Python RootSystem.one方法代码示例

# 需要导入模块: from sage.combinat.root_system.root_system import RootSystem [as 别名]
# 或者: from sage.combinat.root_system.root_system.RootSystem import one [as 别名]
class RationalCherednikAlgebra(CombinatorialFreeModule):
    r"""
    A rational Cherednik algebra.

    Let `k` be a field. Let `W` be a complex reflection group acting on
    a vector space `\mathfrak{h}` (over `k`). Let `\mathfrak{h}^*` denote
    the corresponding dual vector space. Let `\cdot` denote the
    natural action of `w` on `\mathfrak{h}` and `\mathfrak{h}^*`. Let
    `\mathcal{S}` denote the set of reflections of `W` and  `\alpha_s`
    and `\alpha_s^{\vee}` are the associated root and coroot of `s`. Let
    `c = (c_s)_{s \in W}` such that `c_s = c_{tst^{-1}}` for all `t \in W`.

    The *rational Cherednik algebra* is the `k`-algebra
    `H_{c,t}(W) = T(\mathfrak{h} \oplus \mathfrak{h}^*) \otimes kW` with
    parameters `c, t \in k` that is subject to the relations:

    .. MATH::

        \begin{aligned}
        w \alpha & = (w \cdot \alpha) w,
        \\ \alpha^{\vee} w & = w (w^{-1} \cdot \alpha^{\vee}),
        \\ \alpha \alpha^{\vee} & = \alpha^{\vee} \alpha
        + t \langle \alpha^{\vee}, \alpha \rangle
        + \sum_{s \in \mathcal{S}} c_s \frac{\langle \alpha^{\vee},
        \alpha_s \rangle \langle \alpha^{\vee}_s, \alpha \rangle}{
        \langle \alpha^{\vee}, \alpha \rangle} s,
        \end{aligned}

    where `w \in W` and `\alpha \in \mathfrak{h}` and
    `\alpha^{\vee} \in \mathfrak{h}^*`.

    INPUT:

    - ``ct`` -- a finite Cartan type
    - ``c`` -- the parameters `c_s` given as an element or a tuple, where
      the first entry is the one for the long roots and (for
      non-simply-laced types) the second is for the short roots
    - ``t`` -- the parameter `t`
    - ``base_ring`` -- (optional) the base ring
    - ``prefix`` -- (default: ``('a', 's', 'ac')``) the prefixes

    .. TODO::

        Implement a version for complex reflection groups.

    REFERENCES:

    - [GGOR2003]_
    - [EM2001]_
    """
    @staticmethod
    def __classcall_private__(cls, ct, c=1, t=None, base_ring=None, prefix=('a', 's', 'ac')):
        """
        Normalize input to ensure a unique representation.

        EXAMPLES::

            sage: R1 = algebras.RationalCherednik(['B',2], 1, 1, QQ)
            sage: R2 = algebras.RationalCherednik(CartanType(['B',2]), [1,1], 1, QQ, ('a', 's', 'ac'))
            sage: R1 is R2
            True
        """
        ct = CartanType(ct)
        if not ct.is_finite():
            raise ValueError("the Cartan type must be finite")
        if base_ring is None:
            if t is None:
                base_ring = QQ
            else:
                base_ring = t.parent()
        if t is None:
            t = base_ring.one()
        else:
            t = base_ring(t)

        # Normalize the parameter c
        if isinstance(c, (tuple, list)):
            if ct.is_simply_laced():
                if len(c) != 1:
                    raise ValueError("1 parameter c_s must be given for simply-laced types")
                c = (base_ring(c[0]),)
            else:
                if len(c) != 2:
                    raise ValueError("2 parameters c_s must be given for non-simply-laced types")
                c = (base_ring(c[0]), base_ring(c[1]))
        else:
            c = base_ring(c)
            if ct.is_simply_laced():
                c = (c,)
            else:
                c = (c, c)

        return super(RationalCherednikAlgebra, cls).__classcall__(cls, ct, c, t, base_ring, tuple(prefix))

    def __init__(self, ct, c, t, base_ring, prefix):
        r"""
        Initialize ``self``.

        EXAMPLES::

#.........这里部分代码省略.........