Lattice paths and Kazhdan-Lusztig polynomials

Brenti, Francesco¹

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

Journal of the American Mathematical Society, Tome 11 (1998) no. 2, pp. 229-259

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-98-00249-5

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