def spin_polynomial_square(part, weight, length):
"""
Returns the spin polynomial associated with part, weight, and
length, with the substitution t -> t^2 made.
EXAMPLES::
sage: from sage.combinat.ribbon_tableau import spin_polynomial_square
sage: spin_polynomial_square([6,6,6],[4,2],3)
t^12 + t^10 + 2*t^8 + t^6 + t^4
sage: spin_polynomial_square([6,6,6],[4,1,1],3)
t^12 + 2*t^10 + 3*t^8 + 2*t^6 + t^4
sage: spin_polynomial_square([3,3,3,2,1], [2,2], 3)
t^7 + t^5
sage: spin_polynomial_square([3,3,3,2,1], [2,1,1], 3)
2*t^7 + 2*t^5 + t^3
sage: spin_polynomial_square([3,3,3,2,1], [1,1,1,1], 3)
3*t^7 + 5*t^5 + 3*t^3 + t
sage: spin_polynomial_square([5,4,3,2,1,1,1], [2,2,1], 3)
2*t^9 + 6*t^7 + 2*t^5
sage: spin_polynomial_square([[6]*6, [3,3]], [4,4,2], 3)
3*t^18 + 5*t^16 + 9*t^14 + 6*t^12 + 3*t^10
"""
R = ZZ['t']
t = R.gen()
if part in partition.Partitions():
part = skew_partition.SkewPartition([part,partition.Partition_class([])])
elif part in skew_partition.SkewPartitions():
part = skew_partition.SkewPartition(part)
if part == [[],[]] and weight == []:
return t.parent()(1)
return R(graph_implementation_rec(part, weight, length, functools.partial(spin_rec,t))[0])
def _conjugacy_classes_representatives_underlying_group(self):
r"""
Returns a complete list of representatives of conjugacy classes of the underlying symmetric group
EXAMPLES::
sage: SG=SymmetricGroupAlgebra(ZZ,3)
sage: SG._conjugacy_classes_representatives_underlying_group()
[[2, 3, 1], [2, 1, 3], [1, 2, 3]]
"""
return [
permutation.Permutations(self.n).element_in_conjugacy_classes(nu)
for nu in partition.Partitions(self.n)
]
def RibbonTableaux(shape, weight, length):
"""
Returns the combinatorial class of ribbon tableaux of skew shape
shape and weight weight tiled by ribbons of length length.
EXAMPLES::
sage: RibbonTableaux([[2,1],[]],[1,1,1],1)
Ribbon tableaux of shape [[2, 1], []] and weight [1, 1, 1] with 1-ribbons
"""
if shape in partition.Partitions():
shape = partition.Partition(shape)
shape = skew_partition.SkewPartition([shape, shape.core(length)])
else:
shape = skew_partition.SkewPartition(shape)
if shape.size() != length * sum(weight):
raise ValueError
weight = [i for i in weight if i != 0]
return RibbonTableaux_shapeweightlength(shape, weight, length)