Explicit formulae for some Kazhdan-Lusztig polynomials
Let us consider the Kazhdan-Lusztig polynomials \(P_{u,v}(q)\) indexed by permutations \(u,v\). The authors obtain a simplified reccurence relation satisfied by \(P_{u,v}(q)\) when the maximum value of \(v\in S_n\) occurs in position \(n-2\) or \(n-1\). A corollary of this result provides an explicit expresion for \(P_{e,34\ldots n12}(q)\) (\(e\) stands for the identity permutation) as a \(q\)-analogue of a Fibonacci number, namely \(P_{e,34\ldots n12}(q)=F_{n-2}(q)\) for all \(n\geq 3\). This result settles a conjecture of M. Haiman. The second result is also an explicit expresion for \(P_{e,34\ldots (n-2)n(n-1)12}(q)\). At the end of the paper the authors present several conjectures regarding the expression for \(P_{u,v}(q)\) under hypotheses similar to those of the above-mentioned results.
scientific article; zbMATH DE number 1495743