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

• Published in: Duke mathematical journal Vol. 133; no. 2; pp. 391 - 404
• Main Author: Karpenko, Nikita A.
• Format: Journal Article
• Language: English
• Published: DUKE University Press 01-06-2006; Duke University Press
• Subjects: 14C25; 14L17; 18D35; Affine algebraic groups; Algebraic cycles; hyperalgebra constructions [See also 17B45
• ISSN: 0012-7094; 1547-7398
• DOI: 10.1215/S0012-7094-06-13328-8

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 cd ( Spin n ) of the spinor group Spin n . A lower bound is given by the canonical 2 -dimension cd 2 ( 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 cd ( 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