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Type A molecules are Kazhdan–Lusztig

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Abstract

Let (WS) 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 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.

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Acknowledgments

I would like to thank John Stembridge for suggesting the problem. I am also grateful to him as well as to Jonah Blasiak, Shifra Reif, and Elena Yudovina for many useful discussions, and to an anonymous referee for carefully reading the final document and providing valuable suggestions.

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Correspondence to Michael Chmutov.

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Chmutov, M. Type A molecules are Kazhdan–Lusztig. J Algebr Comb 42, 1059–1076 (2015). https://doi.org/10.1007/s10801-015-0616-z

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Keywords

  • Iwahori–Hecke algebra
  • W-graphs
  • W-molecules
  • Dual equivalence graphs
  • Kazhdan–Lusztig cells

Mathematics Subject Classification

  • 05E10
  • 20C08