def CoxeterGroup(cartan_type, implementation=None):
"""
INPUT:
- ``cartan_type`` -- a cartan type (or coercible into; see :class:`CartanType`)
- ``implementation`` -- "permutation", "matrix", "coxeter3", or None (default: None)
Returns an implementation of the Coxeter group of type
``cartan_type``.
EXAMPLES:
If ``implementation`` is not specified, a permutation
representation is returned whenever possible (finite irreducible
Cartan type, with the GAP3 Chevie package available)::
sage: W = CoxeterGroup(["A",2])
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
Otherwise, a Weyl group is returned::
sage: W = CoxeterGroup(["A",3,1])
sage: W
Weyl Group of type ['A', 3, 1] (as a matrix group acting on the root space)
We now use the ``implementation`` option::
sage: W = CoxeterGroup(["A",2], implementation = "permutation") # optional - chevie
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
sage: W.category() # optional - chevie
Join of Category of finite permutation groups and Category of finite coxeter groups
sage: W = CoxeterGroup(["A",2], implementation = "matrix")
sage: W
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
sage: W = CoxeterGroup(["H",3], implementation = "matrix")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['H', 3] as matrix group not implemented
sage: W = CoxeterGroup(["A",4,1], implementation = "permutation")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['A', 4, 1] as permutation group not implemented
"""
assert implementation in ["permutation", "matrix", "coxeter3", None]
cartan_type = CartanType(cartan_type)
if implementation is None:
if cartan_type.is_finite() and cartan_type.is_irreducible(
) and is_chevie_available():
implementation = "permutation"
else:
implementation = "matrix"
if implementation == "coxeter3":
try:
from sage.libs.coxeter3.coxeter_group import CoxeterGroup
except ImportError:
raise RuntimeError, "coxeter3 must be installed"
else:
return CoxeterGroup(cartan_type)
if implementation == "permutation" and is_chevie_available() and \
cartan_type.is_finite() and cartan_type.is_irreducible():
return CoxeterGroupAsPermutationGroup(cartan_type)
elif implementation == "matrix" and cartan_type.is_crystalographic():
return WeylGroup(cartan_type)
else:
raise NotImplementedError, "Coxeter group of type %s as %s group not implemented " % (
cartan_type, implementation)
def CoxeterGroup(cartan_type, implementation=None):
"""
INPUT:
- ``cartan_type`` -- a cartan type (or coercible into; see :class:`CartanType`)
- ``implementation`` -- "permutation", "matrix", "coxeter3", or None (default: None)
Returns an implementation of the Coxeter group of type
``cartan_type``.
EXAMPLES:
If ``implementation`` is not specified, a permutation
representation is returned whenever possible (finite irreducible
Cartan type, with the GAP3 Chevie package available)::
sage: W = CoxeterGroup(["A",2])
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
Otherwise, a Weyl group is returned::
sage: W = CoxeterGroup(["A",3,1])
sage: W
Weyl Group of type ['A', 3, 1] (as a matrix group acting on the root space)
We now use the ``implementation`` option::
sage: W = CoxeterGroup(["A",2], implementation = "permutation") # optional - chevie
sage: W # optional - chevie
Permutation Group with generators [(1,3)(2,5)(4,6), (1,4)(2,3)(5,6)]
sage: W.category() # optional - chevie
Join of Category of finite permutation groups and Category of finite coxeter groups
sage: W = CoxeterGroup(["A",2], implementation = "matrix")
sage: W
Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)
sage: W = CoxeterGroup(["H",3], implementation = "matrix")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['H', 3] as matrix group not implemented
sage: W = CoxeterGroup(["A",4,1], implementation = "permutation")
Traceback (most recent call last):
...
NotImplementedError: Coxeter group of type ['A', 4, 1] as permutation group not implemented
"""
assert implementation in ["permutation", "matrix", "coxeter3", None]
cartan_type = CartanType(cartan_type)
if implementation is None:
if cartan_type.is_finite() and cartan_type.is_irreducible() and is_chevie_available():
implementation = "permutation"
else:
implementation = "matrix"
if implementation == "coxeter3":
try:
from sage.libs.coxeter3.coxeter_group import CoxeterGroup
except ImportError:
raise RuntimeError, "coxeter3 must be installed"
else:
return CoxeterGroup(cartan_type)
if (
implementation == "permutation"
and is_chevie_available()
and cartan_type.is_finite()
and cartan_type.is_irreducible()
):
return CoxeterGroupAsPermutationGroup(cartan_type)
elif implementation == "matrix" and cartan_type.is_crystalographic():
return WeylGroup(cartan_type)
else:
raise NotImplementedError, "Coxeter group of type %s as %s group not implemented " % (
cartan_type,
implementation,
)
