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Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define $$f_{\mathrm{symm}}(x_1,\ldots,x_n) = 2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1} f(\varepsilon_1x_1,\ldots,\varepsilon_n x_n),$$ i.e., the average of $f$ over all sign-changes of the arguments. I know that the resulting function (or a similarly defined measure, random variable, etc) is called "sign invariant", but what is the idempotent operator $f\mapsto f_{\mathrm{symm}}$ called?

Is there any general theory of this operator? It is hard to search without a name.

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    $\begingroup$ I think this may be called a "polarization operator" --- see e.g., page 6 of Gurvits' paper: arxiv.org/abs/math/0510452v3 $\endgroup$
    – Suvrit
    Commented Jan 29, 2015 at 5:10
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    $\begingroup$ It's averaging over an action of a particular finite group, namely $\mathbb{Z}_2^n$. I think in some contexts the more general operation of averaging over an action of a finite group is called the Reynolds operator. $\endgroup$
    – Qiaochu Yuan
    Commented Jan 29, 2015 at 5:58

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