# Elementary symmetric functions¶

class sage.combinat.sf.elementary.SymmetricFunctionAlgebra_elementary(Sym)

A class for methods for the elementary basis of the symmetric functions.

INPUT:

• self – an elementary basis of the symmetric functions
• Sym – an instance of the ring of symmetric functions
class Element
expand(n, alphabet='x')

Expand the symmetric function self as a symmetric polynomial in n variables.

INPUT:

• n – a nonnegative integer
• alphabet – (default: 'x') a variable for the expansion

OUTPUT:

A monomial expansion of self in the $$n$$ variables 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 $$\omega$$ of the ring of symmetric functions that satisfies $$\omega(e_k) = h_k$$ for all positive integers $$k$$ (where $$e_k$$ stands for the $$k$$-th elementary symmetric function, and $$h_k$$ stands for the $$k$$-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 $$p_k$$ to $$(-1)^{k-1} p_k$$ for every positive integer $$k$$.

The images of some bases under the omega automorphism are given by

$\omega(e_{\lambda}) = h_{\lambda}, \qquad \omega(h_{\lambda}) = e_{\lambda}, \qquad \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad \omega(s_{\lambda}) = s_{\lambda^{\prime}},$

where $$\lambda$$ is any partition, where $$\ell(\lambda)$$ denotes the length (length()) of the partition $$\lambda$$, where $$\lambda^{\prime}$$ denotes the conjugate partition (conjugate()) of $$\lambda$$, and where the usual notations for bases are used ($$e$$ = elementary, $$h$$ = complete homogeneous, $$p$$ = powersum, $$s$$ = Schur).

omega_involution() is a synonym for the 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 $$\omega$$ of the ring of symmetric functions that satisfies $$\omega(e_k) = h_k$$ for all positive integers $$k$$ (where $$e_k$$ stands for the $$k$$-th elementary symmetric function, and $$h_k$$ stands for the $$k$$-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 $$p_k$$ to $$(-1)^{k-1} p_k$$ for every positive integer $$k$$.

The images of some bases under the omega automorphism are given by

$\omega(e_{\lambda}) = h_{\lambda}, \qquad \omega(h_{\lambda}) = e_{\lambda}, \qquad \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad \omega(s_{\lambda}) = s_{\lambda^{\prime}},$

where $$\lambda$$ is any partition, where $$\ell(\lambda)$$ denotes the length (length()) of the partition $$\lambda$$, where $$\lambda^{\prime}$$ denotes the conjugate partition (conjugate()) of $$\lambda$$, and where the usual notations for bases are used ($$e$$ = elementary, $$h$$ = complete homogeneous, $$p$$ = powersum, $$s$$ = Schur).

omega_involution() is a synonym for the 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 $$n$$-th Verschiebung operator.

The $$n$$-th Verschiebung operator $$\mathbf{V}_n$$ is defined to be the unique algebra endomorphism $$V$$ of the ring of symmetric functions that satisfies $$V(h_r) = h_{r/n}$$ for every positive integer $$r$$ divisible by $$n$$, and satisfies $$V(h_r) = 0$$ for every positive integer $$r$$ not divisible by $$n$$. This operator $$\mathbf{V}_n$$ is a Hopf algebra endomorphism. For every nonnegative integer $$r$$ with $$n \mid r$$, it satisfies

$\mathbf{V}_n(h_r) = h_{r/n}, \quad \mathbf{V}_n(p_r) = n p_{r/n}, \quad \mathbf{V}_n(e_r) = (-1)^{r - r/n} e_{r/n}$

(where $$h$$ is the complete homogeneous basis, $$p$$ is the powersum basis, and $$e$$ is the elementary basis). For every nonnegative integer $$r$$ with $$n \nmid r$$, it satisfes

$\mathbf{V}_n(h_r) = \mathbf{V}_n(p_r) = \mathbf{V}_n(e_r) = 0.$

The $$n$$-th Verschiebung operator is also called the $$n$$-th Verschiebung endomorphism. Its name derives from the Verschiebung (German for “shift”) endomorphism of the Witt vectors.

The $$n$$-th Verschiebung operator is adjoint to the $$n$$-th Frobenius operator (see frobenius() for its definition) with respect to the Hall scalar product (scalar()).

The action of the $$n$$-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 $$\lambda$$ be a partition. Let $$n$$ be a positive integer. If the $$n$$-core of $$\lambda$$ is nonempty, then $$\mathbf{V}_n(s_\lambda) = 0$$. Otherwise, the following method computes $$\mathbf{V}_n(s_\lambda)$$: Write the partition $$\lambda$$ in the form $$(\lambda_1, \lambda_2, ..., \lambda_{ns})$$ for some nonnegative integer $$s$$. (If $$n$$ does not divide the length of $$\lambda$$, then this is achieved by adding trailing zeroes to $$\lambda$$.) Set $$\beta_i = \lambda_i + ns - i$$ for every $$s \in \{ 1, 2, \ldots, ns \}$$. Then, $$(\beta_1, \beta_2, ..., \beta_{ns})$$ is a strictly decreasing sequence of nonnegative integers. Stably sort the list $$(1, 2, \ldots, ns)$$ in order of (weakly) increasing remainder of $$-1 - \beta_i$$ modulo $$n$$. Let $$\xi$$ be the sign of the permutation that is used for this sorting. Let $$\psi$$ be the sign of the permutation that is used to stably sort the list $$(1, 2, \ldots, ns)$$ in order of (weakly) increasing remainder of $$i - 1$$ modulo $$n$$. (Notice that $$\psi = (-1)^{n(n-1)s(s-1)/4}$$.) Then, $$\mathbf{V}_n(s_\lambda) = \xi \psi \prod_{i=0}^{n-1} s_{\lambda^{(i)}}$$, where $$(\lambda^{(0)}, \lambda^{(1)}, \ldots, \lambda^{(n - 1)})$$ is the $$n$$-quotient of $$\lambda$$.

INPUT:

• n – a positive integer

OUTPUT:

The result of applying the $$n$$-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

coproduct_on_generators(i)

Returns the coproduct on self[i].

INPUT:

• self – an elementary basis of the symmetric functions
• i – a nonnegative integer

OUTPUT:

• returns the coproduct on the elementary generator $$e(i)$$

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[]