r"""
Class functions of groups.
This module implements a wrapper of GAP's ClassFunction function.
NOTE: The ordering of the columns of the character table of a group
corresponds to the ordering of the list. However, in general there is
no way to canonically list (or index) the conjugacy classes of a group.
Therefore the ordering of the columns of the character table of
a group is somewhat random.
AUTHORS:
- Franco Saliola (November 2008): initial version
- Volker Braun (October 2010): Bugfixes, exterior and symmetric power.
"""
from sage.structure.sage_object import SageObject
from sage.interfaces.gap import gap, GapElement
from sage.rings.all import Integer
from sage.rings.all import CyclotomicField
class ClassFunction(SageObject):
"""
A wrapper of GAP's ClassFunction function.
.. NOTE::
It is *not* checked whether the given values describes a character,
since GAP does not do this.
EXAMPLES::
sage: G = CyclicPermutationGroup(4)
sage: values = [1, -1, 1, -1]
sage: chi = ClassFunction(G, values); chi
Character of Cyclic group of order 4 as a permutation group
sage: loads(dumps(chi)) == chi
True
"""
def __init__(self, G, values):
r"""
Returns the character of the group G with values given by the list
values. The order of the values must correspond to the output of
G.conjugacy_classes_representatives().
EXAMPLES::
sage: G = CyclicPermutationGroup(4)
sage: values = [1, -1, 1, -1]
sage: chi = ClassFunction(G, values); chi
Character of Cyclic group of order 4 as a permutation group
"""
self._group = G
if isinstance(values, GapElement) and gap.IsClassFunction(values):
self._gap_classfunction = values
else:
self._gap_classfunction = gap.ClassFunction(G, list(values))
e = self._gap_classfunction.Conductor()
self._base_ring = CyclotomicField(e)
def _gap_init_(self):
r"""
Returns a string showing how to declare / initialize self in Gap.
Stored in the \code{self._gap_string} attribute.
EXAMPLES::
sage: G = CyclicPermutationGroup(4)
sage: values = [1, -1, 1, -1]
sage: ClassFunction(G, values)._gap_init_()
'ClassFunction( CharacterTable( Group( [ (1,2,3,4) ] ) ), [ 1, -1, 1, -1 ] )'
"""
return str(self._gap_classfunction)
def _gap_(self, *args):
r"""
Coerce self into a GAP element.
EXAMPLES::
sage: G = CyclicPermutationGroup(4)
sage: values = [1, -1, 1, -1]
sage: chi = ClassFunction(G, values); chi
Character of Cyclic group of order 4 as a permutation group
sage: type(_)
<class 'sage.groups.class_function.ClassFunction'>
sage: chi._gap_()
ClassFunction( CharacterTable( Group( [ (1,2,3,4) ] ) ), [ 1, -1, 1, -1 ] )
sage: type(_)
<class 'sage.interfaces.gap.GapElement'>
"""
return self._gap_classfunction
def __repr__(self):
r"""
Return a string representation.
OUTPUT:
A string.
EXAMPLES::
sage: G = SymmetricGroup(4)
sage: values = [1, -1, 1, 1, -1]
sage: ClassFunction(G, values)
Character of Symmetric group of order 4! as a permutation group
"""
return "Character of %s" % repr(self._group)
def __iter__(self):
r"""
Iterate through the values of self evaluated on the conjugacy
classes.
EXAMPLES::
sage: xi = ClassFunction(SymmetricGroup(4), [1, -1, 1, 1, -1])
sage: list(xi)
[1, -1, 1, 1, -1]
"""
for v in self._gap_classfunction:
yield self._base_ring(v)
def __cmp__(self, other):
r"""
Rich comparison for class functions.
Compares groups and then the values of the class function on the
conjugacy classes. Otherwise, compares types of objects.
EXAMPLES::
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: chi = G.character([1, 1, 1, 1, 1, 1, 1])
sage: H = PermutationGroup([[(1,2,3),(4,5)]])
sage: xi = H.character([1, 1, 1, 1, 1, 1])
sage: chi == chi
True
sage: xi == xi
True
sage: xi == chi
False
sage: chi < xi
False
sage: xi < chi
True
"""
if isinstance(other, ClassFunction):
return cmp((self._group, self.values()),
(other._group, other.values()))
else:
return cmp(type(self), type(other))
def __reduce__(self):
r"""
Add pickle support.
EXAMPLES::
sage: G = GL(2,7)
sage: values = list(gap(G).CharacterTable().Irr()[2])
sage: chi = ClassFunction(G, values)
sage: loads(dumps(chi)) == chi
True
"""
return ClassFunction, (self._group, self.values())
def domain(self):
r"""
Returns the domain of the self.
OUTPUT:
The underlying group of the class function.
EXAMPLES::
sage: ClassFunction(SymmetricGroup(4), [1,-1,1,1,-1]).domain()
Symmetric group of order 4! as a permutation group
"""
return self._group
def __call__(self, g):
"""
Evaluate the character on the group element g. Returns an error if
g is not in G.
EXAMPLES::
sage: G = GL(2,7)
sage: values = list(gap(G).CharacterTable().Irr()[2])
sage: chi = ClassFunction(G, values)
sage: z = G([[3,0],[0,3]]); z
[3 0]
[0 3]
sage: chi(z)
1
sage: G = GL(2,3)
sage: chi = G.irreducible_characters()[3]
sage: g = G.conjugacy_class_representatives()[6]
sage: chi(g)
zeta8^3 + zeta8
sage: G = SymmetricGroup(3)
sage: h = G((2,3))
sage: triv = G.trivial_character()
sage: triv(h)
1
"""
return self._base_ring(gap(g)._operation("^", self._gap_classfunction))
def __add__(self, other):
r"""
Returns the sum of the characters self and other.
INPUT:
- ``other`` -- a :class:`ClassFunction` of the same group as
``self``.
OUTPUT:
A :class:`ClassFunction`
EXAMPLES::
sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
sage: s = chi+chi
sage: s
Character of Symmetric group of order 4! as a permutation group
sage: s.values()
[6, 2, -2, 0, -2]
"""
if not isinstance(other, ClassFunction):
raise NotImplementedError
s = self._gap_classfunction + other._gap_classfunction
return ClassFunction(self._group, s)
def __sub__(self, other):
r"""
Returns the difference of the characters ``self`` and ``other``.
INPUT:
- ``other`` -- a :class:`ClassFunction` of the same group as
``self``.
OUTPUT:
A :class:`ClassFunction`
EXAMPLES::
sage: G = SymmetricGroup(4)
sage: chi1 = ClassFunction(G, [3, 1, -1, 0, -1])
sage: chi2 = ClassFunction(G, [1, -1, 1, 1, -1])
sage: s = chi1 - chi2
sage: s
Character of Symmetric group of order 4! as a permutation group
sage: s.values()
[2, 2, -2, -1, 0]
"""
if not isinstance(other, ClassFunction):
raise NotImplementedError
s = self._gap_classfunction - other._gap_classfunction
return ClassFunction(self._group, s)
def __mul__(self, other):
r"""
Return the product of the character with ``other``.
INPUT:
- ``other`` -- either a number or a :class:`ClassFunction` of
the same group as ``self``. A number can be anything that
can be converted into GAP: integers, rational, and elements
of certain number fields.
OUTPUT:
A :class:`ClassFunction`
EXAMPLES::
sage: G = SymmetricGroup(4)
sage: chi1 = ClassFunction(G, [3, 1, -1, 0, -1])
sage: 3*chi1
Character of Symmetric group of order 4! as a permutation group
sage: 3*chi1 == chi1+chi1+chi1
True
sage: (3*chi1).values()
[9, 3, -3, 0, -3]
sage: (1/2*chi1).values()
[3/2, 1/2, -1/2, 0, -1/2]
sage: CF3 = CyclotomicField(3)
sage: CF3.inject_variables()
Defining zeta3
sage: (zeta3 * chi1).values()
[3*zeta3, zeta3, -zeta3, 0, -zeta3]
sage: chi2 = ClassFunction(G, [1, -1, 1, 1, -1])
sage: p = chi1*chi2
sage: p
Character of Symmetric group of order 4! as a permutation group
sage: p.values()
[3, -1, -1, 0, 1]
"""
if isinstance(other, ClassFunction):
p = self._gap_classfunction * other._gap_classfunction
return ClassFunction(self._group, p)
else:
return ClassFunction(self._group, other * self._gap_classfunction)
def __rmul__(self, other):
r"""
Return the reverse multiplication of ``self`` and ``other``.
EXAMPLES::
sage: G = SymmetricGroup(4)
sage: chi = ClassFunction(G, [3, 1, -1, 0, -1])
sage: chi * 4 # calls chi.__mul__
Character of Symmetric group of order 4! as a permutation group
sage: 4 * chi # calls chi.__rmul__
Character of Symmetric group of order 4! as a permutation group
sage: (4 * chi).values()
[12, 4, -4, 0, -4]
"""
return self.__mul__(other)
def __pos__(self):
r"""
Return ``self``.
OUTPUT:
A :class:`ClassFunction`
EXAMPLES::
sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
sage: +chi
Character of Symmetric group of order 4! as a permutation group
sage: _.values()
[3, 1, -1, 0, -1]
sage: chi.__pos__() == +chi
True
"""
return ClassFunction(self._group, self._gap_classfunction)
def __neg__(self):
r"""
Return the additive inverse of ``self``.
OUTPUT:
A :class:`ClassFunction`
EXAMPLES::
sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
sage: -chi
Character of Symmetric group of order 4! as a permutation group
sage: _.values()
[-3, -1, 1, 0, 1]
sage: chi.__neg__() == -chi
True
"""
return ClassFunction(self._group, -self._gap_classfunction)
def __pow__(self, other):
r"""
Returns the product of self with itself other times.
EXAMPLES::
sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
sage: p = chi**3
sage: p
Character of Symmetric group of order 4! as a permutation group
sage: p.values()
[27, 1, -1, 0, -1]
"""
if not isinstance(other, (int,Integer)):
raise NotImplementedError
return ClassFunction(self._group, self._gap_classfunction ** other)
def symmetric_power(self, n):
r"""
Returns the symmetrized product of self with itself ``n`` times.
INPUT:
- ``n`` -- a positive integer.
OUTPUT:
The ``n``-th symmetrized power of ``self`` as a
:class:`ClassFunction`.
EXAMPLES::
sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
sage: p = chi.symmetric_power(3)
sage: p
Character of Symmetric group of order 4! as a permutation group
sage: p.values()
[10, 2, -2, 1, 0]
"""
n = Integer(n)
tbl = gap.UnderlyingCharacterTable(self)
return ClassFunction(self._group, gap.SymmetricParts(tbl,[self],n)[1])
def exterior_power(self, n):
r"""
Returns the anti-symmetrized product of self with itself ``n`` times.
INPUT:
- ``n`` -- a positive integer.
OUTPUT:
The ``n``-th anti-symmetrized power of ``self`` as a
:class:`ClassFunction`.
EXAMPLES::
sage: chi = ClassFunction(SymmetricGroup(4), [3, 1, -1, 0, -1])
sage: p = chi.exterior_power(3) # the highest anti-symmetric power for a 3-d character
sage: p
Character of Symmetric group of order 4! as a permutation group
sage: p.values()
[1, -1, 1, 1, -1]
sage: p == chi.determinant_character()
True
"""
n = Integer(n)
tbl = gap.UnderlyingCharacterTable(self)
return ClassFunction(self._group, gap.AntiSymmetricParts(tbl,[self],n)[1])
def scalar_product(self, other):
r"""
Returns the scalar product of self with other.
EXAMPLES::
sage: S4 = SymmetricGroup(4)
sage: irr = S4.irreducible_characters()
sage: [[x.scalar_product(y) for x in irr] for y in irr]
[[1, 0, 0, 0, 0],
[0, 1, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]]
"""
return self._gap_classfunction.ScalarProduct(other)
def is_irreducible(self):
r"""
Returns True if self cannot be written as the sum of two nonzero
characters of self.
EXAMPLES::
sage: S4 = SymmetricGroup(4)
sage: irr = S4.irreducible_characters()
sage: [x.is_irreducible() for x in irr]
[True, True, True, True, True]
"""
return bool(self._gap_classfunction.IsIrreducible())
def degree(self):
r"""
Returns the degree of the character self.
EXAMPLES::
sage: S5 = SymmetricGroup(5)
sage: irr = S5.irreducible_characters()
sage: [x.degree() for x in irr]
[1, 4, 5, 6, 5, 4, 1]
"""
return Integer(self._gap_classfunction.DegreeOfCharacter())
def irreducible_constituents(self):
r"""
Returns a list of the characters that appear in the decomposition
of chi.
EXAMPLES::
sage: S5 = SymmetricGroup(5)
sage: chi = ClassFunction(S5, [22, -8, 2, 1, 1, 2, -3])
sage: irr = chi.irreducible_constituents(); irr
[Character of Symmetric group of order 5! as a permutation group,
Character of Symmetric group of order 5! as a permutation group]
sage: map(list, irr)
[[4, -2, 0, 1, 1, 0, -1], [5, -1, 1, -1, -1, 1, 0]]
sage: G = GL(2,3)
sage: chi = ClassFunction(G, [-1, -1, -1, -1, -1, -1, -1, -1])
sage: chi.irreducible_constituents()
[Character of General Linear Group of degree 2 over Finite Field of size 3]
sage: chi = ClassFunction(G, [1, 1, 1, 1, 1, 1, 1, 1])
sage: chi.irreducible_constituents()
[Character of General Linear Group of degree 2 over Finite Field of size 3]
sage: chi = ClassFunction(G, [2, 2, 2, 2, 2, 2, 2, 2])
sage: chi.irreducible_constituents()
[Character of General Linear Group of degree 2 over Finite Field of size 3]
sage: chi = ClassFunction(G, [-1, -1, -1, -1, 3, -1, -1, 1])
sage: ic = chi.irreducible_constituents(); ic
[Character of General Linear Group of degree 2 over Finite Field of size 3,
Character of General Linear Group of degree 2 over Finite Field of size 3]
sage: map(list, ic)
[[2, -1, 2, -1, 2, 0, 0, 0], [3, 0, 3, 0, -1, 1, 1, -1]]
"""
L = self._gap_classfunction.ConstituentsOfCharacter()
return [ClassFunction(self._group, list(l)) for l in L]
def decompose(self):
r"""
Returns a list of the characters that appear in the decomposition
of chi.
EXAMPLES::
sage: S5 = SymmetricGroup(5)
sage: chi = ClassFunction(S5, [22, -8, 2, 1, 1, 2, -3])
sage: chi.decompose()
[(3, Character of Symmetric group of order 5! as a permutation group),
(2, Character of Symmetric group of order 5! as a permutation group)]
"""
L = []
for irr in self.irreducible_constituents():
L.append((self.scalar_product(irr), irr))
return L
def norm(self):
r"""
Returns the norm of self.
EXAMPLES::
sage: A5 = AlternatingGroup(5)
sage: [x.norm() for x in A5.irreducible_characters()]
[1, 1, 1, 1, 1]
"""
return self._gap_classfunction.Norm()
def values(self):
r"""
Return the list of values of self on the conjugacy classes.
EXAMPLES::
sage: G = GL(2,3)
sage: [x.values() for x in G.irreducible_characters()] #random
[[1, 1, 1, 1, 1, 1, 1, 1],
[1, 1, 1, 1, 1, -1, -1, -1],
[2, -1, 2, -1, 2, 0, 0, 0],
[2, 1, -2, -1, 0, -zeta8^3 - zeta8, zeta8^3 + zeta8, 0],
[2, 1, -2, -1, 0, zeta8^3 + zeta8, -zeta8^3 - zeta8, 0],
[3, 0, 3, 0, -1, -1, -1, 1],
[3, 0, 3, 0, -1, 1, 1, -1],
[4, -1, -4, 1, 0, 0, 0, 0]]
TESTS::
sage: G = GL(2,3)
sage: k = CyclotomicField(8)
sage: zeta8 = k.gen()
sage: v = [tuple(x.values()) for x in G.irreducible_characters()]
sage: set(v) == set([(1, 1, 1, 1, 1, 1, 1, 1), (1, 1, 1, 1, 1, -1, -1, -1), (2, -1, 2, -1, 2, 0, 0, 0), (2, 1, -2, -1, 0, -zeta8^3 - zeta8, zeta8^3 + zeta8, 0), (2, 1, -2, -1, 0, zeta8^3 + zeta8, -zeta8^3 - zeta8, 0), (3, 0, 3, 0, -1, -1, -1, 1), (3, 0, 3, 0, -1, 1, 1, -1), (4, -1, -4, 1, 0, 0, 0, 0)])
True
"""
return list(self)
def central_character(self):
r"""
Returns the central character of self.
EXAMPLES::
sage: t = SymmetricGroup(4).trivial_character()
sage: t.central_character().values()
[1, 6, 3, 8, 6]
"""
return ClassFunction(self._group, self._gap_classfunction.CentralCharacter())
def determinant_character(self):
r"""
Returns the determinant character of self.
EXAMPLES::
sage: t = ClassFunction(SymmetricGroup(4), [1, -1, 1, 1, -1])
sage: t.determinant_character().values()
[1, -1, 1, 1, -1]
"""
return ClassFunction(self._group, self._gap_classfunction.DeterminantOfCharacter())
def tensor_product(self, other):
r"""
EXAMPLES::
sage: S3 = SymmetricGroup(3)
sage: chi1, chi2, chi3 = S3.irreducible_characters()
sage: chi1.tensor_product(chi3).values()
[1, -1, 1]
"""
return ClassFunction(self._group, gap.Tensored([self],[other])[1])
def restrict(self, H):
r"""
Return the restricted character.
INPUT:
- ``H`` -- a subgroup of the underlying group of ``self``.
OUTPUT:
A :class:`ClassFunction` of ``H`` defined by restriction.
EXAMPLES::
sage: G = SymmetricGroup(5)
sage: chi = ClassFunction(G, [3, -3, -1, 0, 0, -1, 3]); chi
Character of Symmetric group of order 5! as a permutation group
sage: H = G.subgroup([(1,2,3), (1,2), (4,5)])
sage: chi.restrict(H)
Character of Subgroup of (Symmetric group of order 5! as a permutation group) generated by [(4,5), (1,2), (1,2,3)]
sage: chi.restrict(H).values()
[3, -3, -3, -1, 0, 0]
"""
rest = self._gap_classfunction.RestrictedClassFunction(H._gap_())
return ClassFunction(H, rest)
def induct(self, G):
r"""
Return the induced character.
INPUT:
- ``G`` -- A supergroup of the underlying group of ``self``.
OUTPUT:
A :class:`ClassFunction` of ``G`` defined by
induction. Induction is the adjoint functor to restriction,
see :meth:`restrict`.
EXAMPLES::
sage: G = SymmetricGroup(5)
sage: H = G.subgroup([(1,2,3), (1,2), (4,5)])
sage: xi = H.trivial_character(); xi
Character of Subgroup of (Symmetric group of order 5! as a permutation group) generated by [(4,5), (1,2), (1,2,3)]
sage: xi.induct(G)
Character of Symmetric group of order 5! as a permutation group
sage: xi.induct(G).values()
[10, 4, 2, 1, 1, 0, 0]
"""
rest = self._gap_classfunction.InducedClassFunction(G._gap_())
return ClassFunction(G, rest)
