Elementary symmetric functions¶
-
class
sage.combinat.sf.elementary.
SymmetricFunctionAlgebra_elementary
(Sym)¶ Bases:
sage.combinat.sf.multiplicative.SymmetricFunctionAlgebra_multiplicative
A class for methods for the elementary basis of the symmetric functions.
INPUT:
self
– an elementary basis of the symmetric functionsSym
– an instance of the ring of symmetric functions
TESTS:
sage: e = SymmetricFunctions(QQ).e() sage: e == loads(dumps(e)) True sage: TestSuite(e).run(skip=['_test_associativity', '_test_distributivity', '_test_prod']) sage: TestSuite(e).run(elements = [e[1,1]+e[2], e[1]+2*e[1,1]])
-
class
Element
(M, x)¶ 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
(n, alphabet='x')¶ Expand the symmetric function
self
as a symmetric polynomial inn
variables.INPUT:
n
– a nonnegative integeralphabet
– (default:'x'
) a variable for the expansion
OUTPUT:
A monomial expansion of
self
in thevariables labelled by
alphabet
.EXAMPLES:
sage: e = SymmetricFunctions(QQ).e() sage: e([2,1]).expand(3) x0^2*x1 + x0*x1^2 + x0^2*x2 + 3*x0*x1*x2 + x1^2*x2 + x0*x2^2 + x1*x2^2 sage: e([1,1,1]).expand(2) x0^3 + 3*x0^2*x1 + 3*x0*x1^2 + x1^3 sage: e([3]).expand(2) 0 sage: e([2]).expand(3) x0*x1 + x0*x2 + x1*x2 sage: e([3]).expand(4,alphabet='x,y,z,t') x*y*z + x*y*t + x*z*t + y*z*t sage: e([3]).expand(4,alphabet='y') y0*y1*y2 + y0*y1*y3 + y0*y2*y3 + y1*y2*y3 sage: e([]).expand(2) 1 sage: e([]).expand(0) 1 sage: (3*e([])).expand(0) 3
-
omega
()¶ 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 :meth`omega()` method.EXAMPLES:
sage: e = SymmetricFunctions(QQ).e() sage: a = e([2,1]); a e[2, 1] sage: a.omega() e[1, 1, 1] - e[2, 1]
sage: h = SymmetricFunctions(QQ).h() sage: h(e([2,1]).omega()) h[2, 1]
-
omega_involution
()¶ 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 :meth`omega()` method.EXAMPLES:
sage: e = SymmetricFunctions(QQ).e() sage: a = e([2,1]); a e[2, 1] sage: a.omega() e[1, 1, 1] - e[2, 1]
sage: h = SymmetricFunctions(QQ).h() sage: h(e([2,1]).omega()) h[2, 1]
-
verschiebung
(n)¶ Return the image of the symmetric function
self
under the-th Verschiebung operator.
The
-th Verschiebung operator
is defined to be the unique algebra endomorphism
of the ring of symmetric functions that satisfies
for every positive integer
divisible by
, and satisfies
for every positive integer
not divisible by
. This operator
is a Hopf algebra endomorphism. For every nonnegative integer
with
, it satisfies
(where
is the complete homogeneous basis,
is the powersum basis, and
is the elementary basis). For every nonnegative integer
with
, it satisfes
The
-th Verschiebung operator is also called the
-th Verschiebung endomorphism. Its name derives from the Verschiebung (German for “shift”) endomorphism of the Witt vectors.
The
-th Verschiebung operator is adjoint to the
-th Frobenius operator (see
frobenius()
for its definition) with respect to the Hall scalar product (scalar()
).The action of the
-th Verschiebung operator on the Schur basis can also be computed explicitly. The following (probably clumsier than necessary) description can be obtained by solving exercise 7.61 in Stanley [STA].
Let
be a partition. Let
be a positive integer. If the
-core of
is nonempty, then
. Otherwise, the following method computes
: Write the partition
in the form
for some nonnegative integer
. (If
does not divide the length of
, then this is achieved by adding trailing zeroes to
.) Set
for every
. Then,
is a strictly decreasing sequence of nonnegative integers. Stably sort the list
in order of (weakly) increasing remainder of
modulo
. Let
be the sign of the permutation that is used for this sorting. Let
be the sign of the permutation that is used to stably sort the list
in order of (weakly) increasing remainder of
modulo
. (Notice that
.) Then,
, where
is the
-quotient of
.
INPUT:
n
– a positive integer
OUTPUT:
The result of applying the
-th Verschiebung operator (on the ring of symmetric functions) to
self
.EXAMPLES:
sage: Sym = SymmetricFunctions(ZZ) sage: e = Sym.e() sage: e[3].verschiebung(2) 0 sage: e[4].verschiebung(4) -e[1]
The Verschiebung endomorphisms are multiplicative:
sage: all( all( e(lam).verschiebung(2) * e(mu).verschiebung(2) ....: == (e(lam) * e(mu)).verschiebung(2) ....: for mu in Partitions(4) ) ....: for lam in Partitions(4) ) True
TESTS:
Let us check that this method on the elementary basis gives the same result as the implementation in :module`sage.combinat.sf.sfa` on the complete homogeneous basis:
sage: Sym = SymmetricFunctions(QQ) sage: e = Sym.e(); h = Sym.h() sage: all( h(e(lam)).verschiebung(3) == h(e(lam).verschiebung(3)) ....: for lam in Partitions(6) ) True sage: all( e(h(lam)).verschiebung(2) == e(h(lam).verschiebung(2)) ....: for lam in Partitions(4) ) True
-
-
SymmetricFunctionAlgebra_elementary.
coproduct_on_generators
(i)¶ Returns the coproduct on
self[i]
.INPUT:
self
– an elementary basis of the symmetric functionsi
– a nonnegative integer
OUTPUT:
- returns the coproduct on the elementary generator
EXAMPLES:
sage: Sym = SymmetricFunctions(QQ) sage: e = Sym.elementary() sage: e.coproduct_on_generators(2) e[] # e[2] + e[1] # e[1] + e[2] # e[] sage: e.coproduct_on_generators(0) e[] # e[]