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# -*- coding: utf-8 -*- r""" Crystal Of Mirković-Vilonen (MV) Polytopes
AUTHORS:
- Dinakar Muthiah, Travis Scrimshaw (2015-05-11): initial version """
#***************************************************************************** # Copyright (C) 2015 Dinakar Muthiah <muthiah at ualberta.ca> # 2015 Travis Scrimshaw <tscrimsh at umn.edu> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from sage.structure.parent import Parent from sage.structure.unique_representation import UniqueRepresentation from sage.categories.highest_weight_crystals import HighestWeightCrystals from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets from sage.combinat.root_system.cartan_type import CartanType from sage.combinat.crystals.pbw_crystal import PBWCrystalElement, PBWCrystal
class MVPolytope(PBWCrystalElement): """ A Mirković-Vilonen (MV) polytope.
EXAMPLES:
We can create an animation showing how the MV polytope changes under a string of crystal operators::
sage: MV = crystals.infinity.MVPolytopes(['C', 2]) sage: u = MV.highest_weight_vector() sage: L = RootSystem(['C',2,1]).ambient_space() sage: s = [1,2,1,2,2,2,1,1,1,1,2,1,2,2,1,2] sage: BB = [[-9, 2], [-10, 2]] sage: p = L.plot(reflection_hyperplanes=False, bounding_box=BB) # long time sage: frames = [p + L.plot_mv_polytope(u.f_string(s[:i]), # long time ....: circle_size=0.1, ....: wireframe='green', ....: fill='purple', ....: bounding_box=BB) ....: for i in range(len(s))] sage: for f in frames: # long time ....: f.axes(False) sage: animate(frames).show(delay=60) # optional -- ImageMagick # long time """ def _repr_(self): """ Return a string representation of ``self``.
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['E', 6]) sage: b = MV.module_generators[0].f_string([1,2,6,4,3,2,5,2]) sage: b MV polytope with Lusztig datum (0, 1, ..., 1, 0, 0, 0, 0, 0, 0, 3, 1) """
def _latex_(self): r""" Return a latex representation of ``self``.
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['C', 2]) sage: b = MV.module_generators[0].f_string([1,2,1,2]) sage: latex(b) \begin{tikzpicture} \draw (0, 0) -- (-1, 1) -- (-1, 1) -- (-2, 0) -- (-2, -2); \draw (0, 0) -- (0, -2) -- (-1, -3) -- (-1, -3) -- (-2, -2); \draw[fill=black] (0, 0) circle (0.1); \draw[fill=black] (-2, -2) circle (0.1); \end{tikzpicture} sage: MV = crystals.infinity.MVPolytopes(['D',4]) sage: b = MV.module_generators[0].f_string([1,2,1,2]) sage: latex(b) \text{\texttt{MV{ }polytope{ }...}}
TESTS::
sage: MV = crystals.infinity.MVPolytopes(['A',2]) sage: u = MV.highest_weight_vector() sage: b = u.f_string([1,2,2,1]) sage: latex(b) \begin{tikzpicture} \draw (0, 0) -- (3/2, -989/1142) -- (3/2, -2967/1142) -- (0, -1978/571); \draw (0, 0) -- (-3/2, -989/1142) -- (-3/2, -2967/1142) -- (0, -1978/571); \draw[fill=black] (0, 0) circle (0.1); \draw[fill=black] (0, -1978/571) circle (0.1); \end{tikzpicture} """
# We need this to use tikz
for root in pbw_data._root_list_from(red)]
def _polytope_vertices(self, P): """ Return a list of the vertices of ``self`` in ``P``.
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['C', 3]) sage: b = MV.module_generators[0].f_string([1,2,1,2]) sage: sorted(b._polytope_vertices(MV.weight_lattice_realization()), key=list) [(0, 0, 0), (2, 0, -2), (0, 2, -2)]
sage: MV = crystals.infinity.MVPolytopes(['D', 4]) sage: b = MV.module_generators[0].f_string([1,2,3,4]) sage: P = RootSystem(['D',4]).weight_lattice() sage: sorted(b._polytope_vertices(P), key=list) # long time [0, -Lambda[1] + Lambda[3] + Lambda[4], Lambda[1] - Lambda[2] + Lambda[3] + Lambda[4], -2*Lambda[2] + 2*Lambda[3] + 2*Lambda[4], -Lambda[2] + 2*Lambda[3], -Lambda[2] + 2*Lambda[4]] """
for root in pbw_data._root_list_from(red)]
def polytope(self, P=None): """ Return a polytope of ``self``.
INPUT:
- ``P`` -- (optional) a space to realize the polytope; default is the weight lattice realization of the crystal
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['C', 3]) sage: b = MV.module_generators[0].f_string([3,2,3,2,1]) sage: P = b.polytope(); P A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 6 vertices sage: P.vertices() (A vertex at (0, 0, 0), A vertex at (0, 1, -1), A vertex at (0, 1, 1), A vertex at (1, -1, 0), A vertex at (1, 1, -2), A vertex at (1, 1, 2)) """
def plot(self, P=None, **options): """ Plot ``self``.
INPUT:
- ``P`` -- (optional) a space to realize the polytope; default is the weight lattice realization of the crystal
.. SEEALSO::
:meth:`~sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods.plot_mv_polytope`
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['C', 2]) sage: b = MV.highest_weight_vector().f_string([1,2,1,2,2,2,1,1,1,1,2,1]) sage: b.plot() Graphics object consisting of 12 graphics primitives
Here is the above example placed inside the ambient space of type `C_2`:
.. PLOT:: :width: 300 px
MV = crystals.infinity.MVPolytopes(['C', 2]) b = MV.highest_weight_vector().f_string([1,2,1,2,2,2,1,1,1,1,2,1]) L = RootSystem(['C', 2, 1]).ambient_space() p = L.plot(reflection_hyperplanes=False, bounding_box=[[-8,2], [-8,2]]) p += b.plot() p.axes(False) sphinx_plot(p) """
class MVPolytopes(PBWCrystal): r""" The crystal of Mirković-Vilonen (MV) polytopes.
Let `W` denote the corresponding Weyl group and `P_{\RR} = \RR \otimes P`. Let `\Gamma = \{ w \Lambda_i \mid w \in W, i \in I \}`. Consider `M = (M_{\gamma} \in \ZZ)_{\gamma \in \Gamma}` that satisfy the *tropical Plücker relations* (see Proposition 7.1 of [BZ01]_). The *MV polytope* is defined as
.. MATH::
P(M) = \{ \alpha \in P_{\RR} \mid \langle \alpha, \gamma \rangle \geq M_{\gamma} \text{ for all } \gamma \in \Gamma \}.
The vertices `\{\mu_w\}_{w \in W}` are given by
.. MATH::
\langle \mu_w, \gamma \rangle = M_{\gamma}
and are known as the GGMS datum of the MV polytope.
Each path from `\mu_e` to `\mu_{w_0}` corresponds to a reduced expression `\mathbf{i} = (i_1, \ldots, i_m)` for `w_0` and the corresponding edge lengths `(n_k)_{k=1}^m` from the Lusztig datum with respect to `\mathbf{i}`. Explicitly, we have
.. MATH::
\begin{aligned} n_k & = -M_{w_{k-1} \Lambda_{i_k}} - M_{w_k \Lambda_{i_k}} - \sum_{j \neq i} a_{ji} M_{w_k \Lambda_j}, \\ \mu_{w_k} - \mu_{w_{k-1}} & = n_k w_{k-1} \alpha_{i_k}, \end{aligned}
where `w_k = s_{i_1} \cdots s_{i_k}` and `(a_{ji})` is the Cartan matrix.
MV polytopes have a crystal structure that corresponds to the crystal structure, which is isomorphic to `\mathcal{B}(\infty)` with `\mu_{w_0} = 0`, on :class:`PBW data <sage.combinat.crystals.pbw_crystal.PBWCrystal>`. Specifically, we have `f_j P(M)` as being the unique MV polytope given by shifting `\mu_e` by `-\alpha_j` and fixing the vertices `\mu_w` when `s_j w < w` (in Bruhat order) and the weight is given by `\mu_e`. Furthermore, the `*`-involution is given by negating `P(M)`.
INPUT:
- ``cartan_type`` -- a Cartan type
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['B', 3]) sage: hw = MV.highest_weight_vector() sage: x = hw.f_string([1,2,2,3,3,1,3,3,2,3,2,1,3,1,2,3,1,2,1,3,2]); x MV polytope with Lusztig datum (1, 1, 1, 3, 1, 0, 0, 1, 1)
Elements are expressed in terms of Lusztig datum for a fixed reduced expression of `w_0`::
sage: MV.default_long_word() [1, 3, 2, 3, 1, 2, 3, 1, 2] sage: MV.set_default_long_word([2,1,3,2,1,3,2,3,1]) sage: x MV polytope with Lusztig datum (3, 1, 1, 0, 1, 0, 1, 3, 4) sage: MV.set_default_long_word([1, 3, 2, 3, 1, 2, 3, 1, 2])
We can construct elements by giving it Lusztig data (with respect to the default long word reduced expression)::
sage: MV([1,1,1,3,1,0,0,1,1]) MV polytope with Lusztig datum (1, 1, 1, 3, 1, 0, 0, 1, 1)
We can also construct elements by passing in a reduced expression for a long word::
sage: x = MV([1,1,1,3,1,0,0,1,1], [3,2,1,3,2,3,2,1,2]); x MV polytope with Lusztig datum (1, 1, 1, 0, 1, 0, 5, 1, 1) sage: x.to_highest_weight()[1] [1, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 1, 3]
The highest weight crystal `B(\lambda) \subseteq B(\infty)` is characterized by the MV polytopes that sit inside of `W \lambda` (translating `\mu_{w_0} \mapsto \lambda`)::
sage: MV = crystals.infinity.MVPolytopes(['A',2]) sage: La = MV.weight_lattice_realization().fundamental_weights() sage: R = crystals.elementary.R(La[1]+La[2]) sage: T = tensor([R, MV]) sage: x = T(R.module_generators[0], MV.highest_weight_vector()) sage: lw = x.to_lowest_weight()[0]; lw [(2, 1, 0), MV polytope with Lusztig datum (1, 1, 1)] sage: lw[1].polytope().vertices() (A vertex at (0, 0, 0), A vertex at (0, 1, -1), A vertex at (1, -1, 0), A vertex at (1, 1, -2), A vertex at (2, -1, -1), A vertex at (2, 0, -2))
.. PLOT:: :width: 300 px
MV = crystals.infinity.MVPolytopes(['A',2]) x = MV.module_generators[0].f_string([1,2,2,1]) L = RootSystem(['A',2,1]).ambient_space() p = L.plot(bounding_box=[[-2,2],[-4,2]]) + x.plot() p.axes(False) sphinx_plot(x.plot())
REFERENCES:
- [Kam2007]_ - [Kam2010]_ """ def __init__(self, cartan_type): """ Initialize ``self``.
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['B', 2]) sage: TestSuite(MV).run() """ "mark_endpoints": True, "P": self.weight_lattice_realization(), "circle_size": 0.1}
def _repr_(self): """ Return a string representation of ``self``.
EXAMPLES::
sage: crystals.infinity.MVPolytopes(['F', 4]) MV polytopes of type ['F', 4] """
def set_latex_options(self, **kwds): r""" Set the latex options for the elements of ``self``.
INPUT:
- ``projection`` -- the projection; set to ``True`` to use the default projection of the specified weight lattice realization (initial: ``True``) - ``P`` -- the weight lattice realization to use (initial: the weight lattice realization of ``self``) - ``mark_endpoints`` -- whether to mark the endpoints (initial: ``True``) - ``circle_size`` -- the size of the endpoint circles (initial: 0.1)
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['C', 2]) sage: P = RootSystem(['C', 2]).weight_lattice() sage: b = MV.highest_weight_vector().f_string([1,2,1,2]) sage: latex(b) \begin{tikzpicture} \draw (0, 0) -- (-1, 1) -- (-1, 1) -- (-2, 0) -- (-2, -2); \draw (0, 0) -- (0, -2) -- (-1, -3) -- (-1, -3) -- (-2, -2); \draw[fill=black] (0, 0) circle (0.1); \draw[fill=black] (-2, -2) circle (0.1); \end{tikzpicture} sage: MV.set_latex_options(P=P, circle_size=float(0.2)) sage: latex(b) \begin{tikzpicture} \draw (0, 0) -- (-2, 1) -- (-2, 1) -- (-2, 0) -- (0, -2); \draw (0, 0) -- (2, -2) -- (2, -3) -- (2, -3) -- (0, -2); \draw[fill=black] (0, 0) circle (0.2); \draw[fill=black] (0, -2) circle (0.2); \end{tikzpicture} sage: MV.set_latex_options(mark_endpoints=False) sage: latex(b) \begin{tikzpicture} \draw (0, 0) -- (-2, 1) -- (-2, 1) -- (-2, 0) -- (0, -2); \draw (0, 0) -- (2, -2) -- (2, -3) -- (2, -3) -- (0, -2); \end{tikzpicture} sage: MV.set_latex_options(P=MV.weight_lattice_realization(), ....: circle_size=0.2, ....: mark_endpoints=True) """ self._latex_options["projection"] = True del kwds["projection"]
raise ValueError("invalid latex option")
def latex_options(self): """ Return the latex options of ``self``.
EXAMPLES::
sage: MV = crystals.infinity.MVPolytopes(['F', 4]) sage: MV.latex_options() {'P': Ambient space of the Root system of type ['F', 4], 'circle_size': 0.1, 'mark_endpoints': True, 'projection': True} """
Element = MVPolytope
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