Molecules of Type-$B$ and Odd Type-$D$ Gelfand $\mathbf m$- and $\mathbf n$-Graphs

Abstract

Gelfand $W$-graphs provide multiplicity-free canonical models for classical Weyl groups, and their molecules record the connectivity generated by bidirected canonical-basis edges. Restricted Beissinger insertion and dual equivalence give a natural tableau language for these edges, while the wall and fork generators require additional local criteria. For both type-$B$ Gelfand $\mathbf m$- and $\mathbf n$-graphs, we prove that the molecules are exactly the connected components of explicit occurrence--wall graphs constructed from split modified Beissinger insertion. In odd type $D$, we likewise classify the $\mathbf m$-molecules by a completion-filtered occurrence--fork graph and obtain the $\mathbf n$-classification through the parity-sensitive type-$D$ duality.