Let us take the group algebra $\mathbb{C} S_N$ and the subgroup $H=Z_N$ generated by the element $\sigma=(123\dots N)$, which is a cyclic shift. What is the structure of the centralizer of $H$ within $\mathbb{C} S_N$? If we are looking at the Lie algebra $\mathcal{L}(\mathbb{C} S_N)$ obtained from the usual commutator of two elements, then what is the centralizer of the commutative sub-algebra $\mathbb{C} H$?

My motivation for this comes from physics, where the special ordering $1,2,3,\dots N$ has a concrete physical meaning, for example actual atoms are placed next to each other. In this case those group algebra elements that commute with $\sigma$ are called ``translationally invariant'' permutation operators, under periodic boundary conditions.

Ultimately I would like to know what is the maximal number of mutually commuting operators within $\mathbb{C} S_N$ that commute with $\mathbb{C} H$, and how to find explicit bases for them. I wanted to approach this through the structure of the Lie algebra $\mathcal{L}(\mathbb{C} S_N)$, decomposition into simple Lie algebras, etc. I understand that the irreducible subspaces in $\mathbb{C} S_N$ correspond to the Young symmetrizers. But it is not so clear what to do from here, how to add the action of the concrete element $\sigma$ into this.