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, in type $D_r$ with $r$ odd, their number is $2^r-1$. We prove a stronger structural statement. For $r\geq 5$ odd, the rational Weyl group elements in $W(D_r)$ are exactly the longest element $w_0$ together with two explicitly described signed cyclic elements $c_I$ and $d_I$ for every non-empty subset $I\subseteq\{1,\ldots,r-1\}$. Consequently the rationality graph $Γ(D_r)$ is two explicitly labelled Boolean-type halves glued at $w_0$, its number of vertices is $2^r-1$, and its only vertices of valency one are $c_{\{1\}}$ and $d_{\{1\}}$. The proof combines an acyclic two-level description of the rationality graphs $Γ(c_I)$ with a rigidity argument for all one-step rational descents from $w_0$. The latter uses Voloshyn's descent lemma, while all type-$D$ exclusions are given by explicit loops or two-cycles in the root-poset rationality graph.