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spins.pyx
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spins.pyx
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# -*- coding: utf-8 -*-
r"""
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`.
"""
#TODO: Do we want the following two representations?
#
#Two other representations are available as attributes
#:meth:`Spin.internal_repn` and :meth:`Spin.signature` of the crystal element.
#
#- A numerical internal representation, an integer `n` such that if `n-1`
# is written in binary and the `1`'s are replaced by ``-``, the `0`'s by
# ``+``
#
#- The signature, which is a list in which ``+`` is replaced by `+1` and
# ``-`` by `-1`.
#*****************************************************************************
# Copyright (C) 2007 Anne Schilling <anne at math.ucdavis.edu>
# Nicolas Thiery <nthiery at users.sf.net>
# Daniel Bump <bump at match.stanford.edu>
# 2019 Travis Scrimshaw <tcscrims at gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# This code is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# The full text of the GPL is available at:
#
# http://www.gnu.org/licenses/
#****************************************************************************
from cpython.object cimport Py_EQ, Py_NE, Py_LE, Py_GE, Py_LT, Py_GT
from cysignals.memory cimport sig_malloc, sig_free
from sage.misc.cachefunc import cached_method
from sage.misc.lazy_attribute import lazy_attribute
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.parent cimport Parent
from sage.structure.element cimport Element, parent
from sage.categories.classical_crystals import ClassicalCrystals
from sage.combinat.root_system.cartan_type import CartanType
from sage.combinat.tableau import Tableau
from sage.rings.integer_ring import ZZ
from sage.typeset.ascii_art import AsciiArt
from sage.typeset.unicode_art import UnicodeArt
#########################
# Type B spin
#########################
def CrystalOfSpins(ct):
r"""
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
:func:`~sage.combinat.crystals.letters.CrystalOfLetters`,
:func:`~sage.combinat.crystals.spins.CrystalOfSpinsPlus`,
and :func:`~sage.combinat.crystals.spins.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
"""
ct = CartanType(ct)
if ct[0] == 'B':
return GenericCrystalOfSpins(ct, Spin_crystal_type_B_element, "spins")
else:
raise NotImplementedError
#########################
# Type D spins
#########################
def CrystalOfSpinsPlus(ct):
r"""
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()
"""
ct = CartanType(ct)
if ct[0] == 'D':
return GenericCrystalOfSpins(ct, Spin_crystal_type_D_element, "plus")
else:
raise NotImplementedError
def CrystalOfSpinsMinus(ct):
r"""
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
"""
ct = CartanType(ct)
if ct[0] == 'D':
return GenericCrystalOfSpins(ct, Spin_crystal_type_D_element, "minus")
else:
raise NotImplementedError
class GenericCrystalOfSpins(UniqueRepresentation, Parent):
"""
A generic crystal of spins.
"""
def __init__(self, ct, element_class, case):
"""
EXAMPLES::
sage: E = crystals.SpinsMinus(['D',4])
sage: TestSuite(E).run()
"""
self._cartan_type = CartanType(ct)
if case == "spins":
self.rename("The crystal of spins for type %s"%ct)
elif case == "plus":
self.rename("The plus crystal of spins for type %s"%ct)
else:
self.rename("The minus crystal of spins for type %s"%ct)
self.Element = element_class
Parent.__init__(self, category=ClassicalCrystals())
if case == "minus":
generator = [1]*(ct[1]-1)
generator.append(-1)
else:
generator = [1]*ct[1]
self.module_generators = (self.element_class(self, tuple(generator)),)
def _element_constructor_(self, value):
"""
Construct an element of ``self`` from ``value``.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: x = C((1,1,1)); x
+++
sage: y = C([1,1,1]); y
+++
sage: x == y
True
"""
return self.element_class(self, tuple(value))
@lazy_attribute
def _digraph_closure(self):
"""
The transitive closure of the digraph associated to ``self``.
EXAMPLES::
sage: crystals.Spins(['B',4])._digraph_closure
Transitive closure of : Digraph on 16 vertices
"""
return self.digraph().transitive_closure()
def lt_elements(self, x, y):
r"""
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
"""
if parent(x) is not self or parent(y) is not self:
raise ValueError("both elements must be in this crystal")
return self._digraph_closure.has_edge(x, y)
cdef class Spin(Element):
"""
A spin letter in the crystal of spins.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: c = C([1,1,1])
sage: c
+++
sage: c.parent()
The crystal of spins for type ['B', 3]
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
"""
# cdef bint* self._value # A + is a 0/False and a - is a 1/True
def __init__(self, parent, tuple val):
"""
Initialize ``self``.
TESTS::
sage: C = crystals.Spins(['B',3])
sage: c = C([1,1,1])
sage: TestSuite(c).run()
"""
cdef int i
self._n = parent.cartan_type().rank()
self._value = <bint*>sig_malloc(self._n*sizeof(bint))
for i in range(self._n):
self._value[i] = (val[i] != 1)
Element.__init__(self, parent)
cdef Spin _new_c(self, bint* value):
r"""
Fast creation of a spin element.
"""
cdef Spin ret = type(self).__new__(type(self))
ret._parent = self._parent
ret._n = self._n
ret._value = value
ret._hash = 0
return ret
def __dealloc__(self):
"""
Deallocate ``self``.
TESTS::
sage: C = crystals.Spins(['B',3])
sage: c = C([1,1,1])
sage: del c
"""
sig_free(self._value)
def __hash__(self):
"""
Return the hash of ``self``.
TESTS::
sage: C = crystals.Spins(['B',3])
sage: len(set(C)) == len(set([hash(x) for x in C]))
True
"""
cdef int i
if self._hash == 0:
self._hash = hash(tuple([-1 if self._value[i] else 1 for i in range(self._n)]))
return self._hash
def __reduce__(self):
r"""
Used to pickle ``self``.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: a = C([1,-1,1])
sage: a.__reduce__()
(The crystal of spins for type ['B', 3], ((1, -1, 1),))
"""
tup = tuple([-1 if self._value[i] else 1 for i in range(self._n)])
return (self._parent, (tup,))
cpdef _richcmp_(left, right, int op):
"""
Return ``True`` if ``left`` compares with ``right`` based on ``op``.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: x = C([1,1,1])
sage: y = C([-1,-1,-1])
sage: x < y
True
sage: x >= y
False
sage: x < x
False
sage: x <= x
True
sage: x != y
True
sage: x == y
False
"""
cdef Spin self, x
cdef int i
self = left
x = right
if op == Py_EQ:
for i in range(self._n):
if self._value[i] != x._value[i]:
return False
return True
if op == Py_NE:
for i in range(self._n):
if self._value[i] != x._value[i]:
return True
return False
if op == Py_LT:
return self._parent._digraph_closure.has_edge(self, x)
if op == Py_GT:
return x._parent._digraph_closure.has_edge(x, self)
if op == Py_LE:
return self == x or self._parent._digraph_closure.has_edge(self, x)
if op == Py_GE:
return self == x or x._parent._digraph_closure.has_edge(x, self)
return False
@property
def value(self):
r"""
Return ``self`` as a tuple with `+1` and `-1`.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: C([1,1,1]).value
(1, 1, 1)
sage: C([1,1,-1]).value
(1, 1, -1)
"""
cdef int i
one = ZZ.one()
return tuple([-one if self._value[i] else one for i in range(self._n)])
def signature(self):
"""
Return the signature of ``self``.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: C([1,1,1]).signature()
'+++'
sage: C([1,1,-1]).signature()
'++-'
"""
cdef int i
cdef str sword = ""
for i in range(self._n):
sword += "+" if self._value[i] != 1 else "-"
return sword
_repr_ = signature
def _repr_diagram(self):
"""
Return a representation of ``self`` as a diagram.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: b = C([1,1,-1])
sage: print(b._repr_diagram())
+
+
-
"""
return '\n'.join(self.signature())
def _ascii_art_(self):
"""
Return an ascii art representation of ``self``.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: b = C([1,1,-1])
sage: ascii_art(b)
+
+
-
"""
return AsciiArt(list(self.signature()))
def _unicode_art_(self):
"""
Return a unicode art representation of ``self``.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: b = C([1,1,-1])
sage: unicode_art(b)
+
+
-
"""
return UnicodeArt(list(self.signature()))
def pp(self):
"""
Pretty print ``self`` as a column.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: b = C([1,1,-1])
sage: b.pp()
+
+
-
"""
print(self._repr_diagram())
def _latex_(self):
r"""
Gives the latex output of a spin column.
EXAMPLES::
sage: C = crystals.Spins(['B',3])
sage: b = C([1,1,-1])
sage: print(b._latex_())
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}
\raisebox{-.6ex}{$\begin{array}[b]{*{1}c}\cline{1-1}
\lr{-}\\\cline{1-1}
\lr{+}\\\cline{1-1}
\lr{+}\\\cline{1-1}
\end{array}$}
}
"""
return Tableau([[i] for i in reversed(self.signature())])._latex_()
def weight(self):
"""
Return the weight of ``self``.
EXAMPLES::
sage: [v.weight() for v in crystals.Spins(['B',3])]
[(1/2, 1/2, 1/2), (1/2, 1/2, -1/2),
(1/2, -1/2, 1/2), (-1/2, 1/2, 1/2),
(1/2, -1/2, -1/2), (-1/2, 1/2, -1/2),
(-1/2, -1/2, 1/2), (-1/2, -1/2, -1/2)]
"""
WLR = self._parent.weight_lattice_realization()
cdef int i
mone = -WLR.base_ring().one()
# The ambient space is indexed by 0,...,n-1
return WLR._from_dict({i: mone**int(self._value[i]) / 2 for i in range(self._n)},
remove_zeros=False, coerce=False)
cdef class Spin_crystal_type_B_element(Spin):
r"""
Type B spin representation crystal element
"""
cpdef Spin e(self, int i):
r"""
Return 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, --+]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
cdef int j
cdef bint* ret
if i == self._n:
if self._value[i-1]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = False
return self._new_c(ret)
return None
if self._value[i-1] and not self._value[i]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = False
ret[i] = True
return self._new_c(ret)
return None
cpdef Spin f(self, int i):
r"""
Return 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]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
cdef int j
cdef bint* ret
if i == self._n:
if not self._value[i-1]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = True
return self._new_c(ret)
return None
if self._value[i] and not self._value[i-1]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = True
ret[i] = False
return self._new_c(ret)
return None
cpdef int epsilon(self, int i):
r"""
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]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
if i == self._n:
return self._value[i-1]
return self._value[i-1] and not self._value[i]
cpdef int phi(self, int i):
r"""
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]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
if i == self._n:
return not self._value[i-1]
return self._value[i] and not self._value[i-1]
cdef class Spin_crystal_type_D_element(Spin):
r"""
Type D spin representation crystal element
"""
cpdef Spin e(self, int i):
r"""
Return 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, --+-]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
cdef int j
cdef bint* ret
if i == self._n:
if self._value[i-1] and self._value[i-2]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = False
ret[i-2] = False
return self._new_c(ret)
return None
if self._value[i-1] and not self._value[i]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = False
ret[i] = True
return self._new_c(ret)
return None
cpdef Spin f(self, int i):
r"""
Return 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]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
cdef int j
cdef bint* ret
if i == self._n:
if not self._value[i-1] and not self._value[i-2]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = True
ret[i-2] = True
return self._new_c(ret)
return None
if self._value[i] and not self._value[i-1]:
ret = <bint*>sig_malloc(self._n*sizeof(bint))
for j in range(self._n):
ret[j] = self._value[j]
ret[i-1] = True
ret[i] = False
return self._new_c(ret)
return None
cpdef int epsilon(self, int i):
r"""
Return `\varepsilon_i` of ``self``.
EXAMPLES::
sage: C = crystals.SpinsMinus(['D',4])
sage: [[C[m].epsilon(i) for i in C.index_set()] for m in range(8)]
[[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0],
[0, 0, 0, 1], [1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
if i == self._n:
return self._value[i-1] and self._value[i-2]
return self._value[i-1] and not self._value[i]
cpdef int phi(self, int i):
r"""
Return `\varphi_i` of ``self``.
EXAMPLES::
sage: C = crystals.SpinsPlus(['D',4])
sage: [[C[m].phi(i) for i in C.index_set()] for m in range(8)]
[[0, 0, 0, 1], [0, 1, 0, 0], [1, 0, 1, 0], [0, 0, 1, 0],
[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0]]
"""
if i < 1 or i > self._n:
raise ValueError("i is not in the index set")
if i == self._n:
return not self._value[i-1] and not self._value[i-2]
return self._value[i] and not self._value[i-1]