Bases: sage.misc.cachefunc.ClearCacheOnPickle, sage.combinat.free_module.CombinatorialFreeModule
The root space of a root system over a given base ring
INPUT:
The root space (or lattice if base_ring is ) of a root
system is the formal free module
generated by the simple roots
of the root system.
This class is also used for coroot spaces (or lattices).
See also
Todo: standardize the variable used for the root space in the examples (P?)
TESTS:
sage: for ct in CartanType.samples(crystallographic=True)+[CartanType(["A",2],["C",5,1])]:
... TestSuite(ct.root_system().root_lattice()).run()
... TestSuite(ct.root_system().root_space()).run()
sage: r = RootSystem(['A',4]).root_lattice()
sage: r.simple_root(1)
alpha[1]
sage: latex(r.simple_root(1))
\alpha_{1}
alias of RootSpaceElement
INPUT:
- ‘’i’‘
Returns the basis element indexed by i
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a', 'b', 'c'])
sage: F.monomial('a')
B['a']
F.monomial is in fact (almost) a map:
sage: F.monomial
Term map from {'a', 'b', 'c'} to Free module generated by {'a', 'b', 'c'} over Rational Field
Returns the nu map to the coroot space over the same base ring, using the symmetrizer of the Cartan matrix
It does not map the root lattice to the coroot lattice, but has the property that any root is mapped to some scalar multiple of its associated coroot.
EXAMPLES:
sage: R = RootSystem(['A',3]).root_space()
sage: alpha = R.simple_roots()
sage: f = R.to_coroot_space_morphism()
sage: f(alpha[1])
alphacheck[1]
sage: f(alpha[1]+alpha[2])
alphacheck[1] + alphacheck[2]
sage: R = RootSystem(['A',3]).root_lattice()
sage: alpha = R.simple_roots()
sage: f = R.to_coroot_space_morphism()
sage: f(alpha[1])
alphacheck[1]
sage: f(alpha[1]+alpha[2])
alphacheck[1] + alphacheck[2]
sage: S = RootSystem(['G',2]).root_space()
sage: alpha = S.simple_roots()
sage: f = S.to_coroot_space_morphism()
sage: f(alpha[1])
alphacheck[1]
sage: f(alpha[1]+alpha[2])
alphacheck[1] + 3*alphacheck[2]
Bases: sage.combinat.free_module.CombinatorialFreeModuleElement
Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.
TESTS:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
sage: f == loads(dumps(f))
True
Returns the coroot associated to this root
OUTPUT:
An element of the coroot space over the same base ring; in particular the result is in the coroot lattice whenever self is in the root lattice.
EXAMPLES:
sage: L = RootSystem(["B", 3]).root_space()
sage: alpha = L.simple_roots()
sage: alpha[1].associated_coroot()
alphacheck[1]
sage: alpha[1].associated_coroot().parent()
Coroot space over the Rational Field of the Root system of type ['B', 3]
sage: L.highest_root()
alpha[1] + 2*alpha[2] + 2*alpha[3]
sage: L.highest_root().associated_coroot()
alphacheck[1] + 2*alphacheck[2] + alphacheck[3]
sage: alpha = RootSystem(["B", 3]).root_lattice().simple_roots()
sage: alpha[1].associated_coroot()
alphacheck[1]
sage: alpha[1].associated_coroot().parent()
Coroot lattice of the Root system of type ['B', 3]
Checks whether an element in the root space lies in the nonnegative cone spanned by the simple roots.
EXAMPLES:
sage: R=RootSystem(['A',3,1]).root_space()
sage: B=R.basis()
sage: w=B[0]+B[3]
sage: w.is_positive_root()
True
sage: w=B[1]-B[2]
sage: w.is_positive_root()
False
Returns the highest positive coroot whose associated root is less than or equal to self.
INPUT:
Returns None for the zero element.
Really self is an element of a coroot lattice and this method returns the highest root whose associated coroot is <= self.
Warning
This implementation only handles finite Cartan types
EXAMPLES:
sage: root_lattice = RootSystem(['C',2]).root_lattice()
sage: root_lattice.from_vector(vector([1,1])).max_coroot_le()
alphacheck[1] + 2*alphacheck[2]
sage: root_lattice.from_vector(vector([2,1])).max_coroot_le()
alphacheck[1] + 2*alphacheck[2]
sage: root_lattice = RootSystem(['B',2]).root_lattice()
sage: root_lattice.from_vector(vector([1,1])).max_coroot_le()
2*alphacheck[1] + alphacheck[2]
sage: root_lattice.from_vector(vector([1,2])).max_coroot_le()
2*alphacheck[1] + alphacheck[2]
sage: root_lattice.zero().max_coroot_le() is None
True
sage: root_lattice.from_vector(vector([-1,0])).max_coroot_le()
Traceback (most recent call last):
...
ValueError: -alpha[1] is not in the positive cone of roots
sage: root_lattice = RootSystem(['A',2,1]).root_lattice()
sage: root_lattice.simple_root(1).max_coroot_le()
Traceback (most recent call last):
...
NotImplementedError: Only implemented for finite Cartan type
Returns a reduced word for the longest element of the Weyl group whose shortest path in the quantum Bruhat graph to the identity, has sum of quantum coroots at most self.
INPUT:
Really self is an element of a coroot lattice.
Warning
This implementation only handles finite Cartan types
EXAMPLES:
sage: Qvee = RootSystem(['C',2]).coroot_lattice()
sage: Qvee.from_vector(vector([1,2])).max_quantum_element()
[2, 1, 2, 1]
sage: Qvee.from_vector(vector([1,1])).max_quantum_element()
[1, 2, 1]
sage: Qvee.from_vector(vector([0,2])).max_quantum_element()
[2]
Returns True if self is a quantum root and False otherwise.
INPUT:
A root is a quantum root if
where
is the length function,
is the reflection across the hyperplane orthogonal to
, and
is the sum of positive roots.
Warning
This implementation only handles finite Cartan types and assumes that self is a root.
Todo
Rename to is_quantum_root
EXAMPLES:
sage: Q = RootSystem(['C',2]).root_lattice()
sage: positive_roots = Q.positive_roots()
sage: for x in positive_roots:
....: print x, x.quantum_root()
alpha[1] True
alpha[2] True
2*alpha[1] + alpha[2] True
alpha[1] + alpha[2] False
The scalar product between the root lattice and the coroot lattice.
EXAMPLES:
sage: L = RootSystem(['B',4]).root_lattice()
sage: alpha = L.simple_roots()
sage: alphacheck = L.simple_coroots()
sage: alpha[1].scalar(alphacheck[1])
2
sage: alpha[1].scalar(alphacheck[2])
-1
The scalar products between the roots and coroots are given by the Cartan matrix:
sage: matrix([ [ alpha[i].scalar(alphacheck[j])
... for i in L.index_set() ]
... for j in L.index_set() ])
[ 2 -1 0 0]
[-1 2 -1 0]
[ 0 -1 2 -1]
[ 0 0 -2 2]
sage: L.cartan_type().cartan_matrix()
[ 2 -1 0 0]
[-1 2 -1 0]
[ 0 -1 2 -1]
[ 0 0 -2 2]