Submission: 2004, Dec 9
Using the theory of non-negative intersections, duality of the Schubert varieties, and Pieri-type formula for a maximal orthogonal grassmannian, we get an upper bound for the canonical dimension $cd(Spin_n)$ of the spinor group $Spin_n$. A lower bound is given by the canonical $2$-dimension $cd_2(Spin_n)$. If $n$ or $n+1$ is a power of $2$, no space is left between these two bounds; therefore the precise value of $cd(Spin_n)$ is obtained for such $n$.
In the appendix, we also produce an upper bound for canonical dimension of the semi-spinor group (giving the precise value of the canonical dimension in the case when the rank of the group is a power of $2$), compute canonical dimension of the projective orthogonal group, and show that the spinor group represents the unique difficulty when trying to compute the canonical dimension of an arbitrary simple split group, possessing a unique torsion prime.
2000 Mathematics Subject Classification: 14L17, 14C2
Keywords and Phrases: Algebraic groups, projective homogeneous varieties, Chow groups
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