| Summary: | 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 admissible $W$-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.
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