Coxeter groups and Billey-Postnikov decompositions

In this chapter, we give an overview of Billey– Postnikov (BP) decompositions which have become an important tool for understanding the geometry and combinatorics of Schubert varieties. BP-decompositions are factorizations of Coxeter group elements with many nice properties in relation to the Bruhat partial order. They have played an important role in the classification and enumeration of smooth Schubert varieties. They have also been used in the study of inversion hyperplane arrangements, Kazhdan– Lusztig theory, and permutation pattern avoidance. We survey many of these applications.