def _latex_(self, scale=0.5):
r"""
Return a latex representation of this Dynkin diagram
EXAMPLES::
sage: latex(DynkinDiagram(['A',3,1]))
\begin{tikzpicture}[scale=0.5]
\draw (-1,0) node[anchor=east] {$A_{3}^{(1)}$};
\draw (0 cm,0) -- (4 cm,0);
\draw (0 cm,0) -- (2.0 cm, 1.2 cm);
\draw (2.0 cm, 1.2 cm) -- (4 cm, 0);
\draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$1$};
\draw[fill=white] (2 cm, 0 cm) circle (.25cm) node[below=4pt]{$2$};
\draw[fill=white] (4 cm, 0 cm) circle (.25cm) node[below=4pt]{$3$};
\draw[fill=white] (2.0 cm, 1.2 cm) circle (.25cm) node[anchor=south east]{$0$};
\end{tikzpicture}
"""
if self.cartan_type() is None:
return "Dynkin diagram of rank {}".format(self.rank())
from sage.graphs.graph_latex import setup_latex_preamble
setup_latex_preamble()
ret = "\\begin{{tikzpicture}}[scale={}]\n".format(scale)
ret += "\\draw (-1,0) node[anchor=east] {{${}$}};\n".format(self.cartan_type()._latex_())
ret += self.cartan_type()._latex_dynkin_diagram()
ret += "\\end{tikzpicture}"
return ret
def _latex_(self,
show_box=False,
colors=["white", "lightgray", "darkgray"]):
r"""
Return latex code for ``self``, which uses TikZ package to draw
the plane partition.
INPUT:
- ``show_box`` -- boolean (default: ``False``); if ``True``,
also shows the visible tiles on the `xy`-, `yz`-, `zx`-planes
- ``colors`` -- (default: ``["white", "lightgray", "darkgray"]``)
list ``[A, B, C]`` of 3 strings representing colors
OUTPUT:
Latex code for drawing the plane partition.
EXAMPLES::
sage: PP = PlanePartition([[1]])
sage: latex(PP)
\begin{tikzpicture}
\draw[fill=white,shift={(210:0)},shift={(-30:0)},shift={(90:1)}]
(0,0)--(-30:1)--(0,-1)--(210:1)--(0,0);
\draw[fill=darkgray,shift={(210:0)},shift={(-30:1)},shift={(90:0)}]
(0,0)--(210:1)--(150:1)--(0,1)--(0,0);
\draw[fill=lightgray,shift={(210:1)},shift={(-30:0)},shift={(90:0)}]
(0,0)--(0,1)--(30:1)--(-30:1)--(0,0);
\end{tikzpicture}
"""
from sage.graphs.graph_latex import setup_latex_preamble
setup_latex_preamble()
ret = "\\begin{tikzpicture}\n"
def add_topside(i, j, k):
return "\\draw[fill={},shift={{(210:{})}},shift={{(-30:{})}},shift={{(90:{})}}]\n(0,0)--(-30:1)--(0,-1)--(210:1)--(0,0);\n".format(
colors[0], i, j, k)
def add_leftside(j, k, i):
return "\\draw[fill={},shift={{(210:{})}},shift={{(-30:{})}},shift={{(90:{})}}]\n(0,0)--(0,1)--(30:1)--(-30:1)--(0,0);\n".format(
colors[1], i, j, k)
def add_rightside(k, i, j):
return "\\draw[fill={},shift={{(210:{})}},shift={{(-30:{})}},shift={{(90:{})}}]\n(0,0)--(210:1)--(150:1)--(0,1)--(0,0);\n".format(
colors[2], i, j, k)
funcs = [add_topside, add_rightside, add_leftside]
tableaux = [self.z_tableau(), self.y_tableau(), self.x_tableau()]
for i in range(3):
f = funcs[i]
tab = tableaux[i]
for r in range(len(tab)):
for c in range(len(tab[r])):
if tab[r][c] > 0 or show_box:
ret += f(r, c, tab[r][c])
return ret + "\\end{tikzpicture}"
def _latex_(self, show_box=False, colors=["white","lightgray","darkgray"]):
r"""
Return latex code for ``self``, which uses TikZ package to draw
the plane partition.
INPUT:
- ``show_box`` -- boolean (default: ``False``); if ``True``,
also shows the visible tiles on the `xy`-, `yz`-, `zx`-planes
- ``colors`` -- (default: ``["white", "lightgray", "darkgray"]``)
list ``[A, B, C]`` of 3 strings representing colors
OUTPUT:
Latex code for drawing the plane partition.
EXAMPLES::
sage: PP = PlanePartition([[1]])
sage: latex(PP)
\begin{tikzpicture}
\draw[fill=white,shift={(210:0)},shift={(-30:0)},shift={(90:1)}]
(0,0)--(-30:1)--(0,-1)--(210:1)--(0,0);
\draw[fill=darkgray,shift={(210:0)},shift={(-30:1)},shift={(90:0)}]
(0,0)--(210:1)--(150:1)--(0,1)--(0,0);
\draw[fill=lightgray,shift={(210:1)},shift={(-30:0)},shift={(90:0)}]
(0,0)--(0,1)--(30:1)--(-30:1)--(0,0);
\end{tikzpicture}
"""
from sage.graphs.graph_latex import setup_latex_preamble
setup_latex_preamble()
x = self._max_x
y = self._max_y
z = self._max_z
ret = "\\begin{tikzpicture}\n"
def add_topside(i,j,k):
return "\\draw[fill={},shift={{(210:{})}},shift={{(-30:{})}},shift={{(90:{})}}]\n(0,0)--(-30:1)--(0,-1)--(210:1)--(0,0);\n".format(colors[0],i,j,k)
def add_leftside(j,k,i):
return "\\draw[fill={},shift={{(210:{})}},shift={{(-30:{})}},shift={{(90:{})}}]\n(0,0)--(0,1)--(30:1)--(-30:1)--(0,0);\n".format(colors[1],i,j,k)
def add_rightside(k,i,j):
return "\\draw[fill={},shift={{(210:{})}},shift={{(-30:{})}},shift={{(90:{})}}]\n(0,0)--(210:1)--(150:1)--(0,1)--(0,0);\n".format(colors[2],i,j,k)
funcs = [add_topside, add_rightside, add_leftside]
tableaux = [self.z_tableau(), self.y_tableau(), self.x_tableau()]
for i in range(3):
f = funcs[i]
tab = tableaux[i]
for r in range(len(tab)):
for c in range(len(tab[r])):
if tab[r][c] > 0 or show_box:
ret += f(r, c, tab[r][c])
return ret + "\\end{tikzpicture}"
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}
"""
latex_options = self.parent()._latex_options
P = latex_options['P']
plot_options = P.plot_parse_options(
projection=latex_options["projection"])
proj = plot_options.projection
if proj(P.zero()).parent().dimension() != 2:
from sage.misc.latex import latex
return latex(repr(self))
# We need this to use tikz
from sage.graphs.graph_latex import setup_latex_preamble
setup_latex_preamble()
pbw_data = self._pbw_datum.parent
W = pbw_data.weyl_group
w0 = W.long_element()
al = P.simple_roots()
ret = "\\begin{tikzpicture}\n"
final = None
for red in w0.reduced_words():
ret += "\\draw "
cur = proj(P.zero())
red = tuple(red)
ret += str(cur)
roots = [
proj(P.sum(c * al[a] for a, c in root))
for root in pbw_data._root_list_from(red)
]
datum = pbw_data.convert_to_new_long_word(self._pbw_datum, red)
for i in reversed(range(len(datum.lusztig_datum))):
cur -= roots[i] * datum.lusztig_datum[i]
ret += " -- " + str(cur)
final = cur
ret += ";\n"
if latex_options["mark_endpoints"]:
circle_size = latex_options["circle_size"]
ret += "\\draw[fill=black] {} circle ({});\n".format(
proj(P.zero()), circle_size)
ret += "\\draw[fill=black] {} circle ({});\n".format(
final, circle_size)
ret += "\\end{tikzpicture}"
return ret
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) -- (0, -2) -- (-1, -3) -- (-1, -3) -- (-2, -2);
\draw (0, 0) -- (-1, 1) -- (-1, 1) -- (-2, 0) -- (-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}
"""
latex_options = self.parent()._latex_options
P = latex_options['P']
plot_options = P.plot_parse_options(projection=latex_options["projection"])
proj = plot_options.projection
if proj(P.zero()).parent().dimension() != 2:
from sage.misc.latex import latex
return latex(repr(self))
# We need this to use tikz
from sage.graphs.graph_latex import setup_latex_preamble
setup_latex_preamble()
pbw_data = self._pbw_datum.parent
W = pbw_data.weyl_group
w0 = W.long_element()
al = P.simple_roots()
ret = "\\begin{tikzpicture}\n"
final = None
for red in sorted(w0.reduced_words()):
ret += "\\draw "
cur = proj(P.zero())
red = tuple(red)
ret += str(cur)
roots = [proj(P.sum(c*al[a] for a,c in root))
for root in pbw_data._root_list_from(red)]
datum = pbw_data.convert_to_new_long_word(self._pbw_datum, red)
for i in reversed(range(len(datum.lusztig_datum))):
cur -= roots[i] * datum.lusztig_datum[i]
ret += " -- " + str(cur)
final = cur
ret += ";\n"
if latex_options["mark_endpoints"]:
circle_size = latex_options["circle_size"]
ret += "\\draw[fill=black] {} circle ({});\n".format(proj(P.zero()), circle_size)
ret += "\\draw[fill=black] {} circle ({});\n".format(final, circle_size)
ret += "\\end{tikzpicture}"
return ret
def latex_state_evolution(self, num, scale=1):
r"""
Return a latex version of the evolution process of
the state ``num``.
.. SEEALSO::
:meth:`state_evolution`, :meth:`print_state_evolution`
EXAMPLES::
sage: B = SolitonCellularAutomata('113123', 3)
sage: B.evolve(3)
sage: B.latex_state_evolution(0)
\begin{tikzpicture}[scale=1]
\node (i0) at (0,0.9) {$1$};
\node (i1) at (-2,0.9) {$1$};
\node (i2) at (-4,0.9) {$3$};
\node (i3) at (-6,0.9) {$1$};
\node (i4) at (-8,0.9) {$2$};
\node (i5) at (-10,0.9) {$3$};
\node (t0) at (0,-1) {$1$};
\node (t1) at (-2,-1) {$3$};
\node (t2) at (-4,-1) {$2$};
\node (t3) at (-6,-1) {$3$};
\node (t4) at (-8,-1) {$1$};
\node (t5) at (-10,-1) {$1$};
\node (u0) at (1,0) {$111$};
\node (u1) at (-1,0) {$111$};
\node (u2) at (-3,0) {$113$};
\node (u3) at (-5,0) {$112$};
\node (u4) at (-7,0) {$123$};
\node (u5) at (-9,0) {$113$};
\node (u6) at (-11,0) {$111$};
\draw[->] (i0) -- (t0);
\draw[->] (u0) -- (u1);
\draw[->] (i1) -- (t1);
\draw[->] (u1) -- (u2);
\draw[->] (i2) -- (t2);
\draw[->] (u2) -- (u3);
\draw[->] (i3) -- (t3);
\draw[->] (u3) -- (u4);
\draw[->] (i4) -- (t4);
\draw[->] (u4) -- (u5);
\draw[->] (i5) -- (t5);
\draw[->] (u5) -- (u6);
\end{tikzpicture}
sage: B.latex_state_evolution(1)
\begin{tikzpicture}[scale=1]
...
\end{tikzpicture}
"""
from sage.graphs.graph_latex import setup_latex_preamble
from sage.misc.latex import latex, LatexExpr
setup_latex_preamble()
u = self.state_evolution(num) # Also evolves as necessary
final = self._states[num + 1]
vacuum = self._vacuum_elt
initial = [vacuum] * (len(final) - len(self._states[num])) + list(
self._states[num])
def simple_repr(x):
return ''.join(repr(x).strip('[]').split(', '))
ret = '\\begin{{tikzpicture}}[scale={}]\n'.format(scale)
for i, val in enumerate(initial):
ret += '\\node (i{}) at ({},0.9) {{${}$}};\n'.format(
i, -2 * i, simple_repr(val))
for i, val in enumerate(final):
ret += '\\node (t{}) at ({},-1) {{${}$}};\n'.format(
i, -2 * i, simple_repr(val))
for i, val in enumerate(u):
ret += '\\node (u{}) at ({},0) {{${}$}};\n'.format(
i, -2 * i + 1, simple_repr(val))
for i in range(len(initial)):
ret += '\\draw[->] (i{}) -- (t{});\n'.format(i, i)
ret += '\\draw[->] (u{}) -- (u{});\n'.format(i, i + 1)
ret += '\\end{tikzpicture}'
return LatexExpr(ret)