The canonical dimension formula for special orthogonal groups

Let $\operatorname{SO}_{2m-1}$ and $\operatorname{SO}_{2m}$ denote the special orthogonal groups of the indicated dimensions, and let $\operatorname{cd}(G)$ denote the canonical dimension of an algebraic group $G$. Canonical dimension conjecture.

$$\operatorname{cd}(\operatorname{SO}_{2m-1}) = \operatorname{cd}(\operatorname{SO}_{2m}) = \frac{m(m-1)}{2}$$

for every $m \ge 1$.

This gives an explicit formula for the canonical dimensions of the two adjacent families of special orthogonal groups. The conjecture was proved by Karpenko, with an alternative proof due to Vishik.