Root lattices and root spaces

class sage.combinat.root_system.root_space.RootSpace(root_system, base_ring)

Bases: sage.misc.cachefunc.ClearCacheOnPickle, sage.combinat.free_module.CombinatorialFreeModule

The root space of a root system over a given base ring

INPUT:

  • root_system - a root system
  • base_ring: a ring \(R\)

The root space (or lattice if base_ring is \(\ZZ\)) of a root system is the formal free module \(\bigoplus_i R \alpha_i\) generated by the simple roots \((\alpha_i)_{i\in I}\) of the root system.

This class is also used for coroot spaces (or lattices).

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}
Element

alias of RootSpaceElement

simple_root()

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
to_coroot_space_morphism()

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]
class sage.combinat.root_system.root_space.RootSpaceElement(M, x)

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
associated_coroot()

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]
is_positive_root()

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
max_coroot_le()

Returns the highest positive coroot whose associated root is less than or equal to self.

INPUT:

  • self – an element of the nonnegative integer span of simple roots.

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
max_quantum_element()

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:

  • self – an element of the nonnegative integer span of simple roots.

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]
quantum_root()

Returns True if self is a quantum root and False otherwise.

INPUT:

  • self – an element of the nonnegative integer span of simple roots.

A root \(\alpha\) is a quantum root if \(\ell(s_\alpha) = \langle 2 \rho, \alpha^\vee \rangle - 1\) where \(\ell\) is the length function, \(s_\alpha\) is the reflection across the hyperplane orthogonal to \(\alpha\), and \(2\rho\) is the sum of positive roots.

Warning

This implementation only handles finite Cartan types and assumes that self is a root.

EXAMPLES:

sage: Q = RootSystem(['C',2]).root_lattice()
sage: [[x, x.quantum_root()] for x in Q.positive_roots_by_height()]
[[alpha[1], True], [alpha[2], True], [alpha[1] + alpha[2], False], [2*alpha[1] + alpha[2], True]]
scalar(lambdacheck)

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]

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