The gamma filtrations for the spin groups

Yagita’s counter-examples and beyond

“…Counter-examples to Conjecture 1.2 have been constructed with G=Spin (17) and G=Spin (19) by N. Yagita in [19]. Later, Yagita's counter-example for Spin (17) has been modified, simplified, and extended to the base field of arbitrary characteristic in [11].…”

“…Later, Yagita's counter-example for Spin (17) has been modified, simplified, and extended to the base field of arbitrary characteristic in [11]. However, an attempt to treat Spin (19) failed at that time.…”

“…In the present paper, we successfully treat Spin (19) by involving a new ingredient -a Pieri type formula for K-theory of highest orthogonal grassmannians. The Pieri formula (3.12) we need is formulated in [4,Theorem 1.2] in a combinatorial way.…”

“…However, it remains widely open for every of the remaining values of n. One of the obstacles to extend the counter-examples to n=11 is the "drop" to 2 5 of the torsion index of Spin(23): the torsion indexes of Spin (19) and Spin(21) are 2 4 and 2 5 , see [16]. (The similar drop for Spin (19) was also the origin of the difficulties with this case.) Generally speaking, it seems that every n needs an individual treatment.…”