Geodesic
Type $A$ molecules are Kazhdan-Lusztig
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 4, pp. 1059-1076.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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). J. R. Stembridge [Represent. Theory 12, 346--368 (2008; Zbl 1195.20043)] 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 dual equivalence graphs in the sense of Assaf and thus are the same as the ones in the Kazhdan-Lusztig cells.
Classification : 05E10, 20C08
Keywords: Iwahori-Hecke algebra, $W$-graphs, $W$-molecules, dual equivalence graphs, Kazhdan-Lusztig cells 
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     author = {Chmutov, Michael},
     title = {Type $A$ molecules are {Kazhdan-Lusztig}},
     journal = {Journal of Algebraic Combinatorics},
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Chmutov, Michael. Type $A$ molecules are Kazhdan-Lusztig. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 4, pp. 1059-1076. http://geodesic.mathdoc.fr/item/JAC_2015__42_4_a4/