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
- class Element¶
Bases:
sage.combinat.sf.classical.SymmetricFunctionAlgebra_classical.Element
- 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 the \(n\) variables labelled byalphabet
.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
- 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 int
.q
(default: \(1\)) – the value to use for \(q\). Ifq
isNone
, then a ring (or fraction field) of polynomials inq
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[2] + 3*e[1] + 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 theomega()
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 theomega()
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 orinfinity
, specifying whether to compute the principal specialization of ordern
or the stable principal specialization.q
(default:None
) – the value to use for \(q\); the default is to create a ring of polynomials inq
(or a field of rational functions inq
) 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[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 functionsi
– 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[]