Boolean elements in Kazhdan–Lusztig theory

Kazhdan–Lusztig polynomials have been proven to play an important role in different fields. Despite this, there are still few explicit formulae for them. Here we give closed product formulae for the Kazhdan–Lusztig polynomials indexed by Boolean elements in a class of Coxeter systems that we call linear. Boolean elements are elements smaller than a reflection that admits a reduced expression of the form s1…sn−1snsn−1…s1 (si S, si≠sj if i≠j). Then we provide several applications of this result concerning the combinatorial invariance of the Kazhdan–Lusztig polynomials, the classification of the pairs (u,v) with u v, the Kazhdan–Lusztig elements and the intersection homology Poincaré polynomials of the Schubert varieties.