), (-1/2, -1/2, -1/2, 1/2, -1/2, 1/2, 0, 0), (1/2, 1/2, 1/2, -1/2,
-1/2, 1/2, 0, 0), (-1/2, -1/2, 1/2, -1/2, -1/2, 1/2, 0, 0), (-1/2, 1/2,
-1/2, -1/2, -1/2, 1/2, 0, 0), (1/2, -1/2, -1/2, -1/2, -1/2, 1/2, 0, 0),
(0, 0, 0, 0, 0, 1, 1/2, -1/2), (0, 0, 0, 0, 0, -1, -1/2, 1/2), (-1/2,
1/2, 1/2, 1/2, 1/2, -1/2, 0, 0), (1/2, -1/2, 1/2, 1/2, 1/2, -1/2, 0, 0),
(1/2, 1/2, -1/2, 1/2, 1/2, -1/2, 0, 0), (-1/2, -1/2, -1/2, 1/2, 1/2,
-1/2, 0, 0), (1/2, 1/2, 1/2, -1/2, 1/2, -1/2, 0, 0), (-1/2, -1/2, 1/2,
-1/2, 1/2, -1/2, 0, 0), (-1/2, 1/2, -1/2, -1/2, 1/2, -1/2, 0, 0), (1/2,
-1/2, -1/2, -1/2, 1/2, -1/2, 0, 0), (0, 0, 0, 0, 1, 0, 1/2, -1/2), (1/2,
1/2, 1/2, 1/2, -1/2, -1/2, 0, 0), (-1/2, -1/2, 1/2, 1/2, -1/2, -1/2, 0,
0), (-1/2, 1/2, -1/2, 1/2, -1/2, -1/2, 0, 0), (1/2, -1/2, -1/2, 1/2,
-1/2, -1/2, 0, 0), (0, 0, 0, 1, 0, 0, 1/2, -1/2), (-1/2, 1/2, 1/2, -1/2,
-1/2, -1/2, 0, 0), (1/2, -1/2, 1/2, -1/2, -1/2, -1/2, 0, 0), (0, 0, 1,
0, 0, 0, 1/2, -1/2), (1/2, 1/2, -1/2, -1/2, -1/2, -1/2, 0, 0), (0, 1, 0,
0, 0, 0, 1/2, -1/2), (1, 0, 0, 0, 0, 0, 1/2, -1/2), (0, -1, 0, 0, 0, 0,
1/2, -1/2), (0, 0, -1, 0, 0, 0, 1/2, -1/2), (0, 0, 0, -1, 0, 0, 1/2,
-1/2), (0, 0, 0, 0, -1, 0, 1/2, -1/2), (0, 0, 0, 0, 0, -1, 1/2, -1/2),
(-1/2, -1/2, -1/2, -1/2, -1/2, -1/2, 0, 0), (-1, 0, 0, 0, 0, 0, 1/2,
-1/2)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_G_element[source]

Bases: Letter

Type \(G_2\) crystal of letters elements.

e(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['G',2])
sage: [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (0, 1, 3),
 (-3, 1, 0),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3)]

>>> from sage.all import *
>>> C = crystals.Letters(['G',Integer(2)])
>>> [(c,i,c.e(i)) for i in C.index_set() for c in C if c.e(i) is not None]
[(2, 1, 1),
 (0, 1, 3),
 (-3, 1, 0),
 (-1, 1, -2),
 (3, 2, 2),
 (-2, 2, -3)]
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['G',2])
sage: [(c,i,c.epsilon(i)) for i in C.index_set() for c in C if c.epsilon(i) != 0]
[(2, 1, 1), (0, 1, 1), (-3, 1, 2), (-1, 1, 1), (3, 2, 1), (-2, 2, 1)]

>>> from sage.all import *
>>> C = crystals.Letters(['G',Integer(2)])
>>> [(c,i,c.epsilon(i)) for i in C.index_set() for c in C if c.epsilon(i) != Integer(0)]
[(2, 1, 1), (0, 1, 1), (-3, 1, 2), (-1, 1, 1), (3, 2, 1), (-2, 2, 1)]
f(i)[source]

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

EXAMPLES:

sage: C = crystals.Letters(['G',2])
sage: [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2),
 (3, 1, 0),
 (0, 1, -3),
 (-2, 1, -1),
 (2, 2, 3),
 (-3, 2, -2)]

>>> from sage.all import *
>>> C = crystals.Letters(['G',Integer(2)])
>>> [(c,i,c.f(i)) for i in C.index_set() for c in C if c.f(i) is not None]
[(1, 1, 2),
 (3, 1, 0),
 (0, 1, -3),
 (-2, 1, -1),
 (2, 2, 3),
 (-3, 2, -2)]
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['G',2])
sage: [(c,i,c.phi(i)) for i in C.index_set() for c in C if c.phi(i) != 0]
[(1, 1, 1), (3, 1, 2), (0, 1, 1), (-2, 1, 1), (2, 2, 1), (-3, 2, 1)]

>>> from sage.all import *
>>> C = crystals.Letters(['G',Integer(2)])
>>> [(c,i,c.phi(i)) for i in C.index_set() for c in C if c.phi(i) != Integer(0)]
[(1, 1, 1), (3, 1, 2), (0, 1, 1), (-2, 1, 1), (2, 2, 1), (-3, 2, 1)]
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['G',2])]
[(1, 0, -1), (1, -1, 0), (0, 1, -1), (0, 0, 0), (0, -1, 1), (-1, 1, 0), (-1, 0, 1)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['G',Integer(2)])]
[(1, 0, -1), (1, -1, 0), (0, 1, -1), (0, 0, 0), (0, -1, 1), (-1, 1, 0), (-1, 0, 1)]
class sage.combinat.crystals.letters.EmptyLetter[source]

Bases: Element

The affine letter \(\emptyset\) thought of as a classical crystal letter in classical type \(B_n\) and \(C_n\).

Warning

This is not a classical letter.

Used in the rigged configuration bijections.

e(i)[source]

Return \(e_i\) of self which is None.

EXAMPLES:

sage: C = crystals.Letters(['C', 3])
sage: C('E').e(1)

>>> from sage.all import *
>>> C = crystals.Letters(['C', Integer(3)])
>>> C('E').e(Integer(1))
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['C', 3])
sage: C('E').epsilon(1)
0

>>> from sage.all import *
>>> C = crystals.Letters(['C', Integer(3)])
>>> C('E').epsilon(Integer(1))
0
f(i)[source]

Return \(f_i\) of self which is None.

EXAMPLES:

sage: C = crystals.Letters(['C', 3])
sage: C('E').f(1)

>>> from sage.all import *
>>> C = crystals.Letters(['C', Integer(3)])
>>> C('E').f(Integer(1))
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['C', 3])
sage: C('E').phi(1)
0

>>> from sage.all import *
>>> C = crystals.Letters(['C', Integer(3)])
>>> C('E').phi(Integer(1))
0
value[source]
weight()[source]

Return the weight of self.

EXAMPLES:

sage: C = crystals.Letters(['C', 3])
sage: C('E').weight()
(0, 0, 0)

>>> from sage.all import *
>>> C = crystals.Letters(['C', Integer(3)])
>>> C('E').weight()
(0, 0, 0)
class sage.combinat.crystals.letters.Letter[source]

Bases: Element

A class for letters.

Like ElementWrapper, plus delegates __lt__ (comparison) to the parent.

EXAMPLES:

sage: from sage.combinat.crystals.letters import Letter
sage: a = Letter(ZZ, 1)
sage: Letter(ZZ, 1).parent()
Integer Ring

sage: Letter(ZZ, 1)._repr_()
'1'

sage: parent1 = ZZ  # Any fake value ...
sage: parent2 = QQ  # Any fake value ...
sage: l11 = Letter(parent1, 1)
sage: l12 = Letter(parent1, 2)
sage: l21 = Letter(parent2, 1)
sage: l22 = Letter(parent2, 2)
sage: l11 == l11
True
sage: l11 == l12
False
sage: l11 == l21 # not tested
False

sage: C = crystals.Letters(['B', 3])
sage: C(0) != C(0)
False
sage: C(1) != C(-1)
True

>>> from sage.all import *
>>> from sage.combinat.crystals.letters import Letter
>>> a = Letter(ZZ, Integer(1))
>>> Letter(ZZ, Integer(1)).parent()
Integer Ring

>>> Letter(ZZ, Integer(1))._repr_()
'1'

>>> parent1 = ZZ  # Any fake value ...
>>> parent2 = QQ  # Any fake value ...
>>> l11 = Letter(parent1, Integer(1))
>>> l12 = Letter(parent1, Integer(2))
>>> l21 = Letter(parent2, Integer(1))
>>> l22 = Letter(parent2, Integer(2))
>>> l11 == l11
True
>>> l11 == l12
False
>>> l11 == l21 # not tested
False

>>> C = crystals.Letters(['B', Integer(3)])
>>> C(Integer(0)) != C(Integer(0))
False
>>> C(Integer(1)) != C(-Integer(1))
True
value[source]
class sage.combinat.crystals.letters.LetterTuple[source]

Bases: Element

Abstract class for type \(E\) letters.

epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['E',6])
sage: C((-6,)).epsilon(1)
0
sage: C((-6,)).epsilon(6)
1

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)])
>>> C((-Integer(6),)).epsilon(Integer(1))
0
>>> C((-Integer(6),)).epsilon(Integer(6))
1
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['E',6])
sage: C((1,)).phi(1)
1
sage: C((1,)).phi(6)
0

>>> from sage.all import *
>>> C = crystals.Letters(['E',Integer(6)])
>>> C((Integer(1),)).phi(Integer(1))
1
>>> C((Integer(1),)).phi(Integer(6))
0
value[source]
class sage.combinat.crystals.letters.LetterWrapped[source]

Bases: Element

Element which uses another crystal implementation and converts those elements to a tuple with \(\pm i\).

e(i)[source]

Return \(e_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['E', 8])
sage: C((-8,)).e(1)
sage: C((-8,)).e(8)
(-7, 8)

>>> from sage.all import *
>>> C = crystals.Letters(['E', Integer(8)])
>>> C((-Integer(8),)).e(Integer(1))
>>> C((-Integer(8),)).e(Integer(8))
(-7, 8)
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['E', 8])
sage: C((-8,)).epsilon(1)
0
sage: C((-8,)).epsilon(8)
1

>>> from sage.all import *
>>> C = crystals.Letters(['E', Integer(8)])
>>> C((-Integer(8),)).epsilon(Integer(1))
0
>>> C((-Integer(8),)).epsilon(Integer(8))
1
f(i)[source]

Return \(f_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['E', 8])
sage: C((8,)).f(6)
sage: C((8,)).f(8)
(7, -8)

>>> from sage.all import *
>>> C = crystals.Letters(['E', Integer(8)])
>>> C((Integer(8),)).f(Integer(6))
>>> C((Integer(8),)).f(Integer(8))
(7, -8)
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Letters(['E', 8])
sage: C((8,)).phi(8)
1
sage: C((8,)).phi(6)
0

>>> from sage.all import *
>>> C = crystals.Letters(['E', Integer(8)])
>>> C((Integer(8),)).phi(Integer(8))
1
>>> C((Integer(8),)).phi(Integer(6))
0
value[source]
class sage.combinat.crystals.letters.QueerLetter_element[source]

Bases: Letter

Queer supercrystal letters elements.

e(i)[source]

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

EXAMPLES:

sage: Q = crystals.Letters(['Q',3])
sage: [(c,i,c.e(i)) for i in Q.index_set() for c in Q if c.e(i) is not None]
[(2, 1, 1), (3, 2, 2), (3, -2, 2), (2, -1, 1)]

>>> from sage.all import *
>>> Q = crystals.Letters(['Q',Integer(3)])
>>> [(c,i,c.e(i)) for i in Q.index_set() for c in Q if c.e(i) is not None]
[(2, 1, 1), (3, 2, 2), (3, -2, 2), (2, -1, 1)]
epsilon(i)[source]

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: Q = crystals.Letters(['Q',3])
sage: [(c,i) for i in Q.index_set() for c in Q if c.epsilon(i) != 0]
[(2, 1), (3, 2), (3, -2), (2, -1)]

>>> from sage.all import *
>>> Q = crystals.Letters(['Q',Integer(3)])
>>> [(c,i) for i in Q.index_set() for c in Q if c.epsilon(i) != Integer(0)]
[(2, 1), (3, 2), (3, -2), (2, -1)]
f(i)[source]

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

EXAMPLES:

sage: Q = crystals.Letters(['Q',3])
sage: [(c,i,c.f(i)) for i in Q.index_set() for c in Q if c.f(i) is not None]
[(1, 1, 2), (2, 2, 3), (2, -2, 3), (1, -1, 2)]

>>> from sage.all import *
>>> Q = crystals.Letters(['Q',Integer(3)])
>>> [(c,i,c.f(i)) for i in Q.index_set() for c in Q if c.f(i) is not None]
[(1, 1, 2), (2, 2, 3), (2, -2, 3), (1, -1, 2)]
phi(i)[source]

Return \(\varphi_i\) of self.

EXAMPLES:

sage: Q = crystals.Letters(['Q',3])
sage: [(c,i) for i in Q.index_set() for c in Q if c.phi(i) != 0]
[(1, 1), (2, 2), (2, -2), (1, -1)]

>>> from sage.all import *
>>> Q = crystals.Letters(['Q',Integer(3)])
>>> [(c,i) for i in Q.index_set() for c in Q if c.phi(i) != Integer(0)]
[(1, 1), (2, 2), (2, -2), (1, -1)]
weight()[source]

Return the weight of self.

EXAMPLES:

sage: [v.weight() for v in crystals.Letters(['Q',4])]
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]

>>> from sage.all import *
>>> [v.weight() for v in crystals.Letters(['Q',Integer(4)])]
[(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)]