Upper Bounds on the Torsion Index of Half-Spin Groups
Sanghoon Baek, Rostislav Devyatov
TL;DR
This work investigates the torsion index $\tau(G)$ for half-spin groups $\operatorname{HSpin}(2n)$, building on Totaro's computations for split spin groups. By leveraging the Chow rings of two-step flag varieties of type $D_n$, Weyl-invariant structures, and explicit subring relations, the authors bound $\tau(\operatorname{HSpin}(2n))$ in terms of $\tau(\operatorname{Spin}(2n))$, revealing that in most cases $\tau(\operatorname{HSpin}(2n))\le 2\tau(\operatorname{Spin}(2n))$ with certain exceptional $n$ allowing a $2^3$ multiplier; numerous instances exhibit equality with the spin case. The main results include precise interval-type equalities and exact values for small $n$ (e.g., $\tau(\mathrm{HSpin}(12))=2^2$ and $\tau(\mathrm{HSpin}(16))=2^6$), and a framework based on Totaro-decomposable subsets to compare top-degree elements across invariant subrings. The methodological contribution provides a general mechanism to bound torsion indices for split simple groups beyond the previously resolved spin and $E_8$ cases, with potential applications to related torsion problems in algebraic groups.
Abstract
The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of torsion indexes for split simple groups. We present general upper bounds for the torsion index of half-spin groups, showing that, except for certain exceptional cases, it is at most twice that of the corresponding spin groups. For these exceptional cases, the torsion index is bounded above by at most $2^3$ times that of the spin groups. Our results also reveal that in many cases, the torsion index of half-spin groups coincides with that of the spin groups.