Karpenko's Conjecture on the Special Schubert Class in the Gamma Filtration Fails for n = 8, 9 and 11: A Short Consequence of Printed Results

Let \(\varphi\) be a non-degenerate quadratic form of dimension \(2n+1\) over a field \(F\) whose even Clifford algebra \(C_0(\varphi)\) is a division algebra, let \(X\) be its maximal orthogonal Grassmannian and \(S\) the Severi–Brauer variety of \(C_0(\varphi)\). N. Karpenko conjectured (Math. Z. 285 (2017), Conjecture 4.7, also Conjecture 3 of his abstract in Oberwolfach Report 5/2016) that for \(n\ge8\) the class \(e_n\)\({}\in\operatorname{K}(X_{F(S)})\) of the special Schubert variety of lowest dimension does not lie in the term number \(n+1-[(n+1)/2]\) of the gamma filtration. We observe that the conjecture is false: over every field of characteristic \(0\), for \(n=8\), \(9\) and \(11\) and for every such \(\varphi\), the class \(e_n\) lies in this term (term \(5\) for \(n=8,9\), term \(6\) for \(n=11\)), and such forms exist. This is a short consequence of printed results, not found stated in print. The proof is Karpenko's own printed argument that Conjecture 4.7 implies his Conjecture 3.3 on indexes of orthogonal Grassmannians, written out in a general form with two implicit steps filled in, combined with Totaro's computation of the torsion indexes of \(\operatorname{Spin}(20)\) and \(\operatorname{Spin}(24)\) (for \(n=9,11\)) and with Karpenko's theorem on \(\operatorname{Spin}(18)\)-torsors (for \(n=8\)). A computer-assisted supplement, kept separate from the proof, finds by an exact computation in the Grothendieck ring of the split Grassmannian that \(e_8\) lies in the seventh but not in the eighth gamma term, and \(e_9\) in the eighth but not in the ninth. The cases \(n=10\) and \(n=12\) are not decided here, nor is any \(n\ge13\). No printed answer to the question was found in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note.