def __init__(self, group, base_ring=IntegerRing()):
r"""
See :class:`GroupAlgebra` for full documentation.
EXAMPLES::
sage: GroupAlgebra(GL(3, GF(7)))
Group algebra of group "General Linear Group of degree 3 over Finite Field of size 7" over base ring Integer Ring
"""
from sage.groups.group import is_Group
if not base_ring.is_commutative():
raise NotImplementedError("Base ring must be commutative")
if not is_Group(group):
raise TypeError('"%s" is not a group' % group)
self._group = group
CombinatorialFreeModule.__init__(self, base_ring, group,
prefix='',
bracket=False,
category=HopfAlgebrasWithBasis(base_ring))
if not base_ring.has_coerce_map_from(group) :
## some matrix groups assume that coercion is only valid to
## other matrix groups. This is a workaround
## call _element_constructor_ to coerce group elements
#try :
self._populate_coercion_lists_(coerce_list=[base_ring, group])
#except TypeError :
# self._populate_coercion_lists_( coerce_list = [base_ring] )
else :
self._populate_coercion_lists_(coerce_list=[base_ring])
def __init__(self, group, base_ring = IntegerRing()):
r""" Create the given group algebra.
INPUT:
-- (Group) group: a generic group.
-- (Ring) base_ring: a commutative ring.
OUTPUT:
-- a GroupAlgebra instance.
EXAMPLES::
sage: from sage.algebras.group_algebra import GroupAlgebra
doctest:1: DeprecationWarning:...
sage: GroupAlgebra(GL(3, GF(7)))
Group algebra of group "General Linear Group of degree 3 over Finite
Field of size 7" over base ring Integer Ring
sage: GroupAlgebra(1)
Traceback (most recent call last):
...
TypeError: "1" is not a group
sage: GroupAlgebra(SU(2, GF(4, 'a')), IntegerModRing(12)).category()
Category of group algebras over Ring of integers modulo 12
"""
if not base_ring.is_commutative():
raise NotImplementedError("Base ring must be commutative")
if not is_Group(group):
raise TypeError('"%s" is not a group' % group)
ParentWithGens.__init__(self, base_ring, category = GroupAlgebras(base_ring))
self._formal_sum_module = FormalSums(base_ring)
self._group = group
def __init__(self, group, base_ring=IntegerRing()):
r"""
See :class:`GroupAlgebra` for full documentation.
EXAMPLES::
sage: GroupAlgebra(GL(3, GF(7)))
Group algebra of group "General Linear Group of degree 3 over Finite Field of size 7" over base ring Integer Ring
"""
from sage.groups.group import is_Group
if not base_ring.is_commutative():
raise NotImplementedError("Base ring must be commutative")
if not is_Group(group):
raise TypeError('"%s" is not a group' % group)
self._group = group
CombinatorialFreeModule.__init__(self,
base_ring,
group,
prefix='',
bracket=False,
category=GroupAlgebras(base_ring))
if not base_ring.has_coerce_map_from(group):
## some matrix groups assume that coercion is only valid to
## other matrix groups. This is a workaround
## call _element_constructor_ to coerce group elements
#try :
self._populate_coercion_lists_(coerce_list=[group])
def _element_constructor_(self, x):
r"""
Try to turn ``x`` into an element of ``self``.
INPUT:
- ``x`` - an element of some group algebra or of a
ring or of a group
OUTPUT: ``x`` as a member of ``self``.
sage: G = KleinFourGroup()
sage: f = G.gen(0)
sage: ZG = GroupAlgebra(G)
sage: ZG(f) # indirect doctest
(3,4)
sage: ZG(1) == ZG(G(1))
True
sage: G = AbelianGroup(1)
sage: ZG = GroupAlgebra(G)
sage: f = ZG.group().gen()
sage: ZG(FormalSum([(1,f), (2, f**2)]))
f + 2*f^2
sage: G = GL(2,7)
sage: OG = GroupAlgebra(G, ZZ[sqrt(5)])
sage: OG(2)
2*[1 0]
[0 1]
sage: OG(G(2)) # conversion is not the obvious one
[2 0]
[0 2]
sage: OG(FormalSum([ (1, G(2)), (2, RR(0.77)) ]) )
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer
sage: OG(OG.base_ring().gens()[1])
sqrt5*[1 0]
[0 1]
"""
from sage.rings.all import is_Ring
from sage.groups.group import is_Group
from sage.structure.formal_sum import FormalSum
k = self.base_ring()
G = self.group()
S = x.parent()
if isinstance(S, GroupAlgebra):
if self.has_coerce_map_from(S):
# coerce monomials, coerce coefficients, reassemble
d = x.monomial_coefficients()
new_d = {}
for g in d:
g1 = G(g)
if g1 in new_d:
new_d[g1] += k(d[g]) + new_d[g1]
else:
new_d[g1] = k(d[g])
return self._from_dict(new_d)
elif is_Ring(S):
# coerce to multiple of identity element
return k(x) * self(1)
elif is_Group(S):
# Check whether group coerces to base_ring first.
if k.has_coerce_map_from(S):
return k(x) * self(1)
if G.has_coerce_map_from(S):
return self.monomial(self.group()(x))
elif isinstance(x, FormalSum) and k.has_coerce_map_from(S.base_ring()):
y = [(G(g), k(coeff)) for coeff,g in x]
return self.sum_of_terms(y)
raise TypeError("Don't know how to create an element of %s from %s" % \
(self, x))
def _element_constructor_(self, x):
r"""
Try to turn ``x`` into an element of ``self``.
INPUT:
- ``x`` - an element of some group algebra or of a
ring or of a group
OUTPUT: ``x`` as a member of ``self``.
sage: G = KleinFourGroup()
sage: f = G.gen(0)
sage: ZG = GroupAlgebra(G)
sage: ZG(f) # indirect doctest
(3,4)
sage: ZG(1) == ZG(G(1))
True
sage: G = AbelianGroup(1)
sage: ZG = GroupAlgebra(G)
sage: f = ZG.group().gen()
sage: ZG(FormalSum([(1,f), (2, f**2)]))
f + 2*f^2
sage: G = GL(2,7)
sage: OG = GroupAlgebra(G, ZZ[sqrt(5)])
sage: OG(2)
2*[1 0]
[0 1]
sage: OG(G(2)) # conversion is not the obvious one
[2 0]
[0 2]
sage: OG(FormalSum([ (1, G(2)), (2, RR(0.77)) ]) )
Traceback (most recent call last):
...
TypeError: Attempt to coerce non-integral RealNumber to Integer
sage: OG(OG.base_ring().basis()[1])
sqrt5*[1 0]
[0 1]
"""
from sage.rings.ring import is_Ring
from sage.groups.group import is_Group
from sage.structure.formal_sum import FormalSum
k = self.base_ring()
G = self.group()
S = x.parent()
if isinstance(S, GroupAlgebra):
if self.has_coerce_map_from(S):
# coerce monomials, coerce coefficients, reassemble
d = x.monomial_coefficients()
new_d = {}
for g in d:
g1 = G(g)
if g1 in new_d:
new_d[g1] += k(d[g]) + new_d[g1]
else:
new_d[g1] = k(d[g])
return self._from_dict(new_d)
elif is_Ring(S):
# coerce to multiple of identity element
return k(x) * self(1)
elif is_Group(S):
# Check whether group coerces to base_ring first.
if k.has_coerce_map_from(S):
return k(x) * self(1)
if G.has_coerce_map_from(S):
return self.monomial(self.group()(x))
elif isinstance(x, FormalSum) and k.has_coerce_map_from(S.base_ring()):
y = [(G(g), k(coeff)) for coeff, g in x]
return self.sum_of_terms(y)
raise TypeError("Don't know how to create an element of %s from %s" % \
(self, x))
