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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Lattice paths and Kazhdan-Lusztig polynomials
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by Francesco Brenti PDF
J. Amer. Math. Soc. 11 (1998), 229-259 Request permission


The purpose of this paper is to present a new non-recursive combinatorial formula for the Kazhdan-Lusztig polynomials of a Coxeter group $W$. More precisely, we show that each directed path in the Bruhat graph of $W$ has a naturally associated set of lattice paths with the property that the Kazhdan-Lusztig polynomial of $u,v$ is the sum, over all the lattice paths associated to all the paths going from $u$ to $v$, of $(-1)^{\Gamma _{\ge 0}+d_+(\Gamma )}q^{(l(v)-l(u)+\Gamma (l(\Gamma )))/2}$ where $\Gamma _{\ge 0}, d_+(\Gamma )$, and $\Gamma (l(\Gamma ))$ are three natural statistics on the lattice path.
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Additional Information
  • Francesco Brenti
  • Affiliation: Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy
  • MR Author ID: 215806
  • Email:
  • Received by editor(s): December 20, 1996
  • Received by editor(s) in revised form: July 28, 1997
  • Additional Notes: Part of this work was carried out while the author was a member of the Mathematical Sciences Research Institute in Berkeley, California, U.S.A., and was partially supported by NSF grant No. DMS 9022140 and EC grant No. CHRX-CT93-0400.
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 229-259
  • MSC (1991): Primary 20F55; Secondary 05E99
  • DOI:
  • MathSciNet review: 1460390