Infinite symmetric group, pseudomanifolds, and combinatorial cobordism-like structures

We consider a category [Formula: see text] whose morphisms are [Formula: see text]-dimensional pseudomanifolds equipped with certain additional structures (coloring and labeling of some cells), multiplication of morphisms is similar to a concatenation of cobordisms. On the other hand, we consider the product [Formula: see text] of [Formula: see text] copies of infinite symmetric group. We construct a correspondence between the sets of morphisms of [Formula: see text] and double coset spaces of [Formula: see text] with respect to certain subgroups. We show that unitary representations of [Formula: see text] produce functors from the category of [Formula: see text] to the category of Hilbert spaces and bounded linear operators.

Centralizers of the infinite symmetric group

International audience We review and introduce several approaches to the study of centralizer algebras of the infinite symmetric group $S_{\infty}$. Our work is led by the double commutant relationship between finite symmetric groups and partition algebras; in the case of $S_{\infty}$, we obtain centralizer algebras that are contained in partition algebras. In view of the theory of symmetric functions in non-commuting variables, we consider representations of $S_{\infty}$ that are faithful and that contain invariant elements; namely, non-unitary representations on sequence spaces. Nous étudions les algèbres du centralisateur du groupe symétrique infini $S_{\infty}$, passant en revue certaines approches et en introduisant de nouvelles. Notre travail est basé sur la relation du double commutant entre le groupe symétrique fini et les algèbres de partition; dans le cas de $S_{\infty}$, nous obtenons des algèbres du centralisateur contenues dans les algèbres de partition. Compte tenu de la théorie des fonctions symétriques en variables non commutatives, nous considérons les représentations de $S_{\infty}$ qui sont fidèles et contiennent les invariants; c’est-à-dire, les représentations non unitaires sur les espaces de suites.