Crystals of letters

Crystals of letters

class sage.combinat.crystals.letters.ClassicalCrystalOfLetters(cartan_type, element_class, element_print_style=None, dual=None)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.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()

Return a list of the elements of self.

EXAMPLES:

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

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

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 [KN94]. 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).

INPUT:

  • T – A Cartan type

REFERENCES:

[KN94]M. Kashiwara and T. Nakashima. Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra 165, no. 2, pp. 295–345, 1994.

EXAMPLES:

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

Bases: sage.combinat.crystals.letters.Letter

Type A crystal of letters elements.

TESTS:

sage: C = crystals.Letters(['A',3])
sage: C.list()
[1, 2, 3, 4]
sage: [ [x < y for y in C] for x in C ]
[[False, True, True, True],
 [False, False, True, True],
 [False, False, False, True],
 [False, False, False, False]]
sage: C = crystals.Letters(['A',5])
sage: C(1) < C(1), C(1) < C(2), C(1) < C(3), C(2) < C(1)
(False, True, True, False)
sage: TestSuite(C).run()
e(i)

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)]
epsilon(i)

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)]
f(i)

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)]
phi(i)

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)]
weight()

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)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_B_element

Bases: sage.combinat.crystals.letters.Letter

Type B crystal of letters elements.

TESTS:

sage: C = crystals.Letters(['B',3])
sage: TestSuite(C).run()
e(i)

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)]
epsilon(i)

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)]
f(i)

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)]
phi(i)

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)]
weight()

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)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_C_element

Bases: sage.combinat.crystals.letters.Letter

Type C crystal of letters elements.

TESTS:

sage: C = crystals.Letters (['C',3])
sage: C.list()
[1, 2, 3, -3, -2, -1]
sage: [ [x < y for y in C] for x in C ]
[[False, True, True, True, True, True],
 [False, False, True, True, True, True],
 [False, False, False, True, True, True],
 [False, False, False, False, True, True],
 [False, False, False, False, False, True],
 [False, False, False, False, False, False]]
sage: TestSuite(C).run()
e(i)

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)]
epsilon(i)

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)]
f(i)

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)]
phi(i)

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)]
weight()

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)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_D_element

Bases: sage.combinat.crystals.letters.Letter

Type D crystal of letters elements.

TESTS:

sage: C = crystals.Letters(['D',4])
sage: C.list()
[1, 2, 3, 4, -4, -3, -2, -1]
sage: TestSuite(C).run()
e(i)

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)]
epsilon(i)

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)]
f(i)

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)]
phi(i)

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)]
weight()

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)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_E6_element

Bases: sage.combinat.crystals.letters.LetterTuple

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

TESTS:

sage: C = crystals.Letters(['E',6])
sage: C.module_generators
((1,),)
sage: C.list()
[(1,), (-1, 3), (-3, 4), (-4, 2, 5), (-2, 5), (-5, 2, 6), (-2, -5, 4, 6),
(-4, 3, 6), (-3, 1, 6), (-1, 6), (-6, 2), (-2, -6, 4), (-4, -6, 3, 5),
(-3, -6, 1, 5), (-1, -6, 5), (-5, 3), (-3, -5, 1, 4), (-1, -5, 4), (-4, 1, 2),
(-1, -4, 2, 3), (-3, 2), (-2, -3, 4), (-4, 5), (-5, 6), (-6,), (-2, 1), (-1, -2, 3)]
sage: TestSuite(C).run()
sage: all(b.f(i).e(i) == b for i in C.index_set() for b in C if b.f(i) is not None)
True
sage: all(b.e(i).f(i) == b for i in C.index_set() for b in C if b.e(i) is not None)
True
sage: G = C.digraph()
sage: G.show(edge_labels=true, figsize=12, vertex_size=1)
e(i)

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)
f(i)

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)
weight()

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)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_E6_element_dual

Bases: sage.combinat.crystals.letters.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.

TESTS:

sage: C = crystals.Letters(['E',6], dual = True)
sage: C.module_generators
((6,),)
sage: all(b==b.retract(b.lift()) for b in C)
True
sage: C.list()
[(6,), (5, -6), (4, -5), (2, 3, -4), (3, -2), (1, 2, -3), (2, -1), (1, 4, -2, -3),
 (4, -1, -2), (1, 5, -4), (3, 5, -1, -4), (5, -3), (1, 6, -5), (3, 6, -1, -5), (4, 6, -3, -5),
 (2, 6, -4), (6, -2), (1, -6), (3, -1, -6), (4, -3, -6), (2, 5, -4, -6), (5, -2, -6), (2, -5),
 (4, -2, -5), (3, -4), (1, -3), (-1,)]
sage: TestSuite(C).run()
sage: all(b.f(i).e(i) == b for i in C.index_set() for b in C if b.f(i) is not None)
True
sage: all(b.e(i).f(i) == b for i in C.index_set() for b in C if b.e(i) is not None)
True
sage: G = C.digraph()
sage: G.show(edge_labels=true, figsize=12, vertex_size=1)
e(i)

Return the action of e_i on self.

EXAMPLES:

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

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)
lift()

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,)
retract(p)

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)
weight()

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)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_E7_element

Bases: sage.combinat.crystals.letters.LetterTuple

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

TESTS:

sage: C = crystals.Letters(['E',7])
sage: C.module_generators
((7,),)
sage: C.list()
[(7,), (-7, 6), (-6, 5), (-5, 4), (-4, 2, 3), (-2, 3), (-3, 1, 2), (-1,
2), (-3, -2, 1, 4), (-1, -2, 4), (-4, 1, 5), (-4, -1, 3, 5), (-3, 5),
(-5, 6, 1), (-5, -1, 3, 6), (-5, -3, 4, 6), (-4, 2, 6), (-2, 6), (-6, 7,
1), (-1, -6, 3, 7), (-6, -3, 7, 4), (-6, -4, 2, 7, 5), (-6, -2, 7, 5),
(-5, 7, 2), (-5, -2, 4, 7), (-4, 7, 3), (-3, 1, 7), (-1, 7), (-7, 1),
(-1, -7, 3), (-7, -3, 4), (-4, -7, 2, 5), (-7, -2, 5), (-5, -7, 6, 2),
(-5, -2, -7, 4, 6), (-7, -4, 6, 3), (-3, -7, 1, 6), (-7, -1, 6), (-6,
2), (-2, -6, 4), (-6, -4, 5, 3), (-3, -6, 1, 5), (-6, -1, 5), (-5, 3),
(-3, -5, 4, 1), (-5, -1, 4), (-4, 1, 2), (-1, -4, 3, 2), (-3, 2), (-2,
-3, 4), (-4, 5), (-5, 6), (-6, 7), (-7,), (-2, 1), (-2, -1, 3)]
sage: TestSuite(C).run()
sage: all(b.f(i).e(i) == b for i in C.index_set() for b in C if b.f(i) is not None)
True
sage: all(b.e(i).f(i) == b for i in C.index_set() for b in C if b.e(i) is not None)
True
sage: G = C.digraph()
sage: G.show(edge_labels=true, figsize=12, vertex_size=1)
e(i)

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,)
f(i)

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)
weight()

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)]
class sage.combinat.crystals.letters.Crystal_of_letters_type_G_element

Bases: sage.combinat.crystals.letters.Letter

Type G_2 crystal of letters elements.

TESTS:

sage: C = crystals.Letters(['G',2])
sage: C.list()
[1, 2, 3, 0, -3, -2, -1]
sage: TestSuite(C).run()
e(i)

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)]
epsilon(i)

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)]
f(i)

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)]
phi(i)

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)]
weight()

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)]
class sage.combinat.crystals.letters.EmptyLetter

Bases: sage.structure.element.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)

Return e_i of self which is None.

EXAMPLES:

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

Return \varepsilon_i of self.

EXAMPLES:

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

Return f_i of self which is None.

EXAMPLES:

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

Return \varphi_i of self.

EXAMPLES:

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

Return the weight of self.

EXAMPLES:

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

Bases: sage.structure.element.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
value
class sage.combinat.crystals.letters.LetterTuple

Bases: sage.structure.element.Element

Abstract class for type E letters.

epsilon(i)

Return \varepsilon_i of self.

EXAMPLES:

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

Return \varphi_i of self.

EXAMPLES:

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

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