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 (Formula presented.), that all such An-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 An-cells consisting of simple (i.e., directed both ways) edges are dual equivalence graphs in the sense of Assaf and thus are the same as the ones in the Kazhdan–Lusztig cells.