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]
['+++', '++-', '+-+', '-++', '+--', '-+-', '--+', '---']
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]
['+++-', '++-+', '+-++', '-+++', '+---', '-+--', '--+-', '---+']
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]
['++++', '++--', '+-+-', '-++-', '+--+', '-+-+', '--++', '----']
sage.combinat.crystals.spins.GenericCrystalOfSpins

A generic crystal of spins.

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]]
pp()

Pretty print self as a column.

EXAMPLES:

sage: C = crystals.Spins(['B',3])
sage: b = C([1,1,-1])
sage: b.pp()
+
+
-
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]]