Crystal Of Mirković-Vilonen (MV) Polytopes

AUTHORS:

  • Dinakar Muthiah, Travis Scrimshaw (2015-05-11): initial version
class sage.combinat.crystals.mv_polytopes.MVPolytope(parent, lusztig_datum, long_word=None)

Bases: sage.combinat.crystals.pbw_crystal.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
plot(P=None, **options)

Plot 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', 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 \(C_2\):

../../../_images/mv_polytopes-1.svg
polytope(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))
class sage.combinat.crystals.mv_polytopes.MVPolytopes(cartan_type)

Bases: sage.combinat.crystals.pbw_crystal.PBWCrystal

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

\[P(M) = \{ \alpha \in P_{\RR} \mid \langle \alpha, \gamma \rangle \geq M_{\gamma} \text{ for all } \gamma \in \Gamma \}.\]

The vertices \(\{\mu_w\}_{w \in W}\) are given by

\[\langle \mu_w, \gamma \rangle = M_{\gamma}\]

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

\[\begin{split}\begin{aligned} n_k & = -M_{w_{k-1} \Lambda_{i_k}} - M_{w_k \Lambda_{i_k}} - \sum_{j \neq i} a_{ji} M_{w_k \Lambda_j}, \\ \mu_{w_k} - \mu_{w_{k-1}} & = n_k w_{k-1} \alpha_{i_k}, \end{aligned}\end{split}\]

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 PBW data. 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))
../../../_images/mv_polytopes-2.svg

REFERENCES:

Element

alias of MVPolytope

latex_options()

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}
set_latex_options(**kwds)

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) -- (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.set_latex_options(P=P, circle_size=float(0.2))
sage: latex(b)
\begin{tikzpicture}
\draw (0, 0) -- (2, -2) -- (2, -3) -- (2, -3) -- (0, -2);
\draw (0, 0) -- (-2, 1) -- (-2, 1) -- (-2, 0) -- (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, -2) -- (2, -3) -- (2, -3) -- (0, -2);
\draw (0, 0) -- (-2, 1) -- (-2, 1) -- (-2, 0) -- (0, -2);
\end{tikzpicture}
sage: MV.set_latex_options(P=MV.weight_lattice_realization(),
....:                      circle_size=0.2,
....:                      mark_endpoints=True)