Package sage :: Package combinat :: Package crystals :: Module crystals :: Class CrystalElement

Class CrystalElement

                      object --+        
                               |        
structure.sage_object.SageObject --+    
                                   |    
           structure.element.Element --+
                                       |
                                      CrystalElement

Known Subclasses:: spins.Spin_crystal_type_B_element, spins.Spin_crystal_type_D_element, letters.Crystal_of_letters_type_A_element, letters.Crystal_of_letters_type_B_element, letters.Crystal_of_letters_type_C_element, letters.Crystal_of_letters_type_D_element, letters.Crystal_of_letters_type_G_element, affine.AffineCrystalFromClassicalElement, affine.AffinizationFromAffineCrystalElement, tensor_product.TensorProductOfCrystalsElement, fast_crystals.FastCrystalElement


The abstract class of crystal elements

Sub classes should implement:
\begin{itemize}
\item x.e(i)        (returning $e_i(x)$)
\item x.f(i)        (returning $f_i(x)$)
\item x.weight()
\end{itemize}

Instance Methods

[hide private]

index_set(self)
EXAMPLES:...

source code

weight(self)
EXAMPLES:...

source code

e(self, i)
Returns $e_i(x)$ if it exists or None otherwise.

source code

f(self, i)
Returns $f_i(x)$ if it exists or None otherwise.

source code

epsilon(self, i)
EXAMPLES:...

source code

phi(self, i)
EXAMPLES:...

source code

Epsilon(self)
EXAMPLES:...

source code

Phi(self)
EXAMPLES:...

source code

s(self, i)
Returns the reflection of self along its $i$-string...

source code

is_highest_weight(self)
Returns True if self is a highest weight.

source code

Inherited from structure.element.Element: __cmp__, __eq__, __ge__, __gt__, __hash__, __init__, __le__, __lt__, __ne__, __new__, __nonzero__, __reduce__, __rxor__, __xor__, _cmp_, _im_gens_, _repr_, _richcmp_, base_base_extend, base_base_extend_canonical_sym, base_extend, base_extend_canonical, base_extend_canonical_sym, base_extend_recursive, base_ring, category, is_zero, n, parent, subs, substitute

Inherited from structure.element.Element (private): _coeff_repr, _is_atomic, _latex_coeff_repr, _make_new_with_parent_c, _set_parent

Inherited from structure.sage_object.SageObject: __repr__, _axiom_, _axiom_init_, _gap_, _gap_init_, _gp_, _gp_init_, _interface_, _interface_init_, _interface_is_cached_, _kash_, _kash_init_, _macaulay2_, _macaulay2_init_, _magma_, _magma_init_, _maple_, _maple_init_, _mathematica_, _mathematica_init_, _maxima_, _maxima_init_, _octave_, _octave_init_, _pari_, _pari_init_, _r_init_, _sage_, _singular_, _singular_init_, db, dump, dumps, plot, rename, reset_name, save, version

Inherited from object: __delattr__, __getattribute__, __reduce_ex__, __setattr__, __str__

Properties

[hide private]

Inherited from object: __class__

Method Details

[hide private]

index_set(self)

source code


EXAMPLES:
    sage: C = CrystalOfLetters(['A',5])
    sage: C(1).index_set()
    [1, 2, 3, 4, 5]

weight(self)

source code


EXAMPLES:
    sage: C = CrystalOfLetters(['A',5])
    sage: C(1).weight()
    (1, 0, 0, 0, 0, 0)

e(self, i)

source code


Returns $e_i(x)$ if it exists or None otherwise.  This is to be
implemented by subclasses of CrystalElement.

TESTS:
    sage: from sage.combinat.crystals.crystals import CrystalElement
    sage: C = CrystalOfLetters(['A',5])
    sage: CrystalElement.e(C(1), 1)
    Traceback (most recent call last):
    ...
    NotImplementedError

f(self, i)

source code


Returns $f_i(x)$ if it exists or None otherwise.  This is to be
implemented by subclasses of CrystalElement.

TESTS:
    sage: from sage.combinat.crystals.crystals import CrystalElement
    sage: C = CrystalOfLetters(['A',5])
    sage: CrystalElement.f(C(1), 1)
    Traceback (most recent call last):
    ...
    NotImplementedError

epsilon(self, i)

source code


EXAMPLES:
    sage: C = CrystalOfLetters(['A',5])
    sage: C(1).epsilon(1)
    0
    sage: C(2).epsilon(1)
    1

phi(self, i)

source code


EXAMPLES:
    sage: C = CrystalOfLetters(['A',5])
    sage: C(1).phi(1)
    1
    sage: C(2).phi(1)
    0

Epsilon(self)

source code


EXAMPLES:
    sage: C = CrystalOfLetters(['A',5])
    sage: C(0).Epsilon()
    (0, 0, 0, 0, 0, 0)
    sage: C(1).Epsilon()
    (0, 0, 0, 0, 0, 0)
    sage: C(2).Epsilon()
    (1, 0, 0, 0, 0, 0)

Phi(self)

source code


EXAMPLES:
    sage: C = CrystalOfLetters(['A',5])
    sage: C(0).Phi()
    (0, 0, 0, 0, 0, 0)
    sage: C(1).Phi()
    (1, 0, 0, 0, 0, 0)
    sage: C(2).Phi()
    (1, 1, 0, 0, 0, 0)

s(self, i)

source code


Returns the reflection of self along its $i$-string

EXAMPLES:
    sage: C = CrystalOfTableaux(['A',2], shape=[2,1])
    sage: b=C(rows=[[1,1],[3]])
    sage: b.s(1)
    [[2, 2], [3]]
    sage: b=C(rows=[[1,2],[3]])
    sage: b.s(2)
    [[1, 2], [3]]
    sage: T=CrystalOfTableaux(['A',2],shape=[4])
    sage: t=T(rows=[[1,2,2,2]])
    sage: t.s(1)
    [[1, 1, 1, 2]]

is_highest_weight(self)

source code


Returns True if self is a highest weight.

EXAMPLES:
    sage: C = CrystalOfLetters(['A',5])
    sage: C(1).is_highest_weight()
    True
    sage: C(2).is_highest_weight()
    False