The center of $\mathbb {Z}[S^{n+1}]$ is the set of symmetric polynomials in $n$ commuting transposition-sums

Abstract:

Let ${S_{n + 1}}$ be the symmetric group on the $n + 1$ symbols $0,1,2, \ldots ,n$. We show that the center of the group-ring $\mathbb {Z}[{S_{n + 1}}]$ coincides with the set of symmetric polynomials with integral coefficients in the $n$ elements ${s_1}, \ldots ,{s_n} \in \mathbb {Z}[{S_{n + 1}}]$, where ${s_k} = \sum \nolimits _{0 \leq i < k} {(i,k)}$ is a sum of $k$ transpositions, $k = 1, \ldots ,n$. In particular, every conjugacy-class-sum of ${S_{n + 1}}$ is a symmetric polynomial in ${s_1}, \ldots ,{s_n}$.