def __classcall_private__(cls, cartan_type, shapes=None, shape=None):
"""
Normalizes the input arguments to ensure unique representation,
and to delegate the construction of spin tableaux.
EXAMPLES::
sage: T1 = CrystalOfTableaux(CartanType(['A',3]), shape = [2,2])
sage: T2 = CrystalOfTableaux(['A',3], shape = (2,2))
sage: T3 = CrystalOfTableaux(['A',3], shapes = ([2,2],))
sage: T2 is T1, T3 is T1
(True, True)
"""
cartan_type = CartanType(cartan_type)
n = cartan_type.rank()
# standardize shape/shapes input into a tuple of tuples
assert operator.xor(shape is not None, shapes is not None)
if shape is not None:
shapes = (shape, )
spin_shapes = tuple(tuple(shape) for shape in shapes)
try:
shapes = tuple(
tuple(trunc(i) for i in shape) for shape in spin_shapes)
except StandardError:
raise ValueError(
"shapes should all be partitions or half-integer partitions")
if spin_shapes == shapes:
return super(CrystalOfTableaux,
cls).__classcall__(cls, cartan_type, shapes)
# Handle the construction of a crystals of spin tableaux
# Caveat: this currently only supports all shapes being half
# integer partitions of length the rank for type B and D. In
# particular, for type D, the spins all have to be plus or all
# minus spins
assert all(len(sh) == n for sh in shapes), \
"the length of all half-integer partition shapes should be the rank"
assert all(2*i % 2 == 1 for shape in spin_shapes for i in shape), \
"shapes should be either all partitions or all half-integer partitions"
if cartan_type.type() == 'D':
if all(i >= 0 for shape in spin_shapes for i in shape):
S = CrystalOfSpinsPlus(cartan_type)
elif all(shape[-1] < 0 for shape in spin_shapes):
S = CrystalOfSpinsMinus(cartan_type)
else:
raise ValueError, "In type D spins should all be positive or negative"
else:
assert all( i >= 0 for shape in spin_shapes for i in shape), \
"shapes should all be partitions"
S = CrystalOfSpins(cartan_type)
B = CrystalOfTableaux(cartan_type, shapes=shapes)
T = TensorProductOfCrystals(S,
B,
generators=[[S.module_generators[0], x]
for x in B.module_generators])
T.rename("The crystal of tableaux of type %s and shape(s) %s" %
(cartan_type, list(list(shape) for shape in spin_shapes)))
T.shapes = spin_shapes
return T
def __classcall_private__(cls, cartan_type):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: B = crystals.infinity.Tableaux(['A',4])
sage: B2 = crystals.infinity.Tableaux(CartanType(['A',4]))
sage: B is B2
True
"""
cartan_type = CartanType(cartan_type)
if cartan_type.type() == 'D':
return InfinityCrystalOfTableauxTypeD(cartan_type)
if cartan_type.type() == 'Q':
return DualInfinityQueerCrystalOfTableaux(cartan_type)
return super(InfinityCrystalOfTableaux, cls).__classcall__(cls, cartan_type)
def __classcall_private__(cls, cartan_type, shapes = None, shape = None):
"""
Normalizes the input arguments to ensure unique representation,
and to delegate the construction of spin tableaux.
EXAMPLES::
sage: T1 = CrystalOfTableaux(CartanType(['A',3]), shape = [2,2])
sage: T2 = CrystalOfTableaux(['A',3], shape = (2,2))
sage: T3 = CrystalOfTableaux(['A',3], shapes = ([2,2],))
sage: T2 is T1, T3 is T1
(True, True)
"""
cartan_type = CartanType(cartan_type)
n = cartan_type.rank()
# standardize shape/shapes input into a tuple of tuples
assert operator.xor(shape is not None, shapes is not None)
if shape is not None:
shapes = (shape,)
spin_shapes = tuple( tuple(shape) for shape in shapes )
try:
shapes = tuple( tuple(trunc(i) for i in shape) for shape in spin_shapes )
except StandardError:
raise ValueError("shapes should all be partitions or half-integer partitions")
if spin_shapes == shapes:
return super(CrystalOfTableaux, cls).__classcall__(cls, cartan_type, shapes)
# Handle the construction of a crystals of spin tableaux
# Caveat: this currently only supports all shapes being half
# integer partitions of length the rank for type B and D. In
# particular, for type D, the spins all have to be plus or all
# minus spins
assert all(len(sh) == n for sh in shapes), \
"the length of all half-integer partition shapes should be the rank"
assert all(2*i % 2 == 1 for shape in spin_shapes for i in shape), \
"shapes should be either all partitions or all half-integer partitions"
if cartan_type.type() == 'D':
if all( i >= 0 for shape in spin_shapes for i in shape):
S = CrystalOfSpinsPlus(cartan_type)
elif all(shape[-1]<0 for shape in spin_shapes):
S = CrystalOfSpinsMinus(cartan_type)
else:
raise ValueError, "In type D spins should all be positive or negative"
else:
assert all( i >= 0 for shape in spin_shapes for i in shape), \
"shapes should all be partitions"
S = CrystalOfSpins(cartan_type)
B = CrystalOfTableaux(cartan_type, shapes = shapes)
T = TensorProductOfCrystals(S,B, generators=[[S.module_generators[0],x] for x in B.module_generators])
T.rename("The crystal of tableaux of type %s and shape(s) %s"%(cartan_type, list(list(shape) for shape in spin_shapes)))
T.shapes = spin_shapes
return T
def __classcall_private__(cls, cartan_type):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: B = crystals.infinity.Tableaux(['A',4])
sage: B2 = crystals.infinity.Tableaux(CartanType(['A',4]))
sage: B is B2
True
"""
cartan_type = CartanType(cartan_type)
if cartan_type.type() == 'D':
return InfinityCrystalOfTableauxTypeD(cartan_type)
return super(InfinityCrystalOfTableaux, cls).__classcall__(cls, cartan_type)
def __classcall_private__(cls, R, cartan_type):
"""
Return the correct parent based on input.
EXAMPLES::
sage: lie_algebras.ClassicalMatrix(QQ, ['A', 4])
Special linear Lie algebra of rank 5 over Rational Field
sage: lie_algebras.ClassicalMatrix(QQ, CartanType(['B',4]))
Special orthogonal Lie algebra of rank 9 over Rational Field
sage: lie_algebras.ClassicalMatrix(QQ, 'C4')
Symplectic Lie algebra of rank 8 over Rational Field
sage: lie_algebras.ClassicalMatrix(QQ, cartan_type=['D',4])
Special orthogonal Lie algebra of rank 8 over Rational Field
"""
if isinstance(cartan_type, (CartanMatrix, DynkinDiagram_class)):
cartan_type = cartan_type.cartan_type()
else:
cartan_type = CartanType(cartan_type)
if not cartan_type.is_finite():
raise ValueError("only for finite types")
if cartan_type.type() == 'A':
return sl(R, cartan_type.rank() + 1)
if cartan_type.type() == 'B':
return so(R, 2*cartan_type.rank() + 1)
if cartan_type.type() == 'C':
return sp(R, 2*cartan_type.rank())
if cartan_type.type() == 'D':
return so(R, 2*cartan_type.rank())
if cartan_type.type() == 'E':
if cartan_type.rank() == 6:
return e6(R)
if cartan_type.rank() in [7,8]:
raise NotImplementedError("not yet implemented")
if cartan_type.type() == 'F' and cartan_type.rank() == 4:
return f4(R)
if cartan_type.type() == 'G' and cartan_type.rank() == 2:
return g2(R)
raise ValueError("invalid Cartan type")
def FundamentalGroupOfExtendedAffineWeylGroup(cartan_type,
prefix='pi',
general_linear=None):
r"""
Factory for the fundamental group of an extended affine Weyl group.
INPUT:
- ``cartan_type`` -- a Cartan type that is either affine or finite, with the latter being a
shorthand for the untwisted affinization
- ``prefix`` (default: 'pi') -- string that labels the elements of the group
- ``general_linear`` -- (default: None, meaning False) In untwisted type A, if True, use the
universal central extension
.. RUBRIC:: Fundamental group
Associated to each affine Cartan type `\tilde{X}` is an extended affine Weyl group `E`.
Its subgroup of length-zero elements is called the fundamental group `F`.
The group `F` can be identified with a subgroup of the group of automorphisms of the
affine Dynkin diagram. As such, every element of `F` can be viewed as a permutation of the
set `I` of affine Dynkin nodes.
Let `0 \in I` be the distinguished affine node; it is the one whose removal produces the
associated finite Cartan type (call it `X`). A node `i \in I` is called *special*
if some automorphism of the affine Dynkin diagram, sends `0` to `i`.
The node `0` is always special due to the identity automorphism.
There is a bijection of the set of special nodes with the fundamental group. We denote the
image of `i` by `\pi_i`. The structure of `F` is determined as follows.
- `\tilde{X}` is untwisted -- `F` is isomorphic to `P^\vee/Q^\vee` where `P^\vee` and `Q^\vee` are the
coweight and coroot lattices of type `X`. The group `P^\vee/Q^\vee` consists of the cosets `\omega_i^\vee + Q^\vee`
for special nodes `i`, where `\omega_0^\vee = 0` by convention. In this case the special nodes `i`
are the *cominuscule* nodes, the ones such that `\omega_i^\vee(\alpha_j)` is `0` or `1` for all `j\in I_0 = I \setminus \{0\}`.
For `i` special, addition by `\omega_i^\vee+Q^\vee` permutes `P^\vee/Q^\vee` and therefore permutes the set of special nodes.
This permutation extends uniquely to an automorphism of the affine Dynkin diagram.
- `\tilde{X}` is dual untwisted -- (that is, the dual of `\tilde{X}` is untwisted) `F` is isomorphic to `P/Q`
where `P` and `Q` are the weight and root lattices of type `X`. The group `P/Q` consists of the cosets
`\omega_i + Q` for special nodes `i`, where `\omega_0 = 0` by convention. In this case the special nodes `i`
are the *minuscule* nodes, the ones such that `\alpha_j^\vee(\omega_i)` is `0` or `1` for all `j \in I_0`.
For `i` special, addition by `\omega_i+Q` permutes `P/Q` and therefore permutes the set of special nodes.
This permutation extends uniquely to an automorphism of the affine Dynkin diagram.
- `\tilde{X}` is mixed -- (that is, not of the above two types) `F` is the trivial group.
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1]); F
Fundamental group of type ['A', 3, 1]
sage: F.cartan_type().dynkin_diagram()
0
O-------+
| |
| |
O---O---O
1 2 3
A3~
sage: F.special_nodes()
(0, 1, 2, 3)
sage: F(1)^2
pi[2]
sage: F(1)*F(2)
pi[3]
sage: F(3)^(-1)
pi[1]
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("B3"); F
Fundamental group of type ['B', 3, 1]
sage: F.cartan_type().dynkin_diagram()
O 0
|
|
O---O=>=O
1 2 3
B3~
sage: F.special_nodes()
(0, 1)
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("C2"); F
Fundamental group of type ['C', 2, 1]
sage: F.cartan_type().dynkin_diagram()
O=>=O=<=O
0 1 2
C2~
sage: F.special_nodes()
(0, 2)
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("D4"); F
Fundamental group of type ['D', 4, 1]
sage: F.cartan_type().dynkin_diagram()
O 4
|
|
O---O---O
1 |2 3
|
O 0
D4~
sage: F.special_nodes()
(0, 1, 3, 4)
sage: (F(4), F(4)^2)
(pi[4], pi[0])
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("D5"); F
Fundamental group of type ['D', 5, 1]
sage: F.cartan_type().dynkin_diagram()
0 O O 5
| |
| |
O---O---O---O
1 2 3 4
D5~
sage: F.special_nodes()
(0, 1, 4, 5)
sage: (F(5), F(5)^2, F(5)^3, F(5)^4)
(pi[5], pi[1], pi[4], pi[0])
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("E6"); F
Fundamental group of type ['E', 6, 1]
sage: F.cartan_type().dynkin_diagram()
O 0
|
|
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6~
sage: F.special_nodes()
(0, 1, 6)
sage: F(1)^2
pi[6]
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['D',4,2]); F
Fundamental group of type ['C', 3, 1]^*
sage: F.cartan_type().dynkin_diagram()
O=<=O---O=>=O
0 1 2 3
C3~*
sage: F.special_nodes()
(0, 3)
We also implement a fundamental group for `GL_n`. It is defined to be the group of integers, which is the
covering group of the fundamental group Z/nZ for affine `SL_n`::
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',2,1], general_linear=True); F
Fundamental group of GL(3)
sage: x = F.an_element(); x
pi[5]
sage: x*x
pi[10]
sage: x.inverse()
pi[-5]
sage: wt = F.cartan_type().classical().root_system().ambient_space().an_element(); wt
(2, 2, 3)
sage: x.act_on_classical_ambient(wt)
(2, 3, 2)
sage: w = WeylGroup(F.cartan_type(),prefix="s").an_element(); w
s0*s1*s2
sage: x.act_on_affine_weyl(w)
s2*s0*s1
"""
cartan_type = CartanType(cartan_type)
if cartan_type.is_finite():
cartan_type = cartan_type.affine()
if not cartan_type.is_affine():
raise NotImplementedError("Cartan type is not affine")
if general_linear is True:
if cartan_type.is_untwisted_affine() and cartan_type.type() == "A":
return FundamentalGroupGL(cartan_type, prefix)
else:
raise ValueError(
"General Linear Fundamental group is untwisted type A")
return FundamentalGroupOfExtendedAffineWeylGroup_Class(cartan_type,
prefix,
finite=True)
def __classcall_private__(cls, cartan_type, shapes=None, shape=None):
"""
Normalizes the input arguments to ensure unique representation,
and to delegate the construction of spin tableaux.
EXAMPLES::
sage: T1 = crystals.Tableaux(CartanType(['A',3]), shape = [2,2])
sage: T2 = crystals.Tableaux(['A',3], shape = (2,2))
sage: T3 = crystals.Tableaux(['A',3], shapes = ([2,2],))
sage: T2 is T1, T3 is T1
(True, True)
sage: T1 = crystals.Tableaux(['A', [1,1]], shape=[3,1,1,1])
sage: T1
Crystal of BKK tableaux of shape [3, 1, 1, 1] of gl(2|2)
sage: T2 = crystals.Tableaux(['A', [1,1]], [3,1,1,1])
sage: T1 is T2
True
"""
cartan_type = CartanType(cartan_type)
if cartan_type.letter == 'A' and isinstance(cartan_type,
SuperCartanType_standard):
if shape is None:
shape = shapes
shape = _Partitions(shape)
from sage.combinat.crystals.bkk_crystals import CrystalOfBKKTableaux
return CrystalOfBKKTableaux(cartan_type, shape=shape)
if cartan_type.letter == 'Q':
if any(shape[i] == shape[i + 1] for i in range(len(shape) - 1)):
raise ValueError("not a strict partition")
shape = _Partitions(shape)
return CrystalOfQueerTableaux(cartan_type, shape=shape)
n = cartan_type.rank()
# standardize shape/shapes input into a tuple of tuples
# of length n, or n+1 in type A
assert operator.xor(shape is not None, shapes is not None)
if shape is not None:
shapes = (shape, )
if cartan_type.type() == "A":
n1 = n + 1
else:
n1 = n
if not all(all(i == 0 for i in shape[n1:]) for shape in shapes):
raise ValueError(
"shapes should all have length at most equal to the rank or the rank + 1 in type A"
)
spin_shapes = tuple((tuple(shape) + (0, ) * (n1 - len(shape)))[:n1]
for shape in shapes)
try:
shapes = tuple(
tuple(trunc(i) for i in shape) for shape in spin_shapes)
except Exception:
raise ValueError(
"shapes should all be partitions or half-integer partitions")
if spin_shapes == shapes:
shapes = tuple(
_Partitions(shape) if shape[n1 - 1] in NN else shape
for shape in shapes)
return super(CrystalOfTableaux,
cls).__classcall__(cls, cartan_type, shapes)
# Handle the construction of a crystals of spin tableaux
# Caveat: this currently only supports all shapes being half
# integer partitions of length the rank for type B and D. In
# particular, for type D, the spins all have to be plus or all
# minus spins
if any(len(sh) != n for sh in shapes):
raise ValueError(
"the length of all half-integer partition shapes should be the rank"
)
if any(2 * i % 2 != 1 for shape in spin_shapes for i in shape):
raise ValueError(
"shapes should be either all partitions or all half-integer partitions"
)
if any(
any(i < j
for i, j in zip(shape, shape[1:-1] + (abs(shape[-1]), )))
for shape in spin_shapes):
raise ValueError("entries of each shape must be weakly decreasing")
if cartan_type.type() == 'D':
if all(i >= 0 for shape in spin_shapes for i in shape):
S = CrystalOfSpinsPlus(cartan_type)
elif all(shape[-1] < 0 for shape in spin_shapes):
S = CrystalOfSpinsMinus(cartan_type)
else:
raise ValueError(
"in type D spins should all be positive or negative")
else:
if any(i < 0 for shape in spin_shapes for i in shape):
raise ValueError("shapes should all be partitions")
S = CrystalOfSpins(cartan_type)
B = CrystalOfTableaux(cartan_type, shapes=shapes)
T = TensorProductOfCrystals(S,
B,
generators=[[S.module_generators[0], x]
for x in B.module_generators])
T.rename("The crystal of tableaux of type %s and shape(s) %s" %
(cartan_type, list(list(shape) for shape in spin_shapes)))
T.shapes = spin_shapes
return T
def FundamentalGroupOfExtendedAffineWeylGroup(cartan_type, prefix='pi', general_linear=None):
r"""
Factory for the fundamental group of an extended affine Weyl group.
INPUT:
- ``cartan_type`` -- a Cartan type that is either affine or finite, with the latter being a
shorthand for the untwisted affinization
- ``prefix`` (default: 'pi') -- string that labels the elements of the group
- ``general_linear`` -- (default: None, meaning False) In untwisted type A, if True, use the
universal central extension
.. RUBRIC:: Fundamental group
Associated to each affine Cartan type `\tilde{X}` is an extended affine Weyl group `E`.
Its subgroup of length-zero elements is called the fundamental group `F`.
The group `F` can be identified with a subgroup of the group of automorphisms of the
affine Dynkin diagram. As such, every element of `F` can be viewed as a permutation of the
set `I` of affine Dynkin nodes.
Let `0 \in I` be the distinguished affine node; it is the one whose removal produces the
associated finite Cartan type (call it `X`). A node `i \in I` is called *special*
if some automorphism of the affine Dynkin diagram, sends `0` to `i`.
The node `0` is always special due to the identity automorphism.
There is a bijection of the set of special nodes with the fundamental group. We denote the
image of `i` by `\pi_i`. The structure of `F` is determined as follows.
- `\tilde{X}` is untwisted -- `F` is isomorphic to `P^\vee/Q^\vee` where `P^\vee` and `Q^\vee` are the
coweight and coroot lattices of type `X`. The group `P^\vee/Q^\vee` consists of the cosets `\omega_i^\vee + Q^\vee`
for special nodes `i`, where `\omega_0^\vee = 0` by convention. In this case the special nodes `i`
are the *cominuscule* nodes, the ones such that `\omega_i^\vee(\alpha_j)` is `0` or `1` for all `j\in I_0 = I \setminus \{0\}`.
For `i` special, addition by `\omega_i^\vee+Q^\vee` permutes `P^\vee/Q^\vee` and therefore permutes the set of special nodes.
This permutation extends uniquely to an automorphism of the affine Dynkin diagram.
- `\tilde{X}` is dual untwisted -- (that is, the dual of `\tilde{X}` is untwisted) `F` is isomorphic to `P/Q`
where `P` and `Q` are the weight and root lattices of type `X`. The group `P/Q` consists of the cosets
`\omega_i + Q` for special nodes `i`, where `\omega_0 = 0` by convention. In this case the special nodes `i`
are the *minuscule* nodes, the ones such that `\alpha_j^\vee(\omega_i)` is `0` or `1` for all `j \in I_0`.
For `i` special, addition by `\omega_i+Q` permutes `P/Q` and therefore permutes the set of special nodes.
This permutation extends uniquely to an automorphism of the affine Dynkin diagram.
- `\tilde{X}` is mixed -- (that is, not of the above two types) `F` is the trivial group.
EXAMPLES::
sage: from sage.combinat.root_system.fundamental_group import FundamentalGroupOfExtendedAffineWeylGroup
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',3,1]); F
Fundamental group of type ['A', 3, 1]
sage: F.cartan_type().dynkin_diagram()
0
O-------+
| |
| |
O---O---O
1 2 3
A3~
sage: F.special_nodes()
(0, 1, 2, 3)
sage: F(1)^2
pi[2]
sage: F(1)*F(2)
pi[3]
sage: F(3)^(-1)
pi[1]
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("B3"); F
Fundamental group of type ['B', 3, 1]
sage: F.cartan_type().dynkin_diagram()
O 0
|
|
O---O=>=O
1 2 3
B3~
sage: F.special_nodes()
(0, 1)
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("C2"); F
Fundamental group of type ['C', 2, 1]
sage: F.cartan_type().dynkin_diagram()
O=>=O=<=O
0 1 2
C2~
sage: F.special_nodes()
(0, 2)
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("D4"); F
Fundamental group of type ['D', 4, 1]
sage: F.cartan_type().dynkin_diagram()
O 4
|
|
O---O---O
1 |2 3
|
O 0
D4~
sage: F.special_nodes()
(0, 1, 3, 4)
sage: (F(4), F(4)^2)
(pi[4], pi[0])
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("D5"); F
Fundamental group of type ['D', 5, 1]
sage: F.cartan_type().dynkin_diagram()
0 O O 5
| |
| |
O---O---O---O
1 2 3 4
D5~
sage: F.special_nodes()
(0, 1, 4, 5)
sage: (F(5), F(5)^2, F(5)^3, F(5)^4)
(pi[5], pi[1], pi[4], pi[0])
sage: F = FundamentalGroupOfExtendedAffineWeylGroup("E6"); F
Fundamental group of type ['E', 6, 1]
sage: F.cartan_type().dynkin_diagram()
O 0
|
|
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6~
sage: F.special_nodes()
(0, 1, 6)
sage: F(1)^2
pi[6]
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['D',4,2]); F
Fundamental group of type ['C', 3, 1]^*
sage: F.cartan_type().dynkin_diagram()
O=<=O---O=>=O
0 1 2 3
C3~*
sage: F.special_nodes()
(0, 3)
We also implement a fundamental group for `GL_n`. It is defined to be the group of integers, which is the
covering group of the fundamental group Z/nZ for affine `SL_n`::
sage: F = FundamentalGroupOfExtendedAffineWeylGroup(['A',2,1], general_linear=True); F
Fundamental group of GL(3)
sage: x = F.an_element(); x
pi[5]
sage: x*x
pi[10]
sage: x.inverse()
pi[-5]
sage: wt = F.cartan_type().classical().root_system().ambient_space().an_element(); wt
(2, 2, 3)
sage: x.act_on_classical_ambient(wt)
(2, 3, 2)
sage: w = WeylGroup(F.cartan_type(),prefix="s").an_element(); w
s0*s1*s2
sage: x.act_on_affine_weyl(w)
s2*s0*s1
"""
cartan_type = CartanType(cartan_type)
if cartan_type.is_finite():
cartan_type = cartan_type.affine()
if not cartan_type.is_affine():
raise NotImplementedError("Cartan type is not affine")
if general_linear is True:
if cartan_type.is_untwisted_affine() and cartan_type.type() == "A":
return FundamentalGroupGL(cartan_type, prefix)
else:
raise ValueError("General Linear Fundamental group is untwisted type A")
return FundamentalGroupOfExtendedAffineWeylGroup_Class(cartan_type,prefix,finite=True)
