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r""" 

Affinization Crystals 

""" 

 

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

# Copyright (C) 2015 Travis Scrimshaw <tscrim at ucdavis.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

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

 

from sage.structure.element import parent 

from sage.structure.parent import Parent 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.structure.element import Element 

from sage.structure.richcmp import richcmp 

from sage.categories.regular_crystals import RegularCrystals 

from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets 

from sage.rings.infinity import Infinity 

 

class AffinizationOfCrystal(UniqueRepresentation, Parent): 

r""" 

An affinization of a crystal. 

 

Let `\mathfrak{g}` be a Kac-Moody algebra of affine type. The 

affinization of a finite `U_q^{\prime}(\mathfrak{g})`-crystal `B` 

is the (infinite) `U_q(\mathfrak{g})`-crystal with underlying set: 

 

.. MATH:: 

 

B^{\mathrm{aff}} = \{ b(m) \mid b \in B, m \in \ZZ \} 

 

and crystal structure determined by: 

 

.. MATH:: 

 

\begin{aligned} 

e_i(b(m)) & = 

\begin{cases} 

(e_0 b)(m+1) & i = 0, \\ 

(e_i b)(m) & i \neq 0, 

\end{cases} \\ 

f_i(b(m)) &= 

\begin{cases} 

(f_0 b)(m-1) & i = 0, \\ 

(f_i b)(m) & i \neq 0, 

\end{cases} \\ 

\mathrm{wt}(b(m)) &= \mathrm{wt}(b) + m \delta. 

\end{aligned} 

 

EXAMPLES: 

 

We first construct a Kirillov-Reshetikhin crystal and then take it's 

corresponding affinization:: 

 

sage: K = crystals.KirillovReshetikhin(['A',2,1], 2, 2) 

sage: A = K.affinization() 

 

Next we construct an affinization crystal from a tensor product of KR 

crystals:: 

 

sage: KT = crystals.TensorProductOfKirillovReshetikhinTableaux(['C',2,1], [[1,2],[2,1]]) 

sage: A = crystals.AffinizationOf(KT) 

 

REFERENCES: 

 

- [HK02]_ Chapter 10 

""" 

def __init__(self, B): 

""" 

Initialize ``self``. 

 

EXAMPLES: 

 

We skip the Stembridge axioms test since this is an abstract crystal:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: TestSuite(A).run(skip="_test_stembridge_local_axioms") # long time 

""" 

if not B.cartan_type().is_affine(): 

raise ValueError("must be an affine crystal") 

if B.cardinality() == Infinity: 

raise ValueError("must be finite crystal") 

self._B = B 

self._cartan_type = B.cartan_type() 

Parent.__init__(self, category=(RegularCrystals(), InfiniteEnumeratedSets())) 

self.module_generators = tuple([self.element_class(self, b, 0) 

for b in B.module_generators]) 

 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

 

EXAMPLES:: 

 

sage: crystals.KirillovReshetikhin(['A',2,1], 1, 1).affinization() 

Affinization of Kirillov-Reshetikhin crystal of type ['A', 2, 1] with (r,s)=(1,1) 

""" 

return "Affinization of {}".format(self._B) 

 

class Element(Element): 

""" 

An element in an affinization crystal. 

""" 

def __init__(self, parent, b, m): 

""" 

Initialize ``self``. 

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: mg = A.module_generators[0] 

sage: TestSuite(mg).run() 

""" 

self._b = b 

self._m = m 

Element.__init__(self, parent) 

 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: A.module_generators[0] 

[[1, 1], [2, 2]](0) 

sage: KT = crystals.TensorProductOfKirillovReshetikhinTableaux(['C',2,1], [[1,2],[2,1]]) 

sage: A = crystals.AffinizationOf(KT) 

sage: A.module_generators[0] 

[[1, 1]] (X) [[1], [2]](0) 

""" 

return "{!r}({})".format(self._b, self._m) 

 

def _latex_(self): 

r""" 

Return a LaTeX representation of ``self``. 

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: latex(A.module_generators[0]) 

{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}} 

\raisebox{-.6ex}{$\begin{array}[b]{*{2}c}\cline{1-2} 

\lr{1}&\lr{1}\\\cline{1-2} 

\lr{2}&\lr{2}\\\cline{1-2} 

\end{array}$} 

} (0) 

""" 

from sage.misc.latex import latex 

return latex(self._b) + "({})".format(self._m) 

 

def __hash__(self): 

r""" 

TESTS:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: mg = A.module_generators[0] 

sage: hash(mg) == hash(mg._b) ^^ hash(mg._m) 

True 

""" 

return hash(self._b) ^ hash(self._m) 

 

def _richcmp_(self, other, op): 

""" 

Comparison. 

 

TESTS:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: mg = A.module_generators[0] 

sage: mg == mg 

True 

sage: mg == mg.f(2).e(2) 

True 

sage: KT = crystals.TensorProductOfKirillovReshetikhinTableaux(['C',2,1], [[1,2],[2,1]]) 

sage: A = crystals.AffinizationOf(KT) 

sage: A(KT.module_generators[3], 1).f(0) == A.module_generators[0] 

True 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: mg = A.module_generators[0] 

sage: mg != mg.f(2) 

True 

sage: mg != mg.f(2).e(2) 

False 

 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2, 2).affinization() 

sage: S = A.subcrystal(max_depth=2) 

sage: sorted(S) 

[[[1, 1], [2, 2]](0), 

[[1, 1], [2, 3]](0), 

[[1, 2], [2, 3]](0), 

[[1, 1], [3, 3]](0), 

[[1, 1], [2, 3]](1), 

[[1, 2], [2, 3]](1), 

[[1, 2], [3, 3]](1), 

[[2, 2], [3, 3]](2)] 

""" 

return richcmp((self._m, self._b), (other._m, other._b), op) 

 

def e(self, i): 

""" 

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

 

INPUT: 

 

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

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization() 

sage: mg = A.module_generators[0] 

sage: mg.e(0) 

[[1, 2], [2, 3]](1) 

sage: mg.e(1) 

sage: mg.e(0).e(1) 

[[1, 1], [2, 3]](1) 

""" 

bp = self._b.e(i) 

if bp is None: 

return None 

if i == 0: 

return self.__class__(self.parent(), bp, self._m+1) 

return self.__class__(self.parent(), bp, self._m) 

 

def f(self, i): 

""" 

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

 

INPUT: 

 

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

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization() 

sage: mg = A.module_generators[0] 

sage: mg.f(2) 

[[1, 1], [2, 3]](0) 

sage: mg.f(2).f(2).f(0) 

sage: mg.f_string([2,1,1]) 

sage: mg.f_string([2,1]) 

[[1, 2], [2, 3]](0) 

sage: mg.f_string([2,1,0]) 

[[1, 1], [2, 2]](-1) 

""" 

bp = self._b.f(i) 

if bp is None: 

return None 

if i == 0: 

return self.__class__(self.parent(), bp, self._m-1) 

return self.__class__(self.parent(), bp, self._m) 

 

def epsilon(self, i): 

r""" 

Return `\varepsilon_i` of ``self``. 

 

INPUT: 

 

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

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization() 

sage: mg = A.module_generators[0] 

sage: mg.epsilon(0) 

2 

sage: mg.epsilon(1) 

0 

""" 

return self._b.epsilon(i) 

 

def phi(self, i): 

r""" 

Return `\varphi_i` of ``self``. 

 

INPUT: 

 

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

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization() 

sage: mg = A.module_generators[0] 

sage: mg.phi(0) 

0 

sage: mg.phi(2) 

2 

""" 

return self._b.phi(i) 

 

def weight(self): 

r""" 

Return the weight of ``self``. 

 

The weight `\mathrm{wt}` of an element is: 

 

.. MATH:: 

 

\mathrm{wt}\bigl( b(m) \bigr) = \mathrm{wt}(b) + m \delta, 

 

where `\delta` is the null root. 

 

EXAMPLES:: 

 

sage: A = crystals.KirillovReshetikhin(['A',2,1], 2,2).affinization() 

sage: mg = A.module_generators[0] 

sage: mg.weight() 

-2*Lambda[0] + 2*Lambda[2] 

sage: mg.e(0).weight() 

-Lambda[1] + Lambda[2] + delta 

sage: mg.e(0).e(0).weight() 

2*Lambda[0] - 2*Lambda[1] + 2*delta 

""" 

WLR = self.parent().weight_lattice_realization() 

La = WLR.fundamental_weights() 

return WLR.sum(c*La[i] for i,c in self._b.weight()) + self._m * WLR.null_root()