Python PolynomialRing.sum方法代码示例

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import sum [as 别名]
    def canonical_scheme(self, t=None):
        """
        Return the canonical scheme.

        This is a scheme that contains this hypergeometric motive in its cohomology.

        EXAMPLES::

            sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
            sage: H = Hyp(cyclotomic=([3],[4]))
            sage: H.gamma_list()
            [-1, 2, 3, -4]
            sage: H.canonical_scheme()
            Spectrum of Quotient of Multivariate Polynomial Ring
            in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring
            in t over Rational Field by the ideal
            (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^2*X1^3 + 27/64*Y0*Y1^4)

            sage: H = Hyp(gamma_list=[-2, 3, 4, -5])
            sage: H.canonical_scheme()
            Spectrum of Quotient of Multivariate Polynomial Ring
            in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring
            in t over Rational Field by the ideal
            (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^3*X1^4 + 1728/3125*Y0^2*Y1^5)

        REFERENCES:

        [Kat1991]_, section 5.4
        """
        if t is None:
            t = FractionField(QQ['t']).gen()
        basering = t.parent()
        gamma_pos = [u for u in self.gamma_list() if u > 0]
        gamma_neg = [u for u in self.gamma_list() if u < 0]
        N_pos = len(gamma_pos)
        N_neg = len(gamma_neg)
        varX = ['X{}'.format(i) for i in range(N_pos)]
        varY = ['Y{}'.format(i) for i in range(N_neg)]
        ring = PolynomialRing(basering, varX + varY)
        gens = ring.gens()
        X = gens[:N_pos]
        Y = gens[N_pos:]
        eq0 = ring.sum(X) - 1
        eq1 = ring.sum(Y) - 1
        eq2_pos = ring.prod(X[i] ** gamma_pos[i] for i in range(N_pos))
        eq2_neg = ring.prod(Y[j] ** -gamma_neg[j] for j in range(N_neg))

        ideal = ring.ideal([eq0, eq1, self.M_value() * eq2_neg - t * eq2_pos])
        return Spec(ring.quotient(ideal))

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import sum [as 别名]
    def parabolic_kazhdan_lusztig_polynomial(self, u, v, J, constant_term_one=True):
        """
        Return the parabolic Kazhdan-Lusztig polynomial `P_{u,v}^{-,J}`.

        INPUT:

        - ``u``, ``v`` -- minimal length coset representatives of `W/W_J` for this Coxeter group `W`
        - ``J`` -- a subset of the index set of ``self`` specifying the parabolic subgroup

        This method implements the parabolic Kazhdan-Lusztig polynomials
        `P^{-,J}_{u,v}` of [Deo1987b]_, which are defined as
        `P^{-,J}_{u,v} = \sum_{z\in W_J} (-1)^{\ell(z)} P_{yz,w}(q)`
        with the conventions in Sage.
        As for :meth:`kazhdan_lusztig_polynomial` the convention
        differs from Theorem 2.7 in [LT1998]_ by:

        .. MATH::

            {}^{LT} P_{y,w}^{-,J}(q) = q^{\ell(w)-\ell(y)} P_{y,w}^{-,J}(q^{-2})

        EXAMPLES::

            sage: W = CoxeterGroup(['A',3], implementation='coxeter3')                      # optional - coxeter3
            sage: W.parabolic_kazhdan_lusztig_polynomial([],[3,2],[1,3])                    # optional - coxeter3
            0
            sage: W.parabolic_kazhdan_lusztig_polynomial([2],[2,1,3,2],[1,3])               # optional - coxeter3
            q

            sage: C = CoxeterGroup(['A',3,1], implementation='coxeter3')                    # optional - coxeter3
            sage: C.parabolic_kazhdan_lusztig_polynomial([],[1],[0])                        # optional - coxeter3
            1
            sage: C.parabolic_kazhdan_lusztig_polynomial([],[1,2,1],[0])                    # optional - coxeter3
            1
            sage: C.parabolic_kazhdan_lusztig_polynomial([],[0,1,0,1,2,1],[0])              # optional - coxeter3
            q
            sage: w=[1, 2, 1, 3, 0, 2, 1, 0, 3, 0, 2]
            sage: v=[1, 2, 1, 3, 0, 1, 2, 1, 0, 3, 0, 2, 1, 0, 3, 0, 2]
            sage: C.parabolic_kazhdan_lusztig_polynomial(w,v,[1,3])                         # optional - coxeter3
            q^2 + q
            sage: C.parabolic_kazhdan_lusztig_polynomial(w,v,[1,3],constant_term_one=False) # optional - coxeter3
            q^4 + q^2

        TESTS::

            sage: W = CoxeterGroup(['A', 3], implementation='coxeter3')                     # optional - coxeter3
            sage: type(W.parabolic_kazhdan_lusztig_polynomial([2],[],[1]))                  # optional - coxeter3
            <type 'sage.rings.polynomial.polynomial_integer_dense_flint.Polynomial_integer_dense_flint'>
        """
        u = self(u)
        v = self(v)
        if any(d in J for d in u.descents()) or any(d in J for d in v.descents()):
            raise ValueError("u and v have to be minimal coset representatives")
        J_set = set(J)
        WOI = self.weak_order_ideal(lambda x: J_set.issuperset(x.descents()))
        if constant_term_one:
            P = PolynomialRing(ZZ, 'q')
            return P.sum((-1)**(z.length()) * self.kazhdan_lusztig_polynomial(u*z,v)
                         for z in WOI if (u*z).bruhat_le(v))
        P = PolynomialRing(ZZ, 'q', sparse=True)
        return P.sum((-1)**(z.length()) * self.kazhdan_lusztig_polynomial(u*z,v, constant_term_one=False).shift(z.length())
                     for z in WOI if (u*z).bruhat_le(v))