---
tag: PAJEHWXW6M
kind: concept
name: Symmetric function
msc: ["05E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Ring_of_symmetric_functions
---

## Summary

A symmetric function is a formal power series of bounded degree in infinitely many variables $x_1, x_2, \ldots$, see the [[definition|NS852SSQ00]].
It does not change under any permutation of the variables.
The symmetric functions form the ring $\Lambda$, which collects the [[symmetric polynomials|0VHSHB9B6J]] in all numbers of variables at once.
The ring has several bases indexed by [[integer partitions|0JV30XN73Z]]: monomial, [[elementary|0X5QVCF06N]], complete homogeneous and [[Schur functions|43VFMJCDYJ]].
Over $\Q$, the [[power sum|0ZNXZB4HNB]] symmetric functions give one more basis.
It is a central object of [[algebraic combinatorics|VPAQZ0XW74]] and of the representation theory of symmetric and general linear groups.

## Explanations

### L1 human

A symmetric polynomial, such as $x_1^2 + x_2^2 + x_3^2$, stays the same when the variables are shuffled.
Identities between symmetric polynomials usually do not depend on the number of variables.
So one works with infinitely many variables at once, as in $x_1^2 + x_2^2 + x_3^2 + \cdots$.
These infinite sums of bounded degree are the symmetric functions.
Setting all but finitely many variables to $0$ gives back the symmetric polynomials.