def __init__(self, cartan_type, prefix, finite=True):
r"""
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1])
sage: F in Groups().Commutative().Finite()
True
sage: TestSuite(F).run()
"""
def leading_support(beta):
r"""
Given a dictionary with one key, return this key
"""
supp = beta.support()
assert len(supp) == 1
return supp[0]
self._cartan_type = cartan_type
self._prefix = prefix
special_node = cartan_type.special_node()
self._special_nodes = cartan_type.special_nodes()
# initialize dictionaries with the entries for the
# distinguished special node
# dictionary of inverse elements
inverse_dict = {}
inverse_dict[special_node] = special_node
# dictionary for the action of special automorphisms by
# permutations of the affine Dynkin nodes
auto_dict = {}
for i in cartan_type.index_set():
auto_dict[special_node, i] = i
# dictionary for the finite Weyl component of the special automorphisms
reduced_words_dict = {}
reduced_words_dict[0] = tuple([])
if cartan_type.dual().is_untwisted_affine():
# this combines the computations for an untwisted affine
# type and its affine dual
cartan_type = cartan_type.dual()
if cartan_type.is_untwisted_affine():
cartan_type_classical = cartan_type.classical()
I = [i for i in cartan_type_classical.index_set()]
Q = RootSystem(cartan_type_classical).root_lattice()
alpha = Q.simple_roots()
omega = RootSystem(
cartan_type_classical).weight_lattice().fundamental_weights()
W = Q.weyl_group(prefix="s")
for i in self._special_nodes:
if i == special_node:
continue
antidominant_weight, reduced_word = omega[
i].to_dominant_chamber(reduced_word=True, positive=False)
reduced_words_dict[i] = tuple(reduced_word)
w0i = W.from_reduced_word(reduced_word)
idual = leading_support(-antidominant_weight)
inverse_dict[i] = idual
auto_dict[i, special_node] = i
for j in I:
if j == idual:
auto_dict[i, j] = special_node
else:
auto_dict[i, j] = leading_support(w0i.action(alpha[j]))
self._action = Family(
self._special_nodes, lambda i: Family(cartan_type.index_set(),
lambda j: auto_dict[i, j]))
self._dual_node = Family(self._special_nodes, inverse_dict.__getitem__)
self._reduced_words = Family(self._special_nodes,
reduced_words_dict.__getitem__)
if finite:
cat = Category.join(
(Groups().Commutative().Finite(), EnumeratedSets()))
else:
cat = Groups().Commutative().Infinite()
Parent.__init__(self, category=cat)
def __init__(self, cartan_type, prefix, finite=True):
r"""
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1])
sage: F in Groups().Commutative().Finite()
True
sage: TestSuite(F).run()
"""
def leading_support(beta):
r"""
Given a dictionary with one key, return this key
"""
supp = beta.support()
assert len(supp) == 1
return supp[0]
self._cartan_type = cartan_type
self._prefix = prefix
special_node = cartan_type.special_node()
self._special_nodes = cartan_type.special_nodes()
# initialize dictionaries with the entries for the distinguished special node
# dictionary of inverse elements
inverse_dict = {}
inverse_dict[special_node] = special_node
# dictionary for the action of special automorphisms by permutations of the affine Dynkin nodes
auto_dict = {}
for i in cartan_type.index_set():
auto_dict[special_node,i] = i
# dictionary for the finite Weyl component of the special automorphisms
reduced_words_dict = {}
reduced_words_dict[0] = tuple([])
if cartan_type.dual().is_untwisted_affine():
# this combines the computations for an untwisted affine type and its affine dual
cartan_type = cartan_type.dual()
if cartan_type.is_untwisted_affine():
cartan_type_classical = cartan_type.classical()
I = [i for i in cartan_type_classical.index_set()]
Q = RootSystem(cartan_type_classical).root_lattice()
alpha = Q.simple_roots()
omega = RootSystem(cartan_type_classical).weight_lattice().fundamental_weights()
W = Q.weyl_group(prefix="s")
for i in self._special_nodes:
if i == special_node:
continue
antidominant_weight, reduced_word = omega[i].to_dominant_chamber(reduced_word=True, positive=False)
reduced_words_dict[i] = tuple(reduced_word)
w0i = W.from_reduced_word(reduced_word)
idual = leading_support(-antidominant_weight)
inverse_dict[i] = idual
auto_dict[i,special_node] = i
for j in I:
if j == idual:
auto_dict[i,j] = special_node
else:
auto_dict[i,j] = leading_support(w0i.action(alpha[j]))
self._action = Family(self._special_nodes, lambda i: Family(cartan_type.index_set(), lambda j: auto_dict[i,j]))
self._dual_node = Family(self._special_nodes, inverse_dict.__getitem__)
self._reduced_words = Family(self._special_nodes, reduced_words_dict.__getitem__)
if finite:
cat = Category.join((Groups().Commutative().Finite(),EnumeratedSets()))
else:
cat = Groups().Commutative().Infinite()
Parent.__init__(self, category = cat)
