Root system data for type A#

class sage.combinat.root_system.type_A.AmbientSpace(root_system, base_ring, index_set=None)#

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()

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])                                                  # optional - sage.symbolic
(1/2, 989/1142)
sage: L._plot_projection(e[1])                                                  # optional - sage.symbolic
(-1, 0)
sage: L._plot_projection(e[2])                                                  # optional - sage.symbolic
(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)                                                     # optional - sage.symbolic
(0, -1121/1189, 7/3)

See also

sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods._plot_projection()

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)

dimension()#

EXAMPLES:

sage: e = RootSystem(["A",3]).ambient_space()
sage: e.dimension()
4

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)}

highest_root()#

EXAMPLES:

sage: e = RootSystem(['A',3]).ambient_lattice()
sage: e.highest_root()
(1, 0, 0, -1)

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)]

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)]

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)

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)}

classmethod smallest_base_ring(cartan_type=None)#

Returns the smallest base ring the ambient space can be defined upon

See also

CartanType()

AmbientSpace#: alias of AmbientSpace

PieriFactors#: alias of PieriFactors_type_A

ascii_art(label=<function CartanType.<lambda> at 0x7fded85b9ea0>, 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

coxeter_number()#

Return the Coxeter number associated with self.

EXAMPLES:

sage: CartanType(['A',4]).coxeter_number()
5

dual_coxeter_number()#

Return the dual Coxeter number associated with self.

EXAMPLES:

sage: CartanType(['A',4]).dual_coxeter_number()
5

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: a.edges(sort=True)
[(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1)]