On $W$-graphs of the symmetric group representations

$W$-graphes are closely related to the Kazhdan–Lusztig polynomials now extensively used in representation theory of semisimple groups. For the symmetric group $W=\sigma_n$ description of Kazhdan–Lusztig polynomials for the grassmanians and of the related $\sigma_n$-graphs is known. The aim of this paper is to describe more general $\sigma_n$-graphs, e. g. the graphs of hook diagrams and all irreducible $\sigma_n$-graphs for $n\leqslant6$.