Denote $S_k^n = \underbrace{S_k \times \dots \times S_k}_{n \text{ times}}$ and let $S_k$ act on $S_k^n$ diagonal manner, so that

$$ \pi (\sigma_1, \dots, \sigma_n)\pi^{-1} := (\pi \sigma_1 \pi^{-1}, \dots, \pi \sigma_n \pi^{-1})\,. $$

What is known about elements of the group algebra $\mathbb{C} S_k^n$ that commute with the conjugate diagonal action of $S_k$?

In other words, what is known about the center $$ Z_{S_k}( \mathbb{C} S_k^n) = \Big\{ \alpha \in \mathbb{C} S_k^n \,\Big|\, \pi \alpha \pi^{-1} = \alpha \,\text{ for all }\, \pi \in S_k \Big\}\,.$$ In particular, is there a decomposition of $Z_{S_k}( \mathbb{C} S_k^n)$ into centrally (with respect to $S_k$) primitive idempotents? If so, how can one find it?

Denote $S_k^n = \underbrace{S_k \times \dots \times S_k}_{n \text{ times}}$ and let $S_k$ act on $S_k^n$ conjugate diagonal, so that

$$ \pi (\sigma_1, \dots, \sigma_n)\pi^{-1} := (\pi \sigma_1 \pi^{-1}, \dots, \pi \sigma_n \pi^{-1})\,. $$

What is known about elements of the group algebra $\mathbb{C} [S_k^n]$ that commute with the conjugate diagonal action of $S_k$?

In other words, what is known about $$ \mathbb{C}[S_k^n]^{S_k} = \Big\{ \alpha \in \mathbb{C} [S_k^n] \,\Big|\, \pi \alpha \pi^{-1} = \alpha \,\text{ for all }\, \pi \in S_k \Big\}\,.$$ In particular, is an explicit decomposition of $\mathbb{C} [S_k^n]^{S_k}$ into centrally (with respect to $S_k$) primitive idempotents known?