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 and gave a combinatorial characterization of all W-graphs that have these features. He conjectured, and checked up to n=9, that all such A_n-cells are Kazhdan-Lusztig cells. The current paper provides a first step toward a potential 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 are the same as in the Kazhdan-Lusztig cells.