Ticket #3660: trac_3660-2.patch

a/sage/combinat/crystals/crystals.py

@@ -4 +4 @@
-Let $T$ be a CartanType with index set $I$, and $W$ be a realization of the
-type $T$ weight lattice.
-
-n
-
+ colored
+from 
-such that:
-\begin{itemize}
-each edge has a label in $I$
-for each $i$ in $
+Each edge is colored with a label in $i \in I$.
+For each $i\in 
-  \begin{itemize}
-
-
+;
+.
-  \end{itemize}
- exists
+y exist
-   \begin{itemize}
-
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+;
+.
-   \end{itemize}
-\end{itemize}
-
-This crystal actually models a representation of a Lie algebra if it
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-
-EXAMPLES:
-
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+
-
-    sage: C = CrystalOfLetters(['A',5]); C
-    The crystal of letters for type ['A', 5]
-
-It has a single highest weight element:
-module_generators
+highest_weight_vectors()
-    [1]
-
-A crystal is a CombinatorialClass; and we can count and list its elements
@@ -57 +61 @@
-    sage: Tab.plot()
-
-
-
-For rank two crystals, there is an alternative method of getting
-metapost pictures. For more information see C.metapost?
-
@@ -80 +83 @@
-  \item RestrictionOfCrystal / DirectSumOfCrystals
-  \item Crystal.crystal_morphism
-  \item Affine crystals
+  \item Kirillov--Reshetikhin crystals
-\end{itemize}
-
-Most of the above features (except Littelmann/alcove paths) are in
@@ -128 +132 @@
-    \item index_set
-        the index set of the cartan type
-    \item module_generators
-
+ using $f_i$
-    \item weight_lattice_realization
-    \end{itemize}
-    """
@@ -514 +518 @@
-        Returns a dot_tex version of self.
-
-        EXAMPLES:
-
-
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-
-        """
-        import re

a/sage/combinat/crystals/letters.py

@@ -30 +30 @@
-    For classical types, this is a combinatorial model for the crystal
-    with highest weight Lambda_1 (the first fundamental weight).
-
-
+the 
-    of the tensor product of several copies of this crystal (plus
-    possibly one copy of the spin crystal, see CrystalOfSpins).
+    See M. Kashiwara, T. Nakashima, \textit{Crystal graphs for representations of 
+    the $q$-analogue of classical Lie algebras}, J. Algebra \textbf{165} (1994), no. 2, 295--345.
-    Elements of this irreducible component have a fixed shape, and can
-    be fit inside a tableau shape. Otherwise said, any irreducible
-    classical crystal is isomorphic to a crystal of tableaux with
@@ -74 +76 @@
-
-    All classical crystals of letters should be instances of this class
-    or of subclasses. To define a new crystal of letters, one only
-
+s
-    Letter and CrystalElement), with appropriate e and f
- to
+s to define the
-    subclass ClassicalCrystalOfLetters for the crystal itself.
-
-    The basic assumption is that crystals of letters are small, but

a/sage/combinat/crystals/tensor_product.py

@@ -1 +1 @@
-r"""
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-"""
-
-#*****************************************************************************
@@ -163 +211 @@
-        return self.sibling(l)
-
-def TensorProductOfCrystals(*crystals, **options):
-    r"""
-    Tensor product of crystals.
-
-    EXAMPLES:
-
-        We construct the type $A_2$-crystal generated by $2\otimes
-        1\otimes 1$:
-    
-        sage: C = CrystalOfLetters(['A',2])
-        sage: T = TensorProductOfCrystals(C,C,C,generators=[[C(2),C(1),C(1)]])
-
-        It has $8$ elements
-
-        sage: T.list()
-        [[2, 1, 1], [2, 1, 2], [2, 1, 3], [3, 1, 3], [3, 2, 3], [3, 1, 1], [3, 1, 2], [3, 2, 2]]
-
-        sage: C = CrystalOfTableaux(['A',3], shape=[1,1,0])
-        sage: D = CrystalOfTableaux(['A',3], shape=[1,0,0])
-        sage: T = TensorProductOfCrystals(C,D, generators=[[C(rows=[[1], [2]]), D(rows=[[1]])], [C(rows=[[2], [3]]), D(rows=[[1]])]])
-        sage: T.count()
-        24
-        sage: T.check()
-        True
-        sage: T.module_generators
-        [[[[1], [2]], [[1]]], [[[2], [3]], [[1]]]]
-        sage: [x.weight() for x in T.module_generators]
-        [(2, 1, 0, 0), (1, 1, 1, 0)]
-
-        If no module generators are specified, we obtain the full tensor product:
-
-        sage: C=CrystalOfLetters(['A',2])
-        sage: T=TensorProductOfCrystals(C,C)
-        sage: T.list()
-        [[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
-        sage: T.count()
-        9
-
-        For a tensor product of crystals without module generators, the default implementation of module_generators
-        contains all elements in the tensor product of the crystals. If there is a subset of elements
-        in the tensor product that still generates the crystal, this needs to be implemented for the specific 
-        crystal separately:
-        
-        sage: T.module_generators.list()
-        [[1, 1], [1, 2], [1, 3], [2, 1], [2, 2], [2, 3], [3, 1], [3, 2], [3, 3]]
-
-        For classical highest weight crystals, it is also possible to list all highest weight elements:
-        
-        sage: C = CrystalOfLetters(['A',2])
-        sage: T = TensorProductOfCrystals(C,C,C,generators=[[C(2),C(1),C(1)],[C(1),C(2),C(1)]])
-        sage: T.highest_weight_vectors()
-        [[2, 1, 1], [1, 2, 1]]
-    """
-    if options.has_key('generators'):
-        if all(isinstance(crystal,ClassicalCrystal) for crystal in crystals):
-            return TensorProductOfClassicalCrystalsWithGenerators(generators=options['generators'], *crystals)
@@ -523 +519 @@
-
-#        sage: Tab.latex() # random
-
-
+.crystals
-
-        Internally, a tableau of a given Cartan type is represented as a tensor 
-        product of letters of the same type. The order in which the tensor factors
-. 
-
+
+(in French notation). 
-        sage: T = CrystalOfTableaux(['A',2], shape = [3,2])
-        sage: T.module_generators[0]
-        [[1, 1, 1], [2, 2]]