Abstract:

The torsion index of an algebraic group G can be defined as the splitting index of a generic G-torsor E. In the case of the spin group G=\Spin(2n+1), this is a 2-power 2^t with an exponent t determined in 2005 by Burt Totaro. The critical exponent i(t) of E is the exponent of the {\em partial splitting index} 2^{i(t)} (also called isotropy index) of E given by the t-th vertex of the Dynkin diagram of G. It plays a central role in the theory of all isotropy indexes of E, has been studied a lot, and was determined for many n in the recent years. The avant-critical exponent i(t-1), corresponding to the (t-1)st vertex of the Dynkin diagram, was also successfully determined for many n (asymptotically for 100 percent of them) and always turned out to be equal to t-1. Here we find the first n (namely, n=16) breaking this apparent rule. The result is based on several previous joint and solo works of the author as well as on some general additions developed on the occasion.