  We consider the subalgebra $\Delta$ in the group algebra of the symmetric
group $G=S_{n_1+\dots+n_\nu}$ consisting of all functions invariant with
respect to left and right shifts by elements of the Young subgroup
$H:=S_{n_1}\times \dots \times S_{n_\nu}$. We discuss structure constants of
the algebra $\Delta$ and construct an algebra with continuous parameters $n_1$
extrapolating algebras $\Delta$, it can be also can be rewritten as an
asymptotic algebra as $n_j\to\infty$ (for fixed $\nu$). We show that there is a
natural map from the Lie algebra of the group of pure braids to $\Delta$ (and
therefore this Lie algebra acts in spaces of multiplicities of the quasiregular
representation of the group $G$ in functions on $G/H$).
