Recursive structures of molecules and cells in Gelfand $S_n$-graphs

$W$-graphs, representing the multiplication action of the standard basis on the canonical basis in the Iwahori-Hecke algebra are introduced by Kazhdan and Lusztig. Marberg defined a generalized $W$-graph, the Gelfand W-graph, corresponding to the Hecke algebra modules instead of Hecke algebras. To classify the molecules and cells of the Gelfand $S_n$-graphs, in this paper, we introduce a recursive structure of $S_n$ and then discuss the action of the recursive structure on the molecules. Using this struction, we show that a specific molecule is indeed a cell.