Python RootSystem.sum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 sum [as 别名]
    def weight_in_root_lattice(self):
        r"""
        Return the weight of ``self`` as an element of the root lattice.

        EXAMPLES::

            sage: M = crystals.infinity.NakajimaMonomials(['F',4])
            sage: m = M.module_generators[0].f_string([3,3,1,2,4])
            sage: m.weight_in_root_lattice()
            -alpha[1] - alpha[2] - 2*alpha[3] - alpha[4]

            sage: M = crystals.infinity.NakajimaMonomials(['B',3,1])
            sage: mg = M.module_generators[0]
            sage: m = mg.f_string([1,3,2,0,1,2,3,0,0,1])
            sage: m.weight_in_root_lattice()
            -3*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]

            sage: M = crystals.infinity.NakajimaMonomials(['C',3,1])
            sage: m = M.module_generators[0].f_string([3,0,1,2,0])
            sage: m.weight_in_root_lattice()
            -2*alpha[0] - alpha[1] - alpha[2] - alpha[3]
        """
        Q = RootSystem(self.parent().cartan_type()).root_lattice()
        al = Q.simple_roots()
        return Q.sum(e*al[k[0]] for k,e in six.iteritems(self._A))

# 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 sum [as 别名]
        def energy_function(self):
            r"""
            Return the energy function of ``self``.

            The energy function `D(\pi)` of the level zero LS path `\pi \in \mathbb{B}_\mathrm{cl}(\lambda)`
            requires a series of definitions; for simplicity the root system is assumed to be untwisted affine.

            The LS path `\pi` is a piecewise linear map from the unit interval `[0,1]` to the weight lattice.
            It is specified by "times" `0=\sigma_0<\sigma_1<\dotsm<\sigma_s=1` and "direction vectors"
            `x_u \lambda` where `x_u \in W/W_J` for `1\le u\le s`, and `W_J` is the
            stabilizer of `\lambda` in the finite Weyl group `W`. Precisely,

            .. MATH::

                \pi(t)=\sum_{u'=1}^{u-1} (\sigma_{u'}-\sigma_{u'-1})x_{u'}\lambda+(t-\sigma_{u-1})x_{u}\lambda

            for `1\le u\le s` and `\sigma_{u-1} \le t \le \sigma_{u}`.

            For any `x,y\in W/W_J` let

            .. MATH::

                d: x= w_{0} \stackrel{\beta_{1}}{\leftarrow}
                w_{1} \stackrel{\beta_{2}}{\leftarrow} \cdots
                \stackrel{\beta_{n}}{\leftarrow} w_{n}=y

            be a shortest directed path in the parabolic quantum Bruhat graph. Define

            .. MATH::

                \mathrm{wt}(d):=\sum_{\substack{1\le k\le n \\  \ell(w_{k-1})<\ell(w_k)}}
                \beta_{k}^{\vee}

            It can be shown that `\mathrm{wt}(d)` depends only on `x,y`;
            call its value `\mathrm{wt}(x,y)`. The energy function `D(\pi)` is defined by

            .. MATH::

                D(\pi)=-\sum_{u=1}^{s-1} (1-\sigma_{u}) \langle \lambda,\mathrm{wt}(x_u,x_{u+1}) \rangle

            For more information, see [LNSSS2013]_.

            REFERENCES:

            .. [LNSSS2013] C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono,
               A uniform model for Kirillov-Reshetikhin crystals. Extended abstract.
               DMTCS proc, to appear ( {{{:arXiv:`1211.6019`}}} )

            .. NOTE::

                In the dual-of-untwisted case the parabolic quantum Bruhat graph that is used is obtained by
                exchanging the roles of roots and coroots. Moreover, in the computation of the
                pairing the short roots must be doubled (or tripled for type `G`). This factor
                is determined by the translation factor of the corresponding root.
                Type `BC` is viewed as untwisted type, whereas the dual of `BC` is viewed as twisted.
                Except for the untwisted cases, these formulas are currently still conjectural.

            EXAMPLES::

                sage: R = RootSystem(['C',3,1])
                sage: La = R.weight_space().basis()
                sage: LS = CrystalOfProjectedLevelZeroLSPaths(La[1]+La[3])
                sage: b = LS.module_generators[0]
                sage: c = b.f(1).f(3).f(2)
                sage: c.energy_function()
                0
                sage: c=b.e(0)
                sage: c.energy_function()
                1

                sage: R = RootSystem(['A',2,1])
                sage: La = R.weight_space().basis()
                sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1])
                sage: b = LS.module_generators[0]
                sage: c = b.e(0)
                sage: c.energy_function()
                1
                sage: [c.energy_function() for c in sorted(LS.list())]
                [0, 1, 0, 0, 0, 1, 0, 1, 0]

            The next test checks that the energy function is constant on classically connected components::

                sage: R = RootSystem(['A',2,1])
                sage: La = R.weight_space().basis()
                sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1]+La[2])
                sage: G = LS.digraph(index_set=[1,2])
                sage: C = G.connected_components()
                sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
                [True, True, True, True]

                sage: R = RootSystem(['D',4,2])
                sage: La = R.weight_space().basis()
                sage: LS = CrystalOfProjectedLevelZeroLSPaths(La[2])
                sage: J = R.cartan_type().classical().index_set()
                sage: hw = [x for x in LS if x.is_highest_weight(J)]
                sage: [(x.weight(), x.energy_function()) for x in hw]
                [(-2*Lambda[0] + Lambda[2], 0), (-2*Lambda[0] + Lambda[1], 1), (0, 2)]
                sage: G = LS.digraph(index_set=J)
                sage: C = G.connected_components()
                sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
#.........这里部分代码省略.........

# 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 sum [as 别名]
        def energy_function(self):
            r"""
            Returns the energy function of ``self`` for untwisted types.

            For level zero LS paths `\pi \in \mathbb{B}_\mathrm{cl}(\lambda)`, the energy function is defined as follows:

            .. MATH::

                D(\pi)=-\sum_{u=1}^{s-1} (1-\sigma_{u}) \langle \lambda,\mathrm{wt}(b_{u}) \rangle

            To make sense of this equation, we first need some definitions.
            Let us write the LS path (or ``self``) as a piecewise linear map

            .. MATH::

                \pi(t)=\sum_{u'=1}^{u-1} (\sigma_{u'}-\sigma_{u'-1})x_{u'}+(t-\sigma_{u-1})x_{u}

            for `\sigma_{u-1} \le t \le \sigma_{u}` and `1 \le u \le s`.
            Here `b_{u}` is a shortest directed path from `x_{u+1}` to `x_{u}` in the parabolic quantum
            Bruhat graph.

            For any `x,y\in W/W_J`, where `W_J` is a parabolic subgroup of `W` (the stabilizer of the weight `\lambda`), let

            .. MATH::

                d: x= w_{0} \stackrel{\beta_{1}}{\leftarrow}
                w_{1} \stackrel{\beta_{2}}{\leftarrow} \cdots
                \stackrel{\beta_{n}}{\leftarrow} w_{n}=y

            be a shortest path in the parabolic quantum Bruhat graph. The weight is defined as

            .. MATH::

                \mathrm{wt}(d):=\sum_{ \begin{subarray}{c}
                1 \le k \le n \text{ such that } \\
                w_{k-1} \stackrel{\beta_{k}}{\leftarrow} w_{k}
                \text{ is a down arrow}
                \end{subarray}
                }
                \beta_{k}^{\vee}.

            For more information, see [LNSSS2013]_.

            REFERENCES:

            .. [LNSSS2013] C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono,
               A uniform model for Kirillov-Reshetikhin crystals. Extended abstract.
               DMTCS proc, to appear ( {{{:arXiv:`1211.6019`}}} )

            .. WARNING::

                The energy function for LS paths is currently only implemented for
                untwisted types!


            EXAMPLES::

                sage: R = RootSystem(['C',3,1])
                sage: La = R.weight_space().basis()
                sage: LS = CrystalOfProjectedLevelZeroLSPaths(La[1]+La[3])
                sage: b = LS.module_generators[0]
                sage: c = b.f(1).f(3).f(2)
                sage: c.energy_function()
                0
                sage: c=b.e(0)
                sage: c.energy_function()
                1

                sage: R = RootSystem(['A',2,1])
                sage: La = R.weight_space().basis()
                sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1])
                sage: b = LS.module_generators[0]
                sage: c = b.e(0)
                sage: c.energy_function()
                1
                sage: [c.energy_function() for c in sorted(LS.list())]
                [0, 1, 0, 0, 0, 1, 0, 1, 0]

            The next test checks that the energy function is constant on classically connected components::

                sage: R = RootSystem(['A',2,1])
                sage: La = R.weight_space().basis()
                sage: LS = CrystalOfProjectedLevelZeroLSPaths(2*La[1]+La[2])
                sage: G = LS.digraph(index_set=[1,2])
                sage: C = G.connected_components()
                sage: [all(c[0].energy_function()==a.energy_function() for a in c) for c in C]
                [True, True, True, True]
            """
            weight = self.parent().weight
            c_weight = weight.parent().classical()(weight)
            cartan = weight.parent().cartan_type().classical()
            W = WeylGroup(cartan, prefix="s")
            R = RootSystem(cartan).coroot_lattice()
            G = W.quantum_bruhat_graph(tuple(weight.weyl_stabilizer()))
            L = self.weyl_group_representation()
            paths = [G.shortest_path(L[i + 1], L[i]) for i in range(len(L) - 1)]
            paths_labels = [
                [G.edge_label(p[i], p[i + 1]) for i in range(len(p) - 1) if p[i].length() + 1 != p[i + 1].length()]
                for p in paths
            ]
#.........这里部分代码省略.........