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Let $r_1, r_2, \ldots, r_k$ be positive integers with or without repetition such that $1\le r_i \le n$ for $i = 1, 2, \ldots, k$. Let $f$ be a continuous multivariate function with the property that the value of $f(r_1, r_2, \ldots, r_k)$ is independent of the order the arguments $r_1, r_2, \ldots, r_k$. The trivial examples are:

$f(r_1, r_2, \ldots, r_k) = g(r_1) + g(r_2) + \ldots + g(r_k)$,

$f(r_1, r_2, \ldots, r_k) = g(r_1)g( r_2) \ldots g(r_k)$ and

$f(r_1, r_2, \ldots, r_k) = c$

where $g$ is a continuous univariate function and $c$ is a constant. I have two questions

Questions:

  1. Are there other non trivial examples of such functions?

  2. Are there infinitely many such non trivial functions $f$?

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3 Answers 3

up vote 3 down vote accepted

You could symmetrize an arbitrary function by averaging it over the symmetric group.

Explicitly: take your favourite function $h$ in $n$ variables, and set $f(r_1, ..., r_n) = \frac{1}{n!}\sum_{\sigma} h(r_{\sigma(1)}, ..., r_{\sigma(n)})$ where the sum is taken over all permutations $\sigma$ of the numbers $1,...,n$.

I'd love to hear of an example which isn't obtained by symmetrization!

edit: heh. Any such example would be its own symmetrization. So I guess in some sense this is a complete, if overly glib, answer.

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I don't understand why you say that $f$ is continuous if its inputs are positive integers. Anyway, you can take any symmetric polynomial in the $g(r_i)$, e.g. $\sum_{i < j} g(r_i) g(r_j)$.

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@Qiaochu: That is because the current application for which I need such a function has discrete arguments (positive integers) but we want to keep it flexible enough to accommodate all positive real values if required in future. –  Nilotpal Sinha Mar 13 '13 at 6:17

Another very famous class of multivariate functions that satisfies this permutation invariance is: Symmetric gauge functions, for example, all the vector $\ell_p$ norms; also in fact their quasi-norm cousins are symmetric.

It is not clear to me where does it matter that the inputs be positive integers.

For example, suppose $r_1,\ldots, r_n$ are $n$ indeterminate elements of some group (possibly non-abelian). Then, using the kind of symmetrization over $S_n$ (that B. Young mentions), we can obtain symmetric functions in an even more general setup.

P.S.: You might also enjoy looking at the book: Symmetric functions and Hall Polynomials by MacDonald.

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@Survit: That practical application for which I need such a function has discrete arguments (positive integers) but we want to keep it flexible enough to accommodate all positive real values if required in future. –  Nilotpal Sinha Mar 13 '13 at 6:18
    
My point is: symmetry as defined above is almost "agnostic" of whether you have integers or not...so you can use these definitions. However, if you are looking to ensure that the symmetric function that you use maps integers to integers, then the number of choices you have decreases... –  Suvrit Mar 13 '13 at 18:50

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