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r""" 

Star-Crystal Structure On `B(\infty)` 

 

AUTHORS: 

 

- Ben Salisbury: Initial version 

 

- Travis Scrimshaw: Initial version 

""" 

 

#***************************************************************************** 

# Copyright (C) 2016 Ben Salisbury <ben DOT salisbury AT cmich DOT edu> 

# Travis Scrimshaw <tscrimsh AT umn DOT 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.structure.element_wrapper import ElementWrapper 

from sage.categories.highest_weight_crystals import HighestWeightCrystals 

from sage.combinat.root_system.cartan_type import CartanType 

from sage.combinat.crystals.elementary_crystals import ElementaryCrystal 

from sage.combinat.crystals.tensor_product import TensorProductOfCrystals 

 

 

class StarCrystal(UniqueRepresentation, Parent): 

r""" 

The star-crystal or `*`-crystal version of a highest weight crystal. 

 

The `*`-crystal structure on `B(\infty)` is the structure induced by 

the algebra antiautomorphism `* \colon U_q(\mathfrak{g}) \longrightarrow 

U_q(\mathfrak{g})` that stabilizes the negative half `U_q^-(\mathfrak{g})`. 

It is defined by 

 

.. MATH:: 

 

E_i^* = E_i , \ \ \ 

F_i^* = F_i , \ \ \ 

q^* = q, \ \ \ 

(q^h)^* = q^{-h}, 

 

where `E_i` and `F_i` are the Chevalley generators of `U_q(\mathfrak{g})` 

and `h` is an element of the Cartan subalgebra. 

 

The induced operation on the crystal `B(\infty)` is called the 

*Kashiwara involution*. Its implementation here is based on the 

recursive algorithm from Theorem 2.2.1 of [Kash95]_, which states 

that for any `i \in I` there is a unique strict crystal embedding 

 

.. MATH:: 

 

\Psi_i\colon B(\infty) \longrightarrow B_i \otimes B(\infty) 

 

such that 

 

- `u_{\infty} \mapsto b_i(0) \otimes u_{\infty}`, where `u_{\infty}` 

is the highest weight vector in `B(\infty)`; 

 

- if `\Psi_i(b) = f_i^mb_i(0) \otimes b_0`, then 

`\Psi_i(f_i^*b) =f_i^{m+1}b_i(0) \otimes b_0` 

and `\varepsilon_i(b^*) = m`; 

 

- the image of `\Psi_i` is `\{f_i^mb_i(0)\otimes b : 

\varepsilon_i(b^*) = 0, \ m\ge 0\}`. 

 

Here, `B_i` is the `i`-th elementary crystal. See 

:class:`~sage.combinat.crystals.elementary_crystals.ElementaryCrystal` 

for more information. 

 

INPUT: 

 

- ``Binf`` -- a crystal from 

:class:`~sage.combinat.crystals.catalog_infinity_crystals` 

 

EXAMPLES:: 

 

sage: B = crystals.infinity.Tableaux(['A',2]) 

sage: Bstar = crystals.infinity.Star(B) 

sage: mg = Bstar.highest_weight_vector() 

sage: mg 

[[1, 1], [2]] 

sage: mg.f_string([1,2,1,2,2]) 

[[1, 1, 1, 1, 1, 2, 2], [2, 3, 3, 3]] 

 

REFERENCES: 

 

.. [Kash95] \M. Kashiwara. 

The crystal base and Littelmann's refined Demazure character formula. 

Duke Math. J. **71** (1993), no. 3, 839-858. 

""" 

def __init__(self, Binf): 

r""" 

Initialize ``self``. 

 

EXAMPLES:: 

 

sage: B = crystals.infinity.Tableaux(['A',2]) 

sage: Bstar = crystals.infinity.Star(B) 

sage: TestSuite(Bstar).run(max_runs=40) 

sage: TestSuite(Bstar).run(max_runs=1000) # long time 

""" 

self._Binf = Binf 

self._cartan_type = Binf.cartan_type() 

Parent.__init__(self, category=HighestWeightCrystals().Infinite()) 

self.module_generators = (self(self._Binf.module_generators[0]),) 

t0 = Binf.highest_weight_vector() 

B = {i: ElementaryCrystal(Binf.cartan_type(),i) for i in self.index_set()} 

self._tens = {i: B[i].tensor(Binf) for i in self.index_set()} 

gens = {i: self._tens[i](B[i](0), t0) for i in self.index_set()} 

self._embedding = {i: Binf.crystal_morphism({t0: gens[i]}) for i in self.index_set()} 

self._pullback = {i: self._tens[i].crystal_morphism({gens[i]: t0}) for i in self.index_set()} 

 

def _repr_(self): 

r""" 

Return a string representation of ``self``. 

 

EXAMPLES:: 

 

sage: Y = crystals.infinity.GeneralizedYoungWalls(3) 

sage: Ystar = crystals.infinity.Star(Y) 

sage: Ystar 

Star-crystal version of Crystal of generalized Young walls of type ['A', 3, 1] 

""" 

return "Star-crystal version of %s" % self._Binf 

 

class Element(ElementWrapper): 

 

def e(self,i): 

r""" 

Return the action of `e_i^*` on ``self``. 

 

INPUT: 

 

- ``i`` -- an element of the index set 

 

EXAMPLES:: 

 

sage: RC = crystals.infinity.RiggedConfigurations(['E',6,1]) 

sage: RCstar = crystals.infinity.Star(RC) 

sage: nuJ = RCstar.module_generators[0].f_string([0,4,6,1,2]) 

sage: ascii_art(nuJ.e(1)) 

-1[ ]-1 (/) 0[ ]1 (/) -1[ ]-1 (/) -2[ ]-1 

 

sage: M = crystals.infinity.NakajimaMonomials(['B',2,1]) 

sage: Mstar = crystals.infinity.Star(M) 

sage: m = Mstar.module_generators[0].f_string([0,1,2,2,1,0]) 

sage: m.e(1) 

Y(0,0)^-1 Y(0,2)^-1 Y(1,1) Y(1,2)^-1 Y(2,1)^2 

""" 

P = self.parent() 

image = P._embedding[i](self.value) 

if image[0].e(i)._m > 0: 

return None 

return P(P._pullback[i]( P._tens[i](image[0].e(i),image[1]) )) 

 

def f(self,i): 

r""" 

Return the action of `f_i^*` on ``self``. 

 

INPUT: 

 

- ``i`` -- an element of the index set 

 

EXAMPLES:: 

 

sage: T = crystals.infinity.Tableaux("G2") 

sage: Tstar = crystals.infinity.Star(T) 

sage: t = Tstar.module_generators[0].f_string([1,2,1,1,2]) 

sage: t 

[[1, 1, 1, 2, 0], [2, 3]] 

 

sage: M = crystals.infinity.NakajimaMonomials(['B',2,1]) 

sage: Mstar = crystals.infinity.Star(M) 

sage: m = Mstar.module_generators[0].f_string([0,1,2,2,1,0]) 

sage: m 

Y(0,0)^-1 Y(0,2)^-1 Y(1,0)^-1 Y(1,2)^-1 Y(2,0)^2 Y(2,1)^2 

""" 

P = self.parent() 

image = P._embedding[i](self.value) 

return P(P._pullback[i]( P._tens[i](image[0].f(i),image[1]) )) 

 

def weight(self): 

r""" 

Return the weight of ``self``. 

 

EXAMPLES:: 

 

sage: RC = crystals.infinity.RiggedConfigurations(['E',6,1]) 

sage: RCstar = crystals.infinity.Star(RC) 

sage: nuJ = RCstar.module_generators[0].f_string([0,4,6,1,2]) 

sage: nuJ.weight() 

-Lambda[0] - 2*Lambda[1] + 2*Lambda[3] - Lambda[4] 

+ 2*Lambda[5] - 2*Lambda[6] - delta 

""" 

return self.value.weight() 

 

def epsilon(self, i): 

r""" 

Return `\varepsilon_i^*` of ``self``. 

 

INPUT: 

 

- ``i`` -- an element of the index set 

 

EXAMPLES:: 

 

sage: Y = crystals.infinity.GeneralizedYoungWalls(3) 

sage: Ystar = crystals.infinity.Star(Y) 

sage: y = Ystar.module_generators[0].f_string([0,1,3,2,1,0]) 

sage: [y.epsilon(i) for i in y.index_set()] 

[1, 0, 1, 0] 

 

sage: RC = crystals.infinity.RiggedConfigurations(['E',6,1]) 

sage: RCstar = crystals.infinity.Star(RC) 

sage: nuJ = RCstar.module_generators[0].f_string([0,4,6,1,2]) 

sage: [nuJ.epsilon(i) for i in nuJ.index_set()] 

[0, 1, 1, 0, 0, 0, 1] 

""" 

ep = -1 

while self is not None: 

ep += 1 

self = self.e(i) 

return ep 

 

def phi(self, i): 

r""" 

Return `\varphi_i^*` of ``self``. 

 

For `b \in B(\infty)`, 

 

.. MATH:: 

 

\varphi_i^*(b) = \varepsilon_i^*(b) + \langle h_i, 

\mathrm{wt}(b) \rangle, 

 

where `h_i` is a simple coroot. 

 

INPUT: 

 

- ``i`` -- an element of the index set 

 

EXAMPLES:: 

 

sage: T = crystals.infinity.Tableaux("A2") 

sage: Tstar = crystals.infinity.Star(T) 

sage: t = Tstar.module_generators[0].f_string([1,2,1,1,2]) 

sage: [t.phi(i) for i in t.index_set()] 

[-3, 1] 

 

sage: M = crystals.infinity.NakajimaMonomials(['B',2,1]) 

sage: Mstar = crystals.infinity.Star(M) 

sage: m = Mstar.module_generators[0].f_string([0,1,2,2,1,0]) 

sage: [m.phi(i) for i in m.index_set()] 

[-1, -1, 4] 

""" 

P = self.parent().weight_lattice_realization() 

ac = P.simple_coroot(i) 

return P(self.weight()).scalar(ac) + self.epsilon(i) 

 

def jump(self, i): 

r""" 

Return the `i`-jump of ``self``. 

 

For `b \in B(\infty)`, 

 

.. MATH:: 

 

\operatorname{jump}_i(b) = \varepsilon_i(b) + \varepsilon_i^*(b) 

+ \langle h_i, \mathrm{wt}(b) \rangle, 

 

where `h_i` is a simple coroot. 

 

INPUT: 

 

- ``i`` -- an element of the index set 

 

EXAMPLES:: 

 

sage: RC = crystals.infinity.RiggedConfigurations("D4") 

sage: RCstar = crystals.infinity.Star(RC) 

sage: nu0star = RCstar.module_generators[0] 

sage: nustar = nu0star.f_string([2,1,3,4,2]) 

sage: [nustar.jump(i) for i in RC.index_set()] 

[0, 1, 0, 0] 

sage: nustar = nu0star.f_string([2,1,3,4,2,2,1,3,2]) # long time 

sage: [nustar.jump(i) for i in RC.index_set()] # long time 

[1, 0, 1, 2] 

""" 

P = self.parent().weight_lattice_realization() 

ac = P.simple_coroot(i) 

return P(self.value.weight()).scalar(ac) + self.epsilon(i) + self.value.epsilon(i)