On the first coefficients in \(q\) of the Kazhdan-Lusztig polynomials
A Kazhdan-Lusztig polynomial \(P_{x,w}\) with respect to two elements \(x\), \(w\) of a Coxeter group \(W\) is a certain polynomial in an indeterminate \(q\) with integer coefficients. The positivity of all the coefficients of the Kazhdan-Lusztig polynomials was conjectured by Kazhdan and Lusztig. This conjecture was verified in the cases when \(W\) is either finite or crystallographic but still remains open in general by now. In the paper under review, the author gives a combinatorial description for the coefficient \(p_ 1(x,w)\) of \(q\) in a Kazhdan-Lusztig polynomial \(P_{x,w}\) by introducing a left subword. As a result, he shows that \(p_ 1(x,w)\) is non-negative when the involved elements \(x, w \in W\) satisfy the length condition \(\ell(w) = \ell(x) + \ell(x^{-1} w)\) (resp. \(\ell(w) = \ell(x) + \ell(wx^{-1}))\).
scientific article; zbMATH DE number 645699