Root lattices and root spaces¶
- class sage.combinat.root_system.root_space.RootSpace(root_system, base_ring)[source]¶
Bases:
CombinatorialFreeModuleThe root space of a root system over a given base ring.
INPUT:
root_system– a root systembase_ring– a ring \(R\)
The root space (or lattice if
base_ringis \(\ZZ\)) 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).
See also
Todo: standardize the variable used for the root space in the examples (P?)
- Element[source]¶
alias of
RootSpaceElement
- simple_root()[source]¶
Return the basis element indexed by
i.INPUT:
i– an element of the index set
EXAMPLES:
sage: F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) sage: F.monomial('a') B['a']
>>> from sage.all import * >>> F = CombinatorialFreeModule(QQ, ['a', 'b', 'c']) >>> F.monomial('a') B['a']
F.monomialis 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
>>> from sage.all import * >>> F.monomial Term map from {'a', 'b', 'c'} to Free module generated by {'a', 'b', 'c'} over Rational Field
- to_ambient_space_morphism()[source]¶
The morphism from
selfto its associated ambient space.EXAMPLES:
sage: CartanType(['A',2]).root_system().root_lattice().to_ambient_space_morphism() Generic morphism: From: Root lattice of the Root system of type ['A', 2] To: Ambient space of the Root system of type ['A', 2]
>>> from sage.all import * >>> CartanType(['A',Integer(2)]).root_system().root_lattice().to_ambient_space_morphism() Generic morphism: From: Root lattice of the Root system of type ['A', 2] To: Ambient space of the Root system of type ['A', 2]
- to_coroot_space_morphism()[source]¶
Return the
numap 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() # needs sage.graphs sage: f(alpha[1]) # needs sage.graphs alphacheck[1] sage: f(alpha[1] + alpha[2]) # needs sage.graphs alphacheck[1] + alphacheck[2] sage: R = RootSystem(['A',3]).root_lattice() sage: alpha = R.simple_roots() sage: f = R.to_coroot_space_morphism() # needs sage.graphs sage: f(alpha[1]) # needs sage.graphs alphacheck[1] sage: f(alpha[1] + alpha[2]) # needs sage.graphs alphacheck[1] + alphacheck[2] sage: S = RootSystem(['G',2]).root_space() sage: alpha = S.simple_roots() sage: f = S.to_coroot_space_morphism() # needs sage.graphs sage: f(alpha[1]) # needs sage.graphs alphacheck[1] sage: f(alpha[1] + alpha[2]) # needs sage.graphs alphacheck[1] + 3*alphacheck[2]
>>> from sage.all import * >>> R = RootSystem(['A',Integer(3)]).root_space() >>> alpha = R.simple_roots() >>> f = R.to_coroot_space_morphism() # needs sage.graphs >>> f(alpha[Integer(1)]) # needs sage.graphs alphacheck[1] >>> f(alpha[Integer(1)] + alpha[Integer(2)]) # needs sage.graphs alphacheck[1] + alphacheck[2] >>> R = RootSystem(['A',Integer(3)]).root_lattice() >>> alpha = R.simple_roots() >>> f = R.to_coroot_space_morphism() # needs sage.graphs >>> f(alpha[Integer(1)]) # needs sage.graphs alphacheck[1] >>> f(alpha[Integer(1)] + alpha[Integer(2)]) # needs sage.graphs alphacheck[1] + alphacheck[2] >>> S = RootSystem(['G',Integer(2)]).root_space() >>> alpha = S.simple_roots() >>> f = S.to_coroot_space_morphism() # needs sage.graphs >>> f(alpha[Integer(1)]) # needs sage.graphs alphacheck[1] >>> f(alpha[Integer(1)] + alpha[Integer(2)]) # needs sage.graphs alphacheck[1] + 3*alphacheck[2]
- class sage.combinat.root_system.root_space.RootSpaceElement[source]¶
Bases:
IndexedFreeModuleElement- associated_coroot()[source]¶
Return 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
selfis in the root lattice.EXAMPLES:
sage: L = RootSystem(["B", 3]).root_space() sage: alpha = L.simple_roots() sage: alpha[1].associated_coroot() # needs sage.graphs alphacheck[1] sage: alpha[1].associated_coroot().parent() # needs sage.graphs Coroot space over the Rational Field of the Root system of type ['B', 3] sage: L.highest_root() # needs sage.graphs alpha[1] + 2*alpha[2] + 2*alpha[3] sage: L.highest_root().associated_coroot() # needs sage.graphs alphacheck[1] + 2*alphacheck[2] + alphacheck[3] sage: alpha = RootSystem(["B", 3]).root_lattice().simple_roots() sage: alpha[1].associated_coroot() # needs sage.graphs alphacheck[1] sage: alpha[1].associated_coroot().parent() # needs sage.graphs Coroot lattice of the Root system of type ['B', 3]
>>> from sage.all import * >>> L = RootSystem(["B", Integer(3)]).root_space() >>> alpha = L.simple_roots() >>> alpha[Integer(1)].associated_coroot() # needs sage.graphs alphacheck[1] >>> alpha[Integer(1)].associated_coroot().parent() # needs sage.graphs Coroot space over the Rational Field of the Root system of type ['B', 3] >>> L.highest_root() # needs sage.graphs alpha[1] + 2*alpha[2] + 2*alpha[3] >>> L.highest_root().associated_coroot() # needs sage.graphs alphacheck[1] + 2*alphacheck[2] + alphacheck[3] >>> alpha = RootSystem(["B", Integer(3)]).root_lattice().simple_roots() >>> alpha[Integer(1)].associated_coroot() # needs sage.graphs alphacheck[1] >>> alpha[Integer(1)].associated_coroot().parent() # needs sage.graphs Coroot lattice of the Root system of type ['B', 3]
- is_positive_root()[source]¶
Check 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
>>> from sage.all import * >>> R = RootSystem(['A',Integer(3),Integer(1)]).root_space() >>> B = R.basis() >>> w = B[Integer(0)] + B[Integer(3)] >>> w.is_positive_root() True >>> w = B[Integer(1)] - B[Integer(2)] >>> w.is_positive_root() False
- max_coroot_le()[source]¶
Return 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
selfis 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: # needs sage.graphs 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 # needs sage.graphs True sage: root_lattice.from_vector(vector([-1,0])).max_coroot_le() # needs sage.graphs 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() # needs sage.graphs Traceback (most recent call last): ... NotImplementedError: Only implemented for finite Cartan type
>>> from sage.all import * >>> # needs sage.graphs >>> root_lattice = RootSystem(['C',Integer(2)]).root_lattice() >>> root_lattice.from_vector(vector([Integer(1),Integer(1)])).max_coroot_le() alphacheck[1] + 2*alphacheck[2] >>> root_lattice.from_vector(vector([Integer(2),Integer(1)])).max_coroot_le() alphacheck[1] + 2*alphacheck[2] >>> root_lattice = RootSystem(['B',Integer(2)]).root_lattice() >>> root_lattice.from_vector(vector([Integer(1),Integer(1)])).max_coroot_le() 2*alphacheck[1] + alphacheck[2] >>> root_lattice.from_vector(vector([Integer(1),Integer(2)])).max_coroot_le() 2*alphacheck[1] + alphacheck[2] >>> root_lattice.zero().max_coroot_le() is None # needs sage.graphs True >>> root_lattice.from_vector(vector([-Integer(1),Integer(0)])).max_coroot_le() # needs sage.graphs Traceback (most recent call last): ... ValueError: -alpha[1] is not in the positive cone of roots >>> root_lattice = RootSystem(['A',Integer(2),Integer(1)]).root_lattice() >>> root_lattice.simple_root(Integer(1)).max_coroot_le() # needs sage.graphs Traceback (most recent call last): ... NotImplementedError: Only implemented for finite Cartan type
- max_quantum_element()[source]¶
Return 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
selfis an element of a coroot lattice.Warning
This implementation only handles finite Cartan types
EXAMPLES:
sage: # needs sage.graphs sage.libs.gap 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]
>>> from sage.all import * >>> # needs sage.graphs sage.libs.gap >>> Qvee = RootSystem(['C',Integer(2)]).coroot_lattice() >>> Qvee.from_vector(vector([Integer(1),Integer(2)])).max_quantum_element() [2, 1, 2, 1] >>> Qvee.from_vector(vector([Integer(1),Integer(1)])).max_quantum_element() [1, 2, 1] >>> Qvee.from_vector(vector([Integer(0),Integer(2)])).max_quantum_element() [2]
- quantum_root()[source]¶
Check whether
selfis a quantum root.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
selfis 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 sorted(positive_roots): # needs sage.graphs ....: print("{} {}".format(x, x.quantum_root())) alpha[1] True alpha[1] + alpha[2] False 2*alpha[1] + alpha[2] True alpha[2] True
>>> from sage.all import * >>> Q = RootSystem(['C',Integer(2)]).root_lattice() >>> positive_roots = Q.positive_roots() >>> for x in sorted(positive_roots): # needs sage.graphs ... print("{} {}".format(x, x.quantum_root())) alpha[1] True alpha[1] + alpha[2] False 2*alpha[1] + alpha[2] True alpha[2] True
- scalar(lambdacheck)[source]¶
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]) # needs sage.graphs 2 sage: alpha[1].scalar(alphacheck[2]) # needs sage.graphs -1
>>> from sage.all import * >>> L = RootSystem(['B',Integer(4)]).root_lattice() >>> alpha = L.simple_roots() >>> alphacheck = L.simple_coroots() >>> alpha[Integer(1)].scalar(alphacheck[Integer(1)]) # needs sage.graphs 2 >>> alpha[Integer(1)].scalar(alphacheck[Integer(2)]) # needs sage.graphs -1
The scalar products between the roots and coroots are given by the Cartan matrix:
sage: matrix([ [ alpha[i].scalar(alphacheck[j]) # needs sage.graphs ....: 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() # needs sage.graphs [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -2 2]
>>> from sage.all import * >>> matrix([ [ alpha[i].scalar(alphacheck[j]) # needs sage.graphs ... 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] >>> L.cartan_type().cartan_matrix() # needs sage.graphs [ 2 -1 0 0] [-1 2 -1 0] [ 0 -1 2 -1] [ 0 0 -2 2]
- to_ambient()[source]¶
Map
selfto the ambient space.EXAMPLES:
sage: alpha = CartanType(['B',2]).root_system().root_lattice().an_element(); alpha 2*alpha[1] + 2*alpha[2] sage: alpha.to_ambient() (2, 0) sage: alphavee = CartanType(['B',2]).root_system().coroot_lattice().an_element(); alphavee 2*alphacheck[1] + 2*alphacheck[2] sage: alphavee.to_ambient() (2, 2)
>>> from sage.all import * >>> alpha = CartanType(['B',Integer(2)]).root_system().root_lattice().an_element(); alpha 2*alpha[1] + 2*alpha[2] >>> alpha.to_ambient() (2, 0) >>> alphavee = CartanType(['B',Integer(2)]).root_system().coroot_lattice().an_element(); alphavee 2*alphacheck[1] + 2*alphacheck[2] >>> alphavee.to_ambient() (2, 2)