Staircase diagrams and enumeration of smooth Schubert varieties

TL;DR

This paper introduces staircase diagrams on graphs to enumerate smooth and rationally smooth Schubert varieties across classical types, extending previous work and providing a combinatorial framework for their classification.

Contribution

The authors define staircase diagrams on graphs and establish a bijection with Schubert varieties having complete BP decompositions, enabling enumeration in classical types.

Findings

Enumerated smooth and rationally smooth Schubert varieties in types A, B, C, D.

Introduced staircase diagrams as a new combinatorial tool.

Connected staircase diagrams to Schubert varieties via BP decompositions.

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.