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Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as

$q(x_1, x_2, \cdots, x_{n+1}) = u(x_1)p(x_2, x_3, \cdots, x_{n+1}) + u(x_2)p(x_1, x_3, \cdots, x_{n+1}) + \cdots \\ \phantom{q(x_1, x_2, \cdots, x_{n+1}) = } \qquad + u(x_{n+1})p(x_1, x_2, \cdots, x_n).$

It is easy to verify that $q$ is a symmetric polynomial. My question is: Is there a name already defined for such a mapping from $(p, u)$ to $q$? Thanks.

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Don't know if there is a name. Possibly this is known to Newton; the inductive proof of Newton's theorem on elementary symmetric polynomials goes along similar lines.

When we start with some polynomial and take the sum over its orbit under $S_{n+1}$ it will be invariant under $S_{n+1}$. You are starting with $u(x_{n+1}) p(x_1,x_2,\ldots, x_n)$, and summing it over the generating set of $n$ transpositions of $S_{n+1}$.

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I don't know if the operation has a name in the context of the classical theory of symmetric functions. However, in mathematical physics this is essentially what is called a creation operator in a Boson Fock space. See, e.g., Reed and Simon "Methods of Modern Mathemtatical Physics" vol 2, page 209, 1975 edition.

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Your sum is (up to a scalar) the transfer or trace of $u(x_{n+1})p(x_1,x_2,\dots,x_n)$. Given any polynomial $h(x_1,x_2,\dots,x_n,x_{n+1})$ its transfer or trace (with respect to the symmetric group, $S_{n+1}$ is the symmetric polynomial $\sum_{\sigma\in S_{n+1}} \sigma \cdot h(x_1,x_2,\dots,x_n,x_{n+1})$. This construction works for any finite group.

More precisely, you have a relative transfer $\sum_{\sigma\in S_{n+1}/S_n} \sigma \cdot h(x_1,x_2,\dots,x_n,x_{n+1})$ where $h=u(x_{n+1})p(x_1,x_2,\dots,x_n)$ is $S_n$-invariant and the sum is over a (any) set of (left) coset representatives of $S_n$ in $S_{n+1}$.

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