Coverage for local/lib/python2.7/site-packages/sage/combinat/root_system/root_space.py : 68%

""" 

Root lattices and root spaces 

""" 

#***************************************************************************** 

# Copyright (C) 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function 

from __future__ import absolute_import 

from sage.misc.cachefunc import cached_method, cached_in_parent_method 

from sage.rings.all import ZZ 

from sage.combinat.free_module import CombinatorialFreeModule 

from .root_lattice_realizations import RootLatticeRealizations 

from sage.misc.cachefunc import cached_in_parent_method 

import functools 

class RootSpace(CombinatorialFreeModule): 

r""" 

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). 

.. SEEALSO:: 

- :meth:`RootSystem` 

- :meth:`RootSystem.root_lattice` and :meth:`RootSystem.root_space` 

- :meth:`~sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations` 

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} 

""" 

def __init__(self, root_system, base_ring): 

""" 

EXAMPLES:: 

sage: P = RootSystem(['A',4]).root_space() 

sage: s = P.simple_reflections() 

""" 

from sage.categories.morphism import SetMorphism 

from sage.categories.homset import Hom 

from sage.categories.sets_with_partial_maps import SetsWithPartialMaps 

self.root_system = root_system 

CombinatorialFreeModule.__init__(self, base_ring, 

root_system.index_set(), 

prefix = "alphacheck" if root_system.dual_side else "alpha", 

latex_prefix = "\\alpha^\\vee" if root_system.dual_side else "\\alpha", 

category = RootLatticeRealizations(base_ring)) 

if base_ring is not ZZ: 

# Register the partial conversion back from ``self`` to the root lattice 

# See :meth:`_to_root_lattice` for tests 

root_lattice = self.root_system.root_lattice() 

SetMorphism(Hom(self, root_lattice, SetsWithPartialMaps()), 

self._to_root_lattice 

).register_as_conversion() 

def _repr_(self): 

""" 

EXAMPLES:: 

sage: RootSystem(['A',4]).root_lattice() # indirect doctest 

Root lattice of the Root system of type ['A', 4] 

sage: RootSystem(['B',4]).root_space() 

Root space over the Rational Field of the Root system of type ['B', 4] 

sage: RootSystem(['A',4]).coroot_lattice() 

Coroot lattice of the Root system of type ['A', 4] 

sage: RootSystem(['B',4]).coroot_space() 

Coroot space over the Rational Field of the Root system of type ['B', 4] 

""" 

return self._name_string() 

def _name_string(self, capitalize=True, base_ring=True, type=True): 

""" 

EXAMPLES:: 

sage: RootSystem(['A',4]).root_space()._name_string() 

"Root space over the Rational Field of the Root system of type ['A', 4]" 

""" 

return self._name_string_helper("root", capitalize=capitalize, base_ring=base_ring, type=type) 

simple_root = CombinatorialFreeModule.monomial 

@cached_method 

def to_coroot_space_morphism(self): 

""" 

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] 

""" 

R = self.base_ring() 

C = self.cartan_type().symmetrizer().map(R) 

return self.module_morphism(diagonal = C.__getitem__, 

codomain=self.coroot_space(R)) 

def _to_root_lattice(self, x): 

""" 

Try to convert ``x`` to the root lattice. 

INPUT: 

- ``x`` -- an element of ``self`` 

EXAMPLES:: 

sage: R = RootSystem(['A',3]) 

sage: root_space = R.root_space() 

sage: x = root_space.an_element(); x 

2*alpha[1] + 2*alpha[2] + 3*alpha[3] 

sage: root_space._to_root_lattice(x) 

2*alpha[1] + 2*alpha[2] + 3*alpha[3] 

sage: root_space._to_root_lattice(x).parent() 

Root lattice of the Root system of type ['A', 3] 

This will fail if ``x`` does not have integral coefficients:: 

sage: root_space._to_root_lattice(x/2) 

Traceback (most recent call last): 

... 

ValueError: alpha[1] + alpha[2] + 3/2*alpha[3] does not have integral coefficients 

.. note:: 

For internal use only; instead use a conversion:: 

sage: R.root_lattice()(x) 

2*alpha[1] + 2*alpha[2] + 3*alpha[3] 

sage: R.root_lattice()(x/2) 

Traceback (most recent call last): 

... 

ValueError: alpha[1] + alpha[2] + 3/2*alpha[3] does not have integral coefficients 

.. todo:: generalize diagonal module morphisms to implement this 

""" 

try: 

return self.root_system.root_lattice().sum_of_terms( (i, ZZ(c)) for (i,c) in x) 

except TypeError: 

raise ValueError("%s does not have integral coefficients"%x) 

@cached_method 

def _to_classical_on_basis(self, i): 

r""" 

Implement the projection onto the corresponding classical root space or lattice, on the basis. 

EXAMPLES:: 

sage: L = RootSystem(["A",3,1]).root_space() 

sage: L._to_classical_on_basis(0) 

-alpha[1] - alpha[2] - alpha[3] 

sage: L._to_classical_on_basis(1) 

alpha[1] 

sage: L._to_classical_on_basis(2) 

alpha[2] 

""" 

if i == self.cartan_type().special_node(): 

return self._classical_alpha_0() 

else: 

return self.classical().simple_root(i) 

@cached_method 

def to_ambient_space_morphism(self): 

r""" 

The morphism from ``self`` to 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] 

""" 

if self.root_system.dual_side: 

L = self.cartan_type().dual().root_system().ambient_space() 

basis = L.simple_coroots() 

else: 

L = self.cartan_type().root_system().ambient_space() 

basis = L.simple_roots() 

def basis_value(basis, i): 

return basis[i] 

return self.module_morphism(on_basis = functools.partial(basis_value, basis) , codomain=L) 

class RootSpaceElement(CombinatorialFreeModule.Element): 

def scalar(self, 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] 

""" 

# Find some better test 

if not (lambdacheck in self.parent().coroot_lattice() or lambdacheck in self.parent().coroot_space()): 

raise TypeError("%s is not in a coroot lattice/space"%(lambdacheck)) 

zero = self.parent().base_ring().zero() 

cartan_matrix = self.parent().dynkin_diagram() 

return sum( (sum( (lambdacheck[i]*s for i,s in cartan_matrix.column(j)), zero) * c for j,c in self), zero) 

def is_positive_root(self): 

""" 

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 

""" 

return all( c>= 0 for c in self.coefficients() ) 

@cached_in_parent_method 

def associated_coroot(self): 

r""" 

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] 

""" 

#assert(self in self.parent().roots() is not False) 

scaled_coroot = self.parent().to_coroot_space_morphism()(self) 

s = self.scalar(scaled_coroot) 

return scaled_coroot.map_coefficients(lambda c: (2*c) // s) 

def quantum_root(self): 

r""" 

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. 

.. 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("{} {}".format(x, x.quantum_root())) 

alpha[1] True 

alpha[2] True 

2*alpha[1] + alpha[2] True 

alpha[1] + alpha[2] False 

""" 

return len(self.associated_reflection()) == -1 + (self.parent().nonparabolic_positive_root_sum(())).scalar(self.associated_coroot()) 

def max_coroot_le(self): 

r""" 

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 

""" 

if not self.parent().cartan_type().is_finite(): 

raise NotImplementedError("Only implemented for finite Cartan type") 

if not self.is_positive_root(): 

raise ValueError("%s is not in the positive cone of roots"%(self)) 

coroots = self.parent().coroot_lattice().positive_roots_by_height(increasing=False) 

for beta in coroots: 

if beta.quantum_root(): 

gamma = self - beta.associated_coroot() 

if gamma.is_positive_root(): 

return beta 

return None 

def max_quantum_element(self): 

r""" 

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] 

""" 

Qvee = self.parent() 

word = [] 

while self != Qvee.zero(): 

beta = self.max_coroot_le() 

word += [x for x in beta.associated_reflection()] 

self = self - beta.associated_coroot() 

W = self.parent().weyl_group() 

return (W.demazure_product(word)).reduced_word() 

def to_ambient(self): 

r""" 

Map ``self`` to 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) 

""" 

return self.parent().to_ambient_space_morphism()(self) 

RootSpace.Element = RootSpaceElement