These are the crystals associated with the three spin
representations: the spin representations of odd orthogonal groups
(or rather their double covers); and the and
spin
representations of the even orthogonal groups.
We follow Kashiwara and Nakashima (Journal of Algebra 165, 1994) in
representing the elements of the spin crystal by sequences of signs
.
Return the spin crystal of the given type .
This is a combinatorial model for the crystal with highest weight
(the
-th fundamental weight). It has
elements, here called Spins. See also
CrystalOfLetters(),
CrystalOfSpinsPlus(),
and CrystalOfSpinsMinus().
INPUT:
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: C.list()
[+++, ++-, +-+, -++, +--, -+-, --+, ---]
sage: C.cartan_type()
['B', 3]
sage: [x.signature() for x in C]
['+++', '++-', '+-+', '-++', '+--', '-+-', '--+', '---']
TESTS:
sage: crystals.TensorProduct(C,C,generators=[[C.list()[0],C.list()[0]]]).cardinality()
35
Return the minus spin crystal of the given type D.
This is the crystal with highest weight
(the
-st fundamental weight).
INPUT:
EXAMPLES:
sage: E = crystals.SpinsMinus(['D',4])
sage: E.list()
[+++-, ++-+, +-++, -+++, +---, -+--, --+-, ---+]
sage: [x.signature() for x in E]
['+++-', '++-+', '+-++', '-+++', '+---', '-+--', '--+-', '---+']
TESTS:
sage: len(crystals.TensorProduct(E,E,generators=[[E[0],E[0]]]).list())
35
sage: D = crystals.SpinsPlus(['D',4])
sage: len(crystals.TensorProduct(D,E,generators=[[D.list()[0],E.list()[0]]]).list())
56
Return the plus spin crystal of the given type D.
This is the crystal with highest weight (the
-th fundamental weight).
INPUT:
EXAMPLES:
sage: D = crystals.SpinsPlus(['D',4])
sage: D.list()
[++++, ++--, +-+-, -++-, +--+, -+-+, --++, ----]
sage: [x.signature() for x in D]
['++++', '++--', '+-+-', '-++-', '+--+', '-+-+', '--++', '----']
TESTS:
sage: TestSuite(D).run()
Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent
A generic crystal of spins.
Return the directed graph associated to self.
EXAMPLES:
sage: crystals.Spins(['B',3]).digraph()
Digraph on 8 vertices
Return a list of the elements of self.
EXAMPLES:
sage: crystals.Spins(['B',3]).list()
[+++, ++-, +-+, -++, +--, -+-, --+, ---]
Return True if and only if there is a path from x to y in the crystal graph.
Because the crystal graph is classical, it is a directed acyclic graph which can be interpreted as a poset. This function implements the comparison function of this poset.
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: x = C([1,1,1])
sage: y = C([-1,-1,-1])
sage: C.lt_elements(x,y)
True
sage: C.lt_elements(y,x)
False
sage: C.lt_elements(x,x)
False
Bases: sage.combinat.crystals.letters.LetterTuple
A spin letter in the crystal of spins.
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: c = C([1,1,1])
sage: TestSuite(c).run()
sage: C([1,1,1]).parent()
The crystal of spins for type ['B', 3]
sage: c = C([1,1,1])
sage: c._repr_()
'+++'
sage: D = crystals.Spins(['B',4])
sage: a = C([1,1,1])
sage: b = C([-1,-1,-1])
sage: c = D([1,1,1,1])
sage: a == a
True
sage: a == b
False
sage: b == c
False
Return of self.
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: [[C[m].epsilon(i) for i in range(1,4)] for m in range(8)]
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [1, 0, 0],
[0, 0, 1], [1, 0, 1], [0, 1, 0], [0, 0, 1]]
Return of self.
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: [[C[m].phi(i) for i in range(1,4)] for m in range(8)]
[[0, 0, 1], [0, 1, 0], [1, 0, 1], [0, 0, 1],
[1, 0, 0], [0, 1, 0], [0, 0, 1], [0, 0, 0]]
Return the signature of self.
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: C([1,1,1]).signature()
'+++'
sage: C([1,1,-1]).signature()
'++-'
Bases: sage.combinat.crystals.spins.Spin
Type B spin representation crystal element
Returns the action of on self.
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: [[C[m].e(i) for i in range(1,4)] for m in range(8)]
[[None, None, None], [None, None, +++], [None, ++-, None], [+-+, None, None],
[None, None, +-+], [+--, None, -++], [None, -+-, None], [None, None, --+]]
Returns the action of on self.
EXAMPLES:
sage: C = crystals.Spins(['B',3])
sage: [[C[m].f(i) for i in range(1,4)] for m in range(8)]
[[None, None, ++-], [None, +-+, None], [-++, None, +--], [None, None, -+-],
[-+-, None, None], [None, --+, None], [None, None, ---], [None, None, None]]
Bases: sage.combinat.crystals.spins.Spin
Type D spin representation crystal element
Returns the action of on self.
EXAMPLES:
sage: D = crystals.SpinsPlus(['D',4])
sage: [[D.list()[m].e(i) for i in range(1,4)] for m in range(8)]
[[None, None, None], [None, None, None], [None, ++--, None], [+-+-, None, None],
[None, None, +-+-], [+--+, None, -++-], [None, -+-+, None], [None, None, None]]
sage: E = crystals.SpinsMinus(['D',4])
sage: [[E[m].e(i) for i in range(1,4)] for m in range(8)]
[[None, None, None], [None, None, +++-], [None, ++-+, None], [+-++, None, None],
[None, None, None], [+---, None, None], [None, -+--, None], [None, None, --+-]]
Returns the action of on self.
EXAMPLES:
sage: D = crystals.SpinsPlus(['D',4])
sage: [[D.list()[m].f(i) for i in range(1,4)] for m in range(8)]
[[None, None, None], [None, +-+-, None], [-++-, None, +--+], [None, None, -+-+],
[-+-+, None, None], [None, --++, None], [None, None, None], [None, None, None]]
sage: E = crystals.SpinsMinus(['D',4])
sage: [[E[m].f(i) for i in range(1,4)] for m in range(8)]
[[None, None, ++-+], [None, +-++, None], [-+++, None, None], [None, None, None],
[-+--, None, None], [None, --+-, None], [None, None, ---+], [None, None, None]]