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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 functions
• Sym – an instance of the ring of symmetric functions

class Element

Bases: sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical.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[]