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Quasisymmetric functions and Kazhdan-Lusztig polynomials

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Abstract

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853-4201, USA

    Louis J. Billera

  2. Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, I-00133, Roma, Italy

    Francesco Brenti

Authors
  1. Louis J. Billera
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  2. Francesco Brenti
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Corresponding author

Correspondence to Louis J. Billera.

Additional information

This work was begun while both authors enjoyed the hospitality of the Mittag-Leffler Institute, Djursholm, Sweden.

The first author was supported in part by NSF grants DMS-0100323 and DMS-0555268.

The second author was partially supported by EU grant CHRT-CT-2001-00272.

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Billera, L.J., Brenti, F. Quasisymmetric functions and Kazhdan-Lusztig polynomials. Isr. J. Math. 184, 317–348 (2011). https://doi.org/10.1007/s11856-011-0070-0

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  • DOI: https://doi.org/10.1007/s11856-011-0070-0

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