Root system data for type A#
- class sage.combinat.root_system.type_A.AmbientSpace(root_system, base_ring, index_set=None)[source]#
Bases:
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() # needs sage.graphs
>>> from sage.all import * >>> R = RootSystem(["A",Integer(3)]) >>> e = R.ambient_space(); e Ambient space of the Root system of type ['A', 3] >>> TestSuite(e).run() # needs sage.graphs
By default, this ambient space uses the barycentric projection for plotting:
sage: # needs sage.symbolic 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)
>>> from sage.all import * >>> # needs sage.symbolic >>> L = RootSystem(["A",Integer(2)]).ambient_space() >>> e = L.basis() >>> L._plot_projection(e[Integer(0)]) (1/2, 989/1142) >>> L._plot_projection(e[Integer(1)]) (-1, 0) >>> L._plot_projection(e[Integer(2)]) (1/2, -989/1142) >>> L = RootSystem(["A",Integer(3)]).ambient_space() >>> l = L.an_element(); l (2, 2, 3, 0) >>> L._plot_projection(l) (0, -1121/1189, 7/3)
See also
sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods._plot_projection()
- det(k=1)[source]#
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)
>>> from sage.all import * >>> e = RootSystem(['A',Integer(3)]).ambient_space() >>> e.det(Integer(1)/Integer(2)) (1/2, 1/2, 1/2, 1/2)
- dimension()[source]#
EXAMPLES:
sage: e = RootSystem(["A",3]).ambient_space() sage: e.dimension() 4
>>> from sage.all import * >>> e = RootSystem(["A",Integer(3)]).ambient_space() >>> e.dimension() 4
- fundamental_weight(i)[source]#
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)}
>>> from sage.all import * >>> e = RootSystem(['A',Integer(3)]).ambient_lattice() >>> e.fundamental_weights() Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1, 1, 1, 0)}
- highest_root()[source]#
EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.highest_root() (1, 0, 0, -1)
>>> from sage.all import * >>> e = RootSystem(['A',Integer(3)]).ambient_lattice() >>> e.highest_root() (1, 0, 0, -1)
- negative_roots()[source]#
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)]
>>> from sage.all import * >>> e = RootSystem(['A',Integer(3)]).ambient_lattice() >>> 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)]
- positive_roots()[source]#
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)]
>>> from sage.all import * >>> e = RootSystem(['A',Integer(3)]).ambient_lattice() >>> 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)]
- root(i, j)[source]#
Note that indexing starts at 0.
EXAMPLES:
sage: e = RootSystem(['A',3]).ambient_lattice() sage: e.root(0,1) (1, -1, 0, 0)
>>> from sage.all import * >>> e = RootSystem(['A',Integer(3)]).ambient_lattice() >>> e.root(Integer(0),Integer(1)) (1, -1, 0, 0)
- simple_root(i)[source]#
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)}
>>> from sage.all import * >>> e = RootSystem(['A',Integer(3)]).ambient_lattice() >>> e.simple_roots() Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}
- classmethod smallest_base_ring(cartan_type=None)[source]#
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
>>> from sage.all import * >>> e = RootSystem(["A",Integer(3)]).ambient_space() >>> e.smallest_base_ring() Integer Ring
- class sage.combinat.root_system.type_A.CartanType(n)[source]#
Bases:
CartanType_standard_finite
,CartanType_simply_laced
,CartanType_simple
Cartan Type \(A_n\)
See also
CartanType()
- AmbientSpace[source]#
alias of
AmbientSpace
- PieriFactors[source]#
alias of
PieriFactors_type_A
- ascii_art(label=None, node=None)[source]#
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
>>> from sage.all import * >>> print(CartanType(['A',Integer(0)]).ascii_art()) >>> print(CartanType(['A',Integer(1)]).ascii_art()) O 1 >>> print(CartanType(['A',Integer(3)]).ascii_art()) O---O---O 1 2 3 >>> print(CartanType(['A',Integer(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 >>> print(CartanType(['A',Integer(5)]).ascii_art(label = lambda x: x+Integer(2))) O---O---O---O---O 3 4 5 6 7 >>> print(CartanType(['A',Integer(5)]).ascii_art(label = lambda x: x-Integer(2))) O---O---O---O---O -1 0 1 2 3
- coxeter_number()[source]#
Return the Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['A',4]).coxeter_number() 5
>>> from sage.all import * >>> CartanType(['A',Integer(4)]).coxeter_number() 5
- dual_coxeter_number()[source]#
Return the dual Coxeter number associated with
self
.EXAMPLES:
sage: CartanType(['A',4]).dual_coxeter_number() 5
>>> from sage.all import * >>> CartanType(['A',Integer(4)]).dual_coxeter_number() 5
- dynkin_diagram()[source]#
Returns the Dynkin diagram of type A.
EXAMPLES:
sage: a = CartanType(['A',3]).dynkin_diagram(); a # needs sage.graphs O---O---O 1 2 3 A3 sage: a.edges(sort=True) # needs sage.graphs [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1)]
>>> from sage.all import * >>> a = CartanType(['A',Integer(3)]).dynkin_diagram(); a # needs sage.graphs O---O---O 1 2 3 A3 >>> a.edges(sort=True) # needs sage.graphs [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1)]