On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups

In the group algebra of the symmetric group , we consider the subalgebra consisting of all functions invariant with respect to left and right shifts by elements of the Young subgroup . We discuss structure constants of the algebra and construct an algebra with continuous parameters extrapolating algebras , this can also be rewritten as an asymptotic algebra as (for fixed ). We show that there is a natural map from the Lie algebra of the group of pure braids to (and therefore this Lie algebra acts in spaces of multiplicities of the quasiregular representation of the group in functions on ).