By this we mean the basis formed of the complete homogeneous
symmetric functions , not an arbitrary graded basis.
Bases: sage.combinat.sf.multiplicative.SymmetricFunctionAlgebra_multiplicative
A class of methods specific to the homogeneous basis of symmetric functions.
INPUT:
TESTS:
sage: h = SymmetricFunctions(QQ).e()
sage: h == loads(dumps(h))
True
sage: TestSuite(h).run(skip=['_test_associativity', '_test_distributivity', '_test_prod'])
sage: TestSuite(h).run(elements = [h[1,1]+h[2], h[1]+2*h[1,1]])
Bases: sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical.Element
Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.
TESTS:
sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
sage: f == loads(dumps(f))
True
Expand the symmetric function self as a symmetric polynomial in n variables.
INPUT:
OUTPUT:
A monomial expansion of self in the variables
labelled by alphabet.
EXAMPLES:
sage: h = SymmetricFunctions(QQ).h()
sage: h([3]).expand(2)
x0^3 + x0^2*x1 + x0*x1^2 + x1^3
sage: h([1,1,1]).expand(2)
x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3
sage: h([2,1]).expand(3)
x0^3 + 2*x0^2*x1 + 2*x0*x1^2 + x1^3 + 2*x0^2*x2 + 3*x0*x1*x2 + 2*x1^2*x2 + 2*x0*x2^2 + 2*x1*x2^2 + x2^3
sage: h([3]).expand(2,alphabet='y')
y0^3 + y0^2*y1 + y0*y1^2 + y1^3
sage: h([3]).expand(2,alphabet='x,y')
x^3 + x^2*y + x*y^2 + y^3
sage: h([3]).expand(3,alphabet='x,y,z')
x^3 + x^2*y + x*y^2 + y^3 + x^2*z + x*y*z + y^2*z + x*z^2 + y*z^2 + z^3
sage: (h([]) + 2*h([1])).expand(3)
2*x0 + 2*x1 + 2*x2 + 1
sage: h([1]).expand(0)
0
sage: (3*h([])).expand(0)
3
Return the image of self under the omega automorphism.
The omega automorphism is defined to be the unique algebra
endomorphism of the ring of symmetric functions that
satisfies
for all positive integers
(where
stands for the
-th elementary symmetric
function, and
stands for the
-th complete homogeneous
symmetric function). It furthermore is a Hopf algebra
endomorphism and an involution, and it is also known as the
omega involution. It sends the power-sum symmetric function
to
for every positive integer
.
The images of some bases under the omega automorphism are given by
where is any partition, where
denotes
the length (length())
of the partition
, where
denotes the
conjugate partition
(conjugate()) of
, and where the usual notations for bases are used
(
= elementary,
= complete homogeneous,
= powersum,
= Schur).
omega_involution() is a synonym for the omega() method.
OUTPUT:
EXAMPLES:
sage: h = SymmetricFunctions(QQ).h()
sage: a = h([2,1]); a
h[2, 1]
sage: a.omega()
h[1, 1, 1] - h[2, 1]
sage: e = SymmetricFunctions(QQ).e()
sage: e(h([2,1]).omega())
e[2, 1]
Return the image of self under the omega automorphism.
The omega automorphism is defined to be the unique algebra
endomorphism of the ring of symmetric functions that
satisfies
for all positive integers
(where
stands for the
-th elementary symmetric
function, and
stands for the
-th complete homogeneous
symmetric function). It furthermore is a Hopf algebra
endomorphism and an involution, and it is also known as the
omega involution. It sends the power-sum symmetric function
to
for every positive integer
.
The images of some bases under the omega automorphism are given by
where is any partition, where
denotes
the length (length())
of the partition
, where
denotes the
conjugate partition
(conjugate()) of
, and where the usual notations for bases are used
(
= elementary,
= complete homogeneous,
= powersum,
= Schur).
omega_involution() is a synonym for the omega() method.
OUTPUT:
EXAMPLES:
sage: h = SymmetricFunctions(QQ).h()
sage: a = h([2,1]); a
h[2, 1]
sage: a.omega()
h[1, 1, 1] - h[2, 1]
sage: e = SymmetricFunctions(QQ).e()
sage: e(h([2,1]).omega())
e[2, 1]
Returns the coproduct on .
INPUT:
OUTPUT:
EXAMPLES:
sage: Sym = SymmetricFunctions(QQ)
sage: h = Sym.homogeneous()
sage: h.coproduct_on_generators(2)
h[] # h[2] + h[1] # h[1] + h[2] # h[]
sage: h.coproduct_on_generators(0)
h[] # h[]