Coxeter groups and Billey-Postnikov decompositions

TL;DR

This paper surveys Billey-Postnikov decompositions of Coxeter group elements, highlighting their significance in understanding Schubert varieties, hyperplane arrangements, and permutation patterns within algebraic combinatorics.

Contribution

It provides a comprehensive overview of BP decompositions, emphasizing their applications in geometry and combinatorics of Schubert varieties and related structures.

Findings

BP decompositions facilitate classification of smooth Schubert varieties

They are instrumental in studying inversion hyperplane arrangements

They aid in permutation pattern avoidance analysis

Abstract

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 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 and permutation pattern avoidance. We survey many of these applications.