Séminaire Lotharingien de Combinatoire, 80B.88 (2018), 12 pp.

Rafael S. González D'León and Joshua Hallam

Whitney Labelings and 0-Hecke Algebra Actions on Graded Posets

Abstract. In the proceedings of FPSAC 2017, the authors introduced the notion of Whitney duality of graded posets. Two graded posets are Whitney dual if their Whitney numbers of the first and second kind are (up to a sign) switched. In this extended abstract, we present new results in the study of Whitney duals. We present new types of edge and chain-edge labelings of graded posets which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual and we show how to explicitly construct a Whitney dual using a technique that involves quotient posets. As an application, we explicitly construct a Whitney dual for the lattice of noncrossing partitions. We also show that a graded poset P with a Whitney labeling admits a local action of the 0-Hecke algebra on the set of maximal chains of P. The characteristic of the associated representation is Ehrenborg's flag quasisymmetric function of P.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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