Spin Crystals

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 \pm.

sage.combinat.crystals.spins.CrystalOfSpins(ct)

Return the spin crystal of the given type B.

This is a combinatorial model for the crystal with highest weight Lambda_n (the n-th fundamental weight). It has 2^n elements, here called Spins. See also CrystalOfLetters(), CrystalOfSpinsPlus(), and CrystalOfSpinsMinus().

INPUT:

  • ['B', n] - A Cartan type B_n.

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
sage.combinat.crystals.spins.CrystalOfSpinsMinus(ct)

Return the minus spin crystal of the given type D.

This is the crystal with highest weight Lambda_{n-1} (the (n-1)-st fundamental weight).

INPUT:

  • ['D', n] - A Cartan type D_n.

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
sage.combinat.crystals.spins.CrystalOfSpinsPlus(ct)

Return the plus spin crystal of the given type D.

This is the crystal with highest weight Lambda_n (the n-th fundamental weight).

INPUT:

  • ['D', n] - A Cartan type D_n.

EXAMPLES:

sage: D = crystals.SpinsPlus(['D',4])
sage: D.list()
[++++, ++--, +-+-, -++-, +--+, -+-+, --++, ----]
sage: [x.signature() for x in D]
['++++', '++--', '+-+-', '-++-', '+--+', '-+-+', '--++', '----']

TESTS:

sage: TestSuite(D).run()
class sage.combinat.crystals.spins.GenericCrystalOfSpins(ct, element_class, case)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

A generic crystal of spins.

digraph()

Return the directed graph associated to self.

EXAMPLES:

sage: crystals.Spins(['B',3]).digraph()
Digraph on 8 vertices
list()

Return a list of the elements of self.

EXAMPLES:

sage: crystals.Spins(['B',3]).list()
[+++, ++-, +-+, -++, +--, -+-, --+, ---]
lt_elements(x, y)

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
class sage.combinat.crystals.spins.Spin

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
epsilon(i)

Return \varepsilon_i 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]]
phi(i)

Return \varphi_i 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]]
signature()

Return the signature of self.

EXAMPLES:

sage: C = crystals.Spins(['B',3])
sage: C([1,1,1]).signature()
'+++'
sage: C([1,1,-1]).signature()
'++-'
class sage.combinat.crystals.spins.Spin_crystal_type_B_element

Bases: sage.combinat.crystals.spins.Spin

Type B spin representation crystal element

e(i)

Returns the action of e_i 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, --+]]
f(i)

Returns the action of f_i 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]]
class sage.combinat.crystals.spins.Spin_crystal_type_D_element

Bases: sage.combinat.crystals.spins.Spin

Type D spin representation crystal element

e(i)

Returns the action of e_i 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, --+-]]
f(i)

Returns the action of f_i 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]]

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