A bound for canonical dimension of the (semi)spinor groups

Abstract

Using the theory of nonnegative intersections, duality of the Schubert varieties, and a Pieri-type formula for the varieties of maximal, totally isotropic subspaces, we get an upper bound for the canonical dimension $\mathrm{cd}({\mathrm{Spin}}_n)$ of the spinor group ${\mathrm{Spin}}_n$. A lower bound is given by the canonical $2$-dimension ${\mathrm{cd}}_2({\mathrm{Spin}}_n)$, computed in [10]. If $n$ or $n+1$ is a power of $2$, no space is left between these two bounds; therefore, the precise value of $\mathrm{cd}({\mathrm{Spin}}_n)$ is obtained for such $n$. We also produce an upper bound for canonical dimension of the semispinor group (giving the precise value of the canonical dimension in the case when the rank of the group is a power of $2$) and show that spinor and semispinor groups are the only open cases of the question about canonical dimension of an arbitrary simple split group possessing a unique torsion prime