Billey-Postnikov posets, rationally smooth Schubert varieties, and Poincar\'e duality

TL;DR

This paper advances the understanding of Billey-Postnikov decompositions in Schubert varieties, linking pattern recognition, Poincaré duality, and rational smoothness, with new classifications and counterexamples in finite and infinite types.

Contribution

It develops the theory of BP decompositions, introduces a poset indexing, and applies these to classify rationally smooth Schubert varieties and their structure constants.

Findings

BP decompositions can be recognized by pattern conditions in finite type.

Rationally smooth, simply laced Schubert varieties have structure constants with a triangularity property.

Counterexamples show rationally smooth Schubert varieties in infinite type may lack Grassmannian BP decompositions.

Abstract

Billey-Postnikov (BP) decompositions govern when Schubert varieties $X(w)$ decompose as bundles of smaller Schubert varieties. We further develop the theory of BP decompositions and show that, in finite type, they can be recognized by pattern conditions and are indexed by the order ideals of a poset $\mathsf{bp}(w)$ that we introduce; we conjecture that this holds in any Coxeter group. We then apply BP decompositions to show that, when $X(w)$ is rationally smooth and $W$ simply laced, the Schubert structure constants $c_{uv}^w$ satisfy a triangularity property, yielding a canonical involution on the Schubert cells of $X(w)$ respecting Poincar\'{e} duality. We also classify the rationally smooth Bruhat intervals in finite type (other than $E$) which admit generalized Lehmer codes, answering questions and conjectures of Billey-Fan-Losonczy, Bolognini-Sentinelli, and Bishop-Mili\'{c}evi\'{c}-Thomas. Finally, we show that rationally smooth Schubert varieties in infinite type need not have Grassmannian BP decompositions, disproving conjectures of Richmond-Slofstra and Oh-Richmond.