Root system data for type A¶
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class
sage.combinat.root_system.type_A.
AmbientSpace
(root_system, base_ring, index_set=None)¶ Bases:
sage.combinat.root_system.ambient_space.AmbientSpace
EXAMPLES:
sage: R = RootSystem(["A",3]) sage: e = R.ambient_space(); e Ambient space of the Root system of type ['A', 3] sage: TestSuite(e).run()
By default, this ambient space uses the barycentric projection for plotting:
sage: L = RootSystem(["A",2]).ambient_space() sage: e = L.basis() sage: L._plot_projection(e[0]) (1/2, 989/1142) sage: L._plot_projection(e[1]) (-1, 0) sage: L._plot_projection(e[2]) (1/2, -989/1142) sage: L = RootSystem(["A",3]).ambient_space() sage: l = L.an_element(); l (2, 2, 3, 0) sage: L._plot_projection(l) (0, -1121/1189, 7/3)
See also
sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods._plot_projection()
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det
(k=1)¶ returns the vector (1, … ,1) which in the [‘A’,r] weight lattice, interpreted as a weight of GL(r+1,CC) is the determinant. If the optional parameter k is given, returns (k, … ,k), the k-th power of the determinant.
EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_space() sage: e.det(1/2) (1/2, 1/2, 1/2, 1/2)
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dimension
()¶ EXAMPLES:
sage: e = RootSystem(["A",3]).ambient_space() sage: e.dimension() 4
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fundamental_weight
(i)¶ EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.fundamental_weights() Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1, 1, 1, 0)}
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highest_root
()¶ EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.highest_root() (1, 0, 0, -1)
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negative_roots
()¶ EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.negative_roots() [(-1, 1, 0, 0), (-1, 0, 1, 0), (-1, 0, 0, 1), (0, -1, 1, 0), (0, -1, 0, 1), (0, 0, -1, 1)]
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positive_roots
()¶ EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.positive_roots() [(1, -1, 0, 0), (1, 0, -1, 0), (0, 1, -1, 0), (1, 0, 0, -1), (0, 1, 0, -1), (0, 0, 1, -1)]
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root
(i, j)¶ Note that indexing starts at 0.
EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.root(0,1) (1, -1, 0, 0)
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simple_root
(i)¶ EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.simple_roots() Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}
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classmethod
smallest_base_ring
(cartan_type=None)¶ Returns the smallest base ring the ambient space can be defined upon
See also
EXAMPLES:
sage: e = RootSystem(["A",3]).ambient_space() sage: e.smallest_base_ring() Integer Ring
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class
sage.combinat.root_system.type_A.
CartanType
(n)¶ Bases:
sage.combinat.root_system.cartan_type.CartanType_standard_finite
,sage.combinat.root_system.cartan_type.CartanType_simply_laced
,sage.combinat.root_system.cartan_type.CartanType_simple
Cartan Type \(A_n\)
See also
CartanType()
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AmbientSpace
¶ alias of
AmbientSpace
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PieriFactors
¶ alias of
sage.combinat.root_system.pieri_factors.PieriFactors_type_A
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ascii_art
(label=<function CartanType.<lambda> at 0x7f96d34de3a0>, node=None)¶ Return an ascii art representation of the Dynkin diagram.
EXAMPLES:
sage: print(CartanType(['A',0]).ascii_art()) sage: print(CartanType(['A',1]).ascii_art()) O 1 sage: print(CartanType(['A',3]).ascii_art()) O---O---O 1 2 3 sage: print(CartanType(['A',12]).ascii_art()) O---O---O---O---O---O---O---O---O---O---O---O 1 2 3 4 5 6 7 8 9 10 11 12 sage: print(CartanType(['A',5]).ascii_art(label = lambda x: x+2)) O---O---O---O---O 3 4 5 6 7 sage: print(CartanType(['A',5]).ascii_art(label = lambda x: x-2)) O---O---O---O---O -1 0 1 2 3
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coxeter_number
()¶ Return the Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['A',4]).coxeter_number() 5
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dual_coxeter_number
()¶ Return the dual Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['A',4]).dual_coxeter_number() 5
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dynkin_diagram
()¶ Returns the Dynkin diagram of type A.
EXAMPLES:
sage: a = CartanType(['A',3]).dynkin_diagram() sage: a O---O---O 1 2 3 A3 sage: sorted(a.edges()) [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1)]
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