Type A molecules are of Kazhdan-Lusztig type

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 of Kazhdan-Lusztig type. In this talk I will discuss a possible first step toward the proof of the conjecture. More concretely, I will describe why the connected subgraphs of A_n-cells consisting of "simple" (i.e. directed both ways) edges are of Kazhdan-Lusztig type.