Staircase diagrams and enumeration of smooth Schubert varieties

2015/10/20 by Edward Richmond, William Slofstra · 12 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Bijection #Combinatorics #Computer science #Decomposition #Enumeration #Graph #Mathematics #Schubert polynomial #Schubert variety #Set (abstract data type) #Type (biology) #math.AG #math.CO #msc:05A15 #msc:14M15 #msc:14N15 #msc:20F55 #msc:22E40

paper · pdf · doi:10.1016/j.jcta.2017.03.009

published in Journal of Combinatorial Theory Series A 150, 328-376 (Elsevier BV) · 42 pages, 3 tables

arxiv created 2015/10/20 · openalex publication_date 2017/04/01 · arxiv updated 2018/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We enumerate smooth and rationally smooth Schubert varieties in the classical finite types A, B, C, and D, extending Haiman's enumeration for type A. To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin-Coxeter graph, the set of "nearly-maximally labelled" staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey-Postnikov (BP) decomposition. We can then use an earlier result of the authors showing that all finite-type rationally smooth Schubert varieties have a complete BP decomposition to finish the enumeration.

Citations

• Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties
• Representations of Coxeter groups and Hecke algebras
• Rationally smooth elements of Coxeter groups and triangle group avoidance

Cited by

• Pattern avoidance and fiber bundle structures on Schubert varieties