Rational Weyl group elements of odd type D

Abstract

Voloshyn introduced rational Weyl group elements in connection with rational normal forms on complex reductive groups and conjectured that their number in type $D_r$, for odd $r$, is $2^r-1$. We prove a stronger structural statement. For every odd integer $r\ge 5$, the rational elements of $W(D_r)$ are precisely the longest element $w_0$ and two explicitly described signed cyclic elements $c_I,d_I$ for each non-empty subset $I\subseteq \{1,\dots ,r-1\}$. Consequently, the rationality graph $\Gamma (D_r)$ is obtained by gluing two explicitly labelled subset-toggle graphs at $w_0$; it has $2^r-1$ vertices, and its only vertices of valency one are $c_{\{1\}}$ and $d_{\{1\}}$. The proof combines a two-level acyclic description of the root-poset graphs ${\Gamma }_{c_I}$ with a rigidity theorem for simple left multiplications of the signed cyclic family. A self-contained simply-laced reflection-preservation lemma and a terminal-layer argument provide the descent step, while every forbidden one-step move from the family is excluded by an explicit loop or two-cycle.