Python CartanType.affine方法代码示例

# 需要导入模块: from sage.combinat.root_system.cartan_type import CartanType [as 别名]
# 或者: from sage.combinat.root_system.cartan_type.CartanType import affine [as 别名]
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])

#.........这里部分代码省略.........