Representations of $\mathfrak{S}_\infty$ admissible with respect to Young subgroups

Let $\mathbb N$ be the set of positive integers and $\mathfrak S_\infty$ the set of finite permutations of $\mathbb N$. For a partition $\Pi$ of the set $\mathbb N$ into infinite parts $\mathbb A_1,\mathbb A_2, \dots$ we denote by $\mathfrak S_\Pi$ the subgroup of $\mathfrak S_\infty$ whose elements leave invariant each of the sets $\mathbb A_j$. We set $\mathfrak S_\infty^{(N)}= \{s\in \mathfrak S_\infty : s(i)=i\ \text{for any}\ i=1,2,\dots,N\}$. A factor representation $T$ of the group $\mathfrak S_\infty$ is said to be $\Pi$-admissible if for some $N$ it contains a nontrivial identity subrepresentation of the subgroup $\mathfrak S_\Pi\cap\mathfrak S_\infty^{(N)}$. In the paper, we obtain a classification of the $\Pi$-admissible factor representations of $\mathfrak S_\infty$.
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