# 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().expand(2)
0
sage: e().expand(3)
x0*x1 + x0*x2 + x1*x2
sage: e().expand(4,alphabet='x,y,z,t')
x*y*z + x*y*t + x*z*t + y*z*t
sage: e().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

exponential_specialization(t=None, q=1)

Return the exponential specialization of a symmetric function (when $$q = 1$$), or the $$q$$-exponential specialization (when $$q \neq 1$$).

The exponential specialization $$ex$$ at $$t$$ is a $$K$$-algebra homomorphism from the $$K$$-algebra of symmetric functions to another $$K$$-algebra $$R$$. It is defined whenever the base ring $$K$$ is a $$\QQ$$-algebra and $$t$$ is an element of $$R$$. The easiest way to define it is by specifying its values on the powersum symmetric functions to be $$p_1 = t$$ and $$p_n = 0$$ for $$n > 1$$. Equivalently, on the homogeneous functions it is given by $$ex(h_n) = t^n / n!$$; see Proposition 7.8.4 of [EnumComb2].

By analogy, the $$q$$-exponential specialization is a $$K$$-algebra homomorphism from the $$K$$-algebra of symmetric functions to another $$K$$-algebra $$R$$ that depends on two elements $$t$$ and $$q$$ of $$R$$ for which the elements $$1 - q^i$$ for all positive integers $$i$$ are invertible. It can be defined by specifying its values on the complete homogeneous symmetric functions to be

$ex_q(h_n) = t^n / [n]_q!,$

where $$[n]_q!$$ is the $$q$$-factorial. Equivalently, for $$q \neq 1$$ and a homogeneous symmetric function $$f$$ of degree $$n$$, we have

$ex_q(f) = (1-q)^n t^n ps_q(f),$

where $$ps_q(f)$$ is the stable principal specialization of $$f$$ (see principal_specialization()). (See (7.29) in [EnumComb2].)

The limit of $$ex_q$$ as $$q \to 1$$ is $$ex$$.

INPUT:

• t (default: None) – the value to use for $$t$$; the default is to create a ring of polynomials in t.

• q (default: $$1$$) – the value to use for $$q$$. If q is None, then a ring (or fraction field) of polynomials in q is created.

EXAMPLES:

sage: e = SymmetricFunctions(QQ).e()
sage: x = e[3,2]
sage: x.exponential_specialization()
1/12*t^5
sage: x = 5*e + 3*e + 1
sage: x.exponential_specialization(t=var("t"), q=var("q"))
5*q*t^2/(q + 1) + 3*t + 1

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]

principal_specialization(n=+ Infinity, q=None)

Return the principal specialization of a symmetric function.

The principal specialization of order $$n$$ at $$q$$ is the ring homomorphism $$ps_{n,q}$$ from the ring of symmetric functions to another commutative ring $$R$$ given by $$x_i \mapsto q^{i-1}$$ for $$i \in \{1,\dots,n\}$$ and $$x_i \mapsto 0$$ for $$i > n$$. Here, $$q$$ is a given element of $$R$$, and we assume that the variables of our symmetric functions are $$x_1, x_2, x_3, \ldots$$. (To be more precise, $$ps_{n,q}$$ is a $$K$$-algebra homomorphism, where $$K$$ is the base ring.) See Section 7.8 of [EnumComb2].

The stable principal specialization at $$q$$ is the ring homomorphism $$ps_q$$ from the ring of symmetric functions to another commutative ring $$R$$ given by $$x_i \mapsto q^{i-1}$$ for all $$i$$. This is well-defined only if the resulting infinite sums converge; thus, in particular, setting $$q = 1$$ in the stable principal specialization is an invalid operation.

INPUT:

• n (default: infinity) – a nonnegative integer or infinity, specifying whether to compute the principal specialization of order n or the stable principal specialization.

• q (default: None) – the value to use for $$q$$; the default is to create a ring of polynomials in q (or a field of rational functions in q) over the given coefficient ring.

We use the formulas from Proposition 7.8.3 of [EnumComb2] (using Gaussian binomial coefficients $$\binom{u}{v}_q$$):

\begin{align}\begin{aligned}ps_{n,q}(e_\lambda) = \prod_i q^{\binom{\lambda_i}{2}} \binom{n}{\lambda_i}_q,\\ps_{n,1}(e_\lambda) = \prod_i \binom{n}{\lambda_i},\\ps_q(e_\lambda) = \prod_i q^{\binom{\lambda_i}{2}} / \prod_{j=1}^{\lambda_i} (1-q^j).\end{aligned}\end{align}

EXAMPLES:

sage: e = SymmetricFunctions(QQ).e()
sage: x = e[3,1]
sage: x.principal_specialization(3)
q^5 + q^4 + q^3
sage: x = 5*e[1,1,1] + 3*e[2,1] + 1
sage: x.principal_specialization(3)
5*q^6 + 18*q^5 + 36*q^4 + 44*q^3 + 36*q^2 + 18*q + 6


By default, we return a rational functions in $$q$$. Sometimes it is better to obtain an element of the symbolic ring:

sage: x.principal_specialization(q=var("q"))
-3*q/((q^2 - 1)*(q - 1)^2) - 5/(q - 1)^3 + 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.verschiebung(2)
0
sage: e.verschiebung(4)
-e


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 + e # e + e # e[]
sage: e.coproduct_on_generators(0)
e[] # e[]