Type a molecules are kazhdan-lusztig

Let (W; S) be a Coxeter system. A W-graph is an encoding of a representation of the corresponding Iwahori-Hecke algebra. Especially important examples include the W-graph corresponding to the action of the Iwahori-Hecke algebra on the Kazhdan-Lusztig basis as well as this graph's strongly connected components (cells). In 2008, Stembridge identified some common features of the Kazhdan-Lusztig graphs ("admissibility") and gave combinatorial rules for detecting admissibleW-graphs. He conjectured, and checked up to n = 9, that all admissible A_n-cells are Kazhdan-Lusztig cells. The current paper provides a possible first step toward a proof of the conjecture. More concretely, we prove that the connected subgraphs of A_n-cells consisting of simple (i.e. directed both ways) edges do fit into the Kazhdan-Lusztig cells.