def _cl_term(n, R=RationalField()):
"""
Compute the order-n term of the cycle index series of the virtual species `\Omega`,
the compositional inverse of the species `E^{+}` of nonempty sets.
EXAMPLES::
sage: from sage.combinat.species.generating_series import _cl_term
sage: [_cl_term(i) for i in range(4)]
[0, p[1], -1/2*p[1, 1] - 1/2*p[2], 1/3*p[1, 1, 1] - 1/3*p[3]]
"""
n = Integer(n) #check that n is an integer
p = SymmetricFunctions(R).power()
res = p.zero()
if n == 1:
res = p([1])
elif n > 1:
res = 1 / n * ((-1)**(n - 1) * p([1])**n -
sum(d * p([Integer(n / d)]).plethysm(_cl_term(d, R))
for d in divisors(n)[:-1]))
return res
def _DualGrothendieck(self,la):
r"""
Returns the expansion of the K-`k`-Schur function in the homogeneous basis. This
method is here for caching purposes.
INPUT:
- ``la`` -- A `k`-bounded partition.
OUTPUT:
- A symmetric function in the homogeneous basis.
EXAMPLES::
sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
sage: g._DualGrothendieck(Partition([2,1]))
h[2] + h[2, 1] - h[3]
sage: g._DualGrothendieck(Partition([]))
h[]
sage: g._DualGrothendieck(Partition([4,1]))
0
"""
m = la.size()
h = SymmetricFunctions(self.base_ring()).h()
M = self._DualGrothMatrix(m)
vec = []
for i in range(m+1):
for x in Partitions(m-i, max_part=self.k):
if x == la:
vec.append(1)
else:
vec.append(0)
from sage.modules.free_module_element import vector
vec = vector(vec)
sol = M.solve_right(vec)
new_function = h.zero()
count = 0
for i in range(m+1):
for x in Partitions(m-i, max_part=self.k):
new_function+= h(x) * sol[count]
count += 1
return new_function
def _DualGrothendieck(self,la):
r"""
Returns the expansion of the K-`k`-Schur function in the homogeneous basis. This
method is here for caching purposes.
INPUT:
- ``la`` -- A `k`-bounded partition.
OUTPUT:
- A symmetric function in the homogeneous basis.
EXAMPLES::
sage: g = SymmetricFunctions(QQ).kBoundedSubspace(3,1).K_kschur()
sage: g._DualGrothendieck(Partition([2,1]))
h[2] + h[2, 1] - h[3]
sage: g._DualGrothendieck(Partition([]))
h[]
sage: g._DualGrothendieck(Partition([4,1]))
0
"""
m = la.size()
h = SymmetricFunctions(self.base_ring()).h()
M = self._DualGrothMatrix(m)
vec = []
for i in range(m+1):
for x in Partitions(m-i, max_part=self.k):
if x == la:
vec.append(1)
else:
vec.append(0)
from sage.modules.free_module_element import vector
vec = vector(vec)
sol = M.solve_right(vec)
new_function = h.zero()
count = 0
for i in range(m+1):
for x in Partitions(m-i, max_part=self.k):
new_function+= h(x) * sol[count]
count += 1
return new_function
def _cl_term(n, R = RationalField()):
r"""
Compute the order-n term of the cycle index series of the virtual species `\Omega`,
the compositional inverse of the species `E^{+}` of nonempty sets.
EXAMPLES::
sage: from sage.combinat.species.generating_series import _cl_term
sage: [_cl_term(i) for i in range(4)]
[0, p[1], -1/2*p[1, 1] - 1/2*p[2], 1/3*p[1, 1, 1] - 1/3*p[3]]
"""
n = Integer(n) # check that n is an integer
p = SymmetricFunctions(R).power()
res = p.zero()
if n == 1:
res = p([1])
elif n > 1:
res = 1/n * ((-1)**(n-1) * p([1])**n - sum(d * p([n // d]).plethysm(_cl_term(d, R)) for d in divisors(n)[:-1]))
return res
def GL_irreducible_character(n, mu, KK):
r"""
Return the character of the irreducible module indexed by ``mu``
of `GL(n)` over the field ``KK``.
INPUT:
- ``n`` -- a positive integer
- ``mu`` -- a partition of at most ``n`` parts
- ``KK`` -- a field
OUTPUT:
a symmetric function which should be interpreted in ``n``
variables to be meaningful as a character
EXAMPLES:
Over `\QQ`, the irreducible character for `\mu` is the Schur
function associated to `\mu`, plus garbage terms (Schur
functions associated to partitions with more than `n` parts)::
sage: from sage.algebras.schur_algebra import GL_irreducible_character
sage: sbasis = SymmetricFunctions(QQ).s()
sage: z = GL_irreducible_character(2, [2], QQ)
sage: sbasis(z)
s[2]
sage: z = GL_irreducible_character(4, [3, 2], QQ)
sage: sbasis(z)
-5*s[1, 1, 1, 1, 1] + s[3, 2]
Over a Galois field, the irreducible character for `\mu` will
in general be smaller.
In characteristic `p`, for a one-part partition `(r)`, where
`r = a_0 + p a_1 + p^2 a_2 + \dots`, the result is (see [Gr2007]_,
after 5.5d) the product of `h[a_0], h[a_1]( pbasis[p]), h[a_2]
( pbasis[p^2]), \dots,` which is consistent with the following ::
sage: from sage.algebras.schur_algebra import GL_irreducible_character
sage: GL_irreducible_character(2, [7], GF(3))
m[4, 3] + m[6, 1] + m[7]
"""
mbasis = SymmetricFunctions(QQ).m()
r = sum(mu)
M = SchurTensorModule(KK, n, r)
A = M._schur
SGA = M._sga
#make ST the superstandard tableau of shape mu
from sage.combinat.tableau import from_shape_and_word
ST = from_shape_and_word(mu, list(range(1, r + 1)), convention='English')
#make ell the reading word of the highest weight tableau of shape mu
ell = [i + 1 for i, l in enumerate(mu) for dummy in range(l)]
e = M.basis()[tuple(ell)] # the element e_l
# This is the notation `\{X\}` from just before (5.3a) of [Gr2007]_.
S = SGA._indices
BracC = SGA._from_dict(
{S(x.tuple()): x.sign()
for x in ST.column_stabilizer()},
remove_zeros=False)
f = e * BracC # M.action_by_symmetric_group_algebra(e, BracC)
# [Green, Theorem 5.3b] says that a basis of the Carter-Lusztig
# module V_\mu is given by taking this f, and multiplying by all
# xi_{i,ell} with ell as above and i semistandard.
carter_lusztig = []
for T in SemistandardTableaux(mu, max_entry=n):
i = tuple(flatten(T))
schur_rep = schur_representative_from_index(i, tuple(ell))
y = A.basis(
)[schur_rep] * e # M.action_by_Schur_alg(A.basis()[schur_rep], e)
carter_lusztig.append(y.to_vector())
#Therefore, we now have carter_lusztig as a list giving the basis
#of `V_\mu`
#We want to think of expressing this character as a sum of monomial
#symmetric functions.
#We will determine a basis element for each m_\lambda in the
#character, and we want to keep track of them by \lambda.
#That means that we only want to pick out the basis elements above for
#those semistandard words whose content is a partition.
contents = Partitions(r, max_length=n).list()
# all partitions of r, length at most n
# JJ will consist of a list for each element of `contents`,
# recording the list
# of semistandard tableaux words with that content
# graded_basis will consist of the a corresponding basis element
graded_basis = []
JJ = []
for i in range(len(contents)):
graded_basis.append([])
JJ.append([])
for T in SemistandardTableaux(mu, max_entry=n):
i = tuple(flatten(T))
# Get the content of T
con = [0] * n
for a in i:
con[a - 1] += 1
try:
P = Partition(con)
P_index = contents.index(P)
JJ[P_index].append(i)
schur_rep = schur_representative_from_index(i, tuple(ell))
x = A.basis(
)[schur_rep] * f # M.action_by_Schur_alg(A.basis()[schur_rep], f)
graded_basis[P_index].append(x.to_vector())
except ValueError:
pass
#There is an inner product on the Carter-Lusztig module V_\mu; its
#maximal submodule is exactly the kernel of the inner product.
#Now, for each possible partition content, we look at the graded piece of
#that degree, and we record how these elements pair with each of the
#elements of carter_lusztig.
#The kernel of this pairing is the part of this graded piece which is
#not in the irreducible module for \mu.
length = len(carter_lusztig)
phi = mbasis.zero()
for aa in range(len(contents)):
mat = []
for kk in range(len(JJ[aa])):
temp = []
for j in range(length):
temp.append(graded_basis[aa][kk].inner_product(
carter_lusztig[j]))
mat.append(temp)
angle = Matrix(mat)
phi += (len(JJ[aa]) - angle.nullity()) * mbasis(contents[aa])
return phi
def GL_irreducible_character(n, mu, KK):
r"""
Return the character of the irreducible module indexed by ``mu``
of `GL(n)` over the field ``KK``.
INPUT:
- ``n`` -- a positive integer
- ``mu`` -- a partition of at most ``n`` parts
- ``KK`` -- a field
OUTPUT:
a symmetric function which should be interpreted in ``n``
variables to be meaningful as a character
EXAMPLES:
Over `\QQ`, the irreducible character for `\mu` is the Schur
function associated to `\mu`, plus garbage terms (Schur
functions associated to partitions with more than `n` parts)::
sage: from sage.algebras.schur_algebra import GL_irreducible_character
sage: sbasis = SymmetricFunctions(QQ).s()
sage: z = GL_irreducible_character(2, [2], QQ)
sage: sbasis(z)
s[2]
sage: z = GL_irreducible_character(4, [3, 2], QQ)
sage: sbasis(z)
-5*s[1, 1, 1, 1, 1] + s[3, 2]
Over a Galois field, the irreducible character for `\mu` will
in general be smaller.
In characteristic `p`, for a one-part partition `(r)`, where
`r = a_0 + p a_1 + p^2 a_2 + \dots`, the result is (see [GreenPoly]_,
after 5.5d) the product of `h[a_0], h[a_1]( pbasis[p]), h[a_2]
( pbasis[p^2]), \dots,` which is consistent with the following ::
sage: from sage.algebras.schur_algebra import GL_irreducible_character
sage: GL_irreducible_character(2, [7], GF(3))
m[4, 3] + m[6, 1] + m[7]
"""
mbasis = SymmetricFunctions(QQ).m()
r = sum(mu)
M = SchurTensorModule(KK, n, r)
A = M._schur
SGA = M._sga
# make ST the superstandard tableau of shape mu
from sage.combinat.tableau import from_shape_and_word
ST = from_shape_and_word(mu, range(1, r + 1), convention="English")
# make ell the reading word of the highest weight tableau of shape mu
ell = [i + 1 for i, l in enumerate(mu) for dummy in range(l)]
e = M.basis()[tuple(ell)] # the element e_l
# This is the notation `\{X\}` from just before (5.3a) of [GreenPoly]_.
S = SGA._indices
BracC = SGA._from_dict({S(x.tuple()): x.sign() for x in ST.column_stabilizer()}, remove_zeros=False)
f = e * BracC # M.action_by_symmetric_group_algebra(e, BracC)
# [Green, Theorem 5.3b] says that a basis of the Carter-Lusztig
# module V_\mu is given by taking this f, and multiplying by all
# xi_{i,ell} with ell as above and i semistandard.
carter_lusztig = []
for T in SemistandardTableaux(mu, max_entry=n):
i = tuple(flatten(T))
schur_rep = schur_representative_from_index(i, tuple(ell))
y = A.basis()[schur_rep] * e # M.action_by_Schur_alg(A.basis()[schur_rep], e)
carter_lusztig.append(y.to_vector())
# Therefore, we now have carter_lusztig as a list giving the basis
# of `V_\mu`
# We want to think of expressing this character as a sum of monomial
# symmetric functions.
# We will determine a basis element for each m_\lambda in the
# character, and we want to keep track of them by \lambda.
# That means that we only want to pick out the basis elements above for
# those semistandard words whose content is a partition.
contents = Partitions(r, max_length=n).list()
# all partitions of r, length at most n
# JJ will consist of a list for each element of `contents`,
# recording the list
# of semistandard tableaux words with that content
# graded_basis will consist of the a corresponding basis element
graded_basis = []
JJ = []
for i in range(len(contents)):
graded_basis.append([])
JJ.append([])
for T in SemistandardTableaux(mu, max_entry=n):
i = tuple(flatten(T))
# Get the content of T
con = [0] * n
for a in i:
con[a - 1] += 1
try:
P = Partition(con)
P_index = contents.index(P)
JJ[P_index].append(i)
schur_rep = schur_representative_from_index(i, tuple(ell))
x = A.basis()[schur_rep] * f # M.action_by_Schur_alg(A.basis()[schur_rep], f)
graded_basis[P_index].append(x.to_vector())
except ValueError:
pass
# There is an inner product on the Carter-Lusztig module V_\mu; its
# maximal submodule is exactly the kernel of the inner product.
# Now, for each possible partition content, we look at the graded piece of
# that degree, and we record how these elements pair with each of the
# elements of carter_lusztig.
# The kernel of this pairing is the part of this graded piece which is
# not in the irreducible module for \mu.
length = len(carter_lusztig)
phi = mbasis.zero()
for aa in range(len(contents)):
mat = []
for kk in range(len(JJ[aa])):
temp = []
for j in range(length):
temp.append(graded_basis[aa][kk].inner_product(carter_lusztig[j]))
mat.append(temp)
angle = Matrix(mat)
phi += (len(JJ[aa]) - angle.nullity()) * mbasis(contents[aa])
return phi