Completing the Reduced Word Criterion in Finite Weyl Types

Hodges and Yong [‘Coxeter combinatorics and spherical Schubert geometry’, J. Lie Theory 32 (2022), 447–474] proposed a reduced word criterion for Levi spherical Schubert varieties. It was proved in type A and in rank two, while counterexamples were found in types D 4 D₄ upper D 4 and F 4 F₄ upper F 4 . A later standard Coxeter element criterion settled Levi sphericity in every finite Lie type, but whether the original criterion remained valid in types B and C was still open. We prove that the two criteria agree in both families. These are the last unresolved infinite Weyl families. Consequently, the original reduced word criterion characterises Levi sphericity for every Weyl group element and every allowed Levi subgroup precisely in types A , B , C and G 2 G₂ upper G 2 . The paper completes the classification of irreducible finite Weyl types for which the original criterion is valid.