---
tag: NS852SSQ00
kind: definition
name: Symmetric function
msc: ["05E05"]
defines: PAJEHWXW6M
uses: [033WGN00BF, 0VHSHB9B6J, 0JV30XN73Z]
sources:
  - url: https://en.wikipedia.org/wiki/Ring_of_symmetric_functions
---

## Definition

Let $x_1, x_2, \ldots$ be countably many commuting variables, and consider [[formal power series|033WGN00BF]] in them with integer coefficients.
A *symmetric function* is such a series $f$ with these properties:

1. $f$ has bounded degree: for some integer $d$, every monomial of $f$ has total degree at most $d$;
2. $f$ is invariant under every permutation of the variables.
   Thus $x_{i_1}^{a_1} \cdots x_{i_r}^{a_r}$ and $x_{j_1}^{a_1} \cdots x_{j_r}^{a_r}$ have equal coefficients when the $i_s$ are distinct and the $j_s$ are distinct.

The symmetric functions form a ring, written $\Lambda$.

## Remarks

The ring $\Lambda$ is graded, $\Lambda = \bigoplus_{k \ge 0} \Lambda^k$, where $\Lambda^k$ consists of the homogeneous symmetric functions of degree $k$.
For a partition $\lambda = (\lambda_1, \ldots, \lambda_r)$, the monomial symmetric function $m_\lambda$ is the sum of all distinct monomials $x_{i_1}^{\lambda_1} \cdots x_{i_r}^{\lambda_r}$ with distinct indices.
The $m_\lambda$ with $\lambda$ a partition of $k$ form a $\Z$-basis of $\Lambda^k$.
Setting $x_{n+1} = x_{n+2} = \cdots = 0$ maps $\Lambda$ onto the ring of [[symmetric polynomials|0VHSHB9B6J]] in $x_1, \ldots, x_n$.
Equivalently, $\Lambda^k$ is the inverse limit of the degree-$k$ symmetric polynomials in $n$ variables, as $n \to \infty$.

Example: the power sum $p_2 = x_1^2 + x_2^2 + x_3^2 + \cdots$ is a symmetric function of degree $2$.
Both clauses hold: all monomials have degree $2$, and a permutation of the variables only permutes the terms.

Non-example: the product $\prod_{i \ge 1} (1 + x_i) = 1 + e_1 + e_2 + \cdots$ is invariant under all permutations of the variables, but it fails clause 1.
It contains the monomial $x_1 x_2 \cdots x_k$ of degree $k$ for every $k$.

The elementary symmetric functions $e_k = \sum_{i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}$ are algebraically independent, and $\Lambda = \Z[e_1, e_2, \ldots]$.
The same holds for the complete homogeneous symmetric functions $h_k$, the sum of all monomials of degree $k$.