Séminaire Lotharingien de Combinatoire, 95B.78 (2026), 12 pp.
Christian Gaetz and Yibo Gao
Billey-Postnikov Posets
Abstract.
Billey-Postnikov (BP) decompositions govern when Schubert
varieties X(w) are 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 bp(w) that we
introduce and study. We then give three new applications of BP
decompositions: we show that when W is of finite simply laced
type and X(w) is rationally smooth, Schubert structure
constants have a triangularity property which gives a canonical
bijection realizing rank-symmetry of [e,w]; we classify
the finite types (other than E) for which rationally smooth
Bruhat intervals admit generalized Lehmer codes, answering
questions and conjectures of Billey-Fan-Losonczy,
Bolognini-Sentinelli, and Bishop-Milićević-Thomas; and
we show that rationally smooth Schubert varieties in infinite type
need not have Grassmannian BP decompositions, disproving conjectures
of Richmond-Slofstra and Oh-Richmond.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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