Crystals of letters¶
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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 subclassClassicalCrystalOfLetters
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
, andf
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]
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lt_elements
(x, y)¶ Return
True
if and only if there is a path fromx
toy
in the crystal graph, whenx
is not equal toy
.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
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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]
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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()
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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)]
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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)]
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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)]
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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)]
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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)]
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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()
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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)]
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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)]
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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)]
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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)]
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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)]
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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()
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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)]
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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)]
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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)]
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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)]
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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)]
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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()
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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)]
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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)]
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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)]
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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)]
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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)]
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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)
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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)
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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)
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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)]
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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)
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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)
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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)
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lift
()¶ Lift an element of
self
to the crystal of letterscrystals.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,)
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retract
(p)¶ Retract element
p
, which is an element incrystals.Letters(['E',6])
to an element incrystals.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)
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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)]
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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)
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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,)
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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)
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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)]
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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()
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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)]
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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)]
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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)]
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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)]
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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)]
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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.
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e
(i)¶ Return \(e_i\) of
self
which isNone
.EXAMPLES:
sage: C = crystals.Letters(['C', 3]) sage: C('E').e(1)
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epsilon
(i)¶ Return \(\varepsilon_i\) of
self
.EXAMPLES:
sage: C = crystals.Letters(['C', 3]) sage: C('E').epsilon(1) 0
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f
(i)¶ Return \(f_i\) of
self
which isNone
.EXAMPLES:
sage: C = crystals.Letters(['C', 3]) sage: C('E').f(1)
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phi
(i)¶ Return \(\varphi_i\) of
self
.EXAMPLES:
sage: C = crystals.Letters(['C', 3]) sage: C('E').phi(1) 0
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value
¶
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weight
()¶ Return the weight of
self
.EXAMPLES:
sage: C = crystals.Letters(['C', 3]) sage: C('E').weight() (0, 0, 0)
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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
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value
¶
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class
sage.combinat.crystals.letters.
LetterTuple
¶ Bases:
sage.structure.element.Element
Abstract class for type \(E\) letters.
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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
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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
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value
¶
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