Michael Chmutov - Type $A$ molecules are Kazhdan-Lusztig

dmtcs:12814 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.12814
Type $A$ molecules are Kazhdan-LusztigArticle

Authors: Michael Chmutov 1

  • 1 Department of Mathematics [Ann Arbor]

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.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: Iwahori-Hecke algebra,W-graphs,W-molecules,dual equivalence graphs,Kazhdan-Lusztig cells,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]

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