# Crystals of letters¶

Crystals of letters

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

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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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

Type $$A$$ crystal of letters elements.

TESTS:

sage: C = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 CrystalOfLetters(['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

Type $$B$$ crystal of letters elements.

TESTS:

sage: C = CrystalOfLetters(['B',3])
sage: TestSuite(C).run()

e(i)

Return the action of $$e_i$$ on self.

EXAMPLES:

sage: C = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 CrystalOfLetters(['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

Type $$C$$ crystal of letters elements.

TESTS:

sage: C = CrystalOfLetters (['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 CrystalOfLetters(['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

Type $$D$$ crystal of letters elements.

TESTS:

sage: C = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 CrystalOfLetters(['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

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

TESTS:

sage: C = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 CrystalOfLetters(['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

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

f(i)

Return the action of $$f_i$$ on self.

EXAMPLES:

sage: C = CrystalOfLetters(['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 CrystalOfLetters(['E',6]) by taking its inverse weight.

EXAMPLES:

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

retract(p)

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

EXAMPLES:

sage: C = CrystalOfLetters(['E',6])
sage: Cd = CrystalOfLetters(['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 = CrystalOfLetters(['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

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

TESTS:

sage: C = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 CrystalOfLetters(['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

Type $$G_2$$ crystal of letters elements.

TESTS:

sage: C = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 = CrystalOfLetters(['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 CrystalOfLetters(['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

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

epsilon(i)

Return $$\varepsilon_i$$ of self.

EXAMPLES:

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

f(i)

Return $$f_i$$ of self which is None.

EXAMPLES:

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

phi(i)

Return $$\varphi_i$$ of self.

EXAMPLES:

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

value
weight()

Return the weight of self.

EXAMPLES:

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

class sage.combinat.crystals.letters.Letter

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

value
class sage.combinat.crystals.letters.LetterTuple

Abstract class for type $$E$$ letters.

epsilon(i)

Return $$\varepsilon_i$$ of self.

EXAMPLES:

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

phi(i)

Return $$\varphi_i$$ of self.

EXAMPLES:

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

value

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