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 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)
