Kazhdan-Lusztig polynomials of parabolic type
Let \((W,S)\) be a Coxeter system. For \(J\subseteq S\), let \(W_J\) be the subgroup of \(W\) generated by \(J\) and let \(W^J\) be the set of all the elements \(w\in W\) such that \(w\) has the minimal length in the coset \(wW_J\). An \(R\)-polynomial \({\mathcal R}^J_{x,w}(q)_u\) of parabolic type (resp. a Kazhdan-Lusztig polynomial \(P^J_{x,w}(q)_u\) of parabolic type) is a certain polynomial in an indeterminate \(q\) indexed by two elements \(x,w\in W^J\), where \(u=-1\) or \(u=q\). The positivity of the coefficients of the polynomials \(P^J_{x,w}(q)_u\) is a long-standing conjecture. It has been shown only when \(W\) is a crystallographic Coxeter group. The author gives a formula for the polynomials \(P^J_{x,w}(q)_u\) in terms of \(R\)-polynomials. This formula has a pure combinatorial nature since the \({\mathcal R}_{x,w}^J(q)_u\)'s are defined in terms of multichains in \(W^J\) under the Bruhat order. The author applies the formula to show the non-negativity of the coefficient of \(q\) in \(P^J_{x,w}(q)_u\) under a certain condition on the length of elements. Note that the formula for the polynomials \(P^J_{x,w}(q)_u\) was also obtained independently by \textit{V. Deodhar} [see J. Algebra 190, No. 1, 214-225 (1997; Zbl 0889.20021)].
scientific article; zbMATH DE number 1139778
scientific article