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Lattice paths and Kazhdan-Lusztig polynomials

Author: Francesco Brenti
Journal: J. Amer. Math. Soc. 11 (1998), 229-259
MSC (1991): Primary 20F55; Secondary 05E99
MathSciNet review: 1460390
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Abstract: 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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Francesco Brenti
Affiliation: Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy
MR Author ID: 215806

Keywords: Coxeter group, Bruhat order, Kazhdan-Lusztig polynomial, Eulerian poset, lattice path, generalized $h$-vector, Bayer-Billera relations
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.
Article copyright: © Copyright 1998 American Mathematical Society