A decomposition of \(R\)-polynomials and Kazhdan-Lusztig polynomials
The \(R\)-polynomial is defined for two elements in an arbitrary Coxeter group. These polynomials are intimately related to Kazhdan-Lusztig polynomials introduced by \textit{D. Kazhdan} and \textit{G. Lusztig} in 1979 [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. For example, it is well known that \[ q^{l(w)-l(x)}P_{x,w}\biggl(\frac1 q\biggr)=\sum_{x\leq y\leq w}R_{x,y}(q)P_{y,w}(q), \] where \(P_{x,w}(q)\) (resp. \(R_{x,w}(q)\)) is the Kazhdan-Lusztig polynomial (resp. the \(R\)-polynomial). In [Combinatorial properties of the Kazhdan-Lusztig polynomials for \(S_n\), Adv. Math. (to appear)] \textit{F. Brenti} found a decomposition formula of \(R\)-polynomials for symmetric groups and he showed that products of \(R\)-polynomials for symmetric groups are also \(R\)-polynomials for symmetric groups. The purpose of this article is to find a decomposition formula of \(R\)-polynomials and Kazhdan-Lusztig polynomials for arbitrary Coxeter groups in extension of Brenti's result.
scientific article; zbMATH DE number 851720