# “Commutative” functions

Say a function is commutative if it remains unchanged under any permutation of its arguments. E.g. $f(0,1)=f(1,0)$. (Alternatively we could describe these as functions over multi-sets, or say that they are reflective about any hyperplane $x_i=x_j$). Some examples are sum, product and average.

1. Is there a name for these functions? Google searches for "commutative" and "reflective" functions don't turn up anything.
2. Can we say anything interesting about these functions? For example, I note that any commutative function which is linear must be the sum function, multiplied by some constant. (i.e. $f(x)=c\sum x_i$). Also I see that the functions make up a field under the obvious operations.
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## 2 Answers

These are called symmetric functions (of two variables.) There is a large literature, that mostly concentrates on symmetric polynomials. Any symmetric polynomial in two variables $x$, $y$ is a polynomial in the variables $x+y$ and $xy$. There is an important analogue for symmetric polynomials in more variables.

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As a response to your second question:

Every symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials, the proof of which you can find on most Abstract Algebra texts (e.g. Dummit and Foote)

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