The question below came into my mind when I was thinking about this one: A nice generating set for the symmetric power of an algebra.

Let $A$ be a commutative, associative unital algebra over a field of char. $0$. Suppose $A$ is generated (as algebra) by $x^\alpha\;(\alpha \in I)$. Then the algebra $B := A \otimes_k \cdots \otimes_k A$ (n times) is generated by $x^\alpha_i= 1 \otimes \cdots \otimes x^\alpha \otimes \cdots \otimes 1$. The symmetric group $S_n$ acts on $B$ by permuting the factors. Define the *generalized elementary symmetric functions* by

$$e_k(\alpha_1,...,\alpha_k)=\sum_{\sigma\in S_n} \sigma\cdot x_1^{\alpha_1}\cdots x_k^{\alpha_k}\quad(1\le k \le n,\;\alpha_1,...,\alpha_k \in I)$$

Is it known if the invariants $B^{S_n}$ are generated by these generalized elementary symmetric functions ? Do such generalized elementary symmetric functions appear in the literature ?