KMS States on $\mathfrak{S}_\infty$ Invariant with Respect to the Young Subgroups

Let $\mathfrak{S}_\mathbb{X}$ be the group of all finite permutations on a countable set $\mathbb {X}$, and let $\Pi=({}^1\mathbb{X},\dots,{}^n\mathbb{X})$ be a partition of $\mathbb{X}$ into disjoint subsets such that $|{}^i\mathbb{X}|=\infty$ for all $i$. We set $\mathfrak{S}_\Pi=\{s\in\mathfrak{S}_\mathbb{X}\mid s({}^i\mathbb{X})={}^i\mathbb{X}$ for all $i\}$. A positive definite function $\varphi$ on $\mathfrak{S}_\mathbb{X}$ is called a KMS state if the corresponding vector in the space of the GNS representation is cyclic for the commutant of this representation. A complete description of all factor KMS states which are invariant (central) with respect to the subgroup $\mathfrak{S}_\Pi$ is obtained.