On 14-dimensional quadratic forms, their spinors, and the difference of two octonion algebras

by Markus Rost (Preprint, May 94, 5 pages)

The isotropy group of a generic 14-dimensional spinor is G₂×G₂. This fact leads to the following consequence:

A 14-dimensional quadratic form with trivial discriminant and trivial Hasse-Witt invariant is isomorphic to the trace of the pure subform of a 3-fold Pfister form (times a trace zero element) over some quadratic extension.

The text originates from an email in May 1994. It has been revised on September 18, 1996 and March 9, 1999 (but still may contain some inaccuracies).

Full text: [tex] [pdf]

Related 1:

At first I erroneously claimed that any such form is the difference of two 3-fold Pfister forms (the special case of a split quadratic extension). For counterexamples see:

• [Corollary 17.5, page 690] Izhboldin, Oleg T.; Karpenko, Nikita A.. Some new examples in the theory of quadratic forms. Math. Z. 234 (2000), no. 4, 647-695. MR 1778404, Zbl 0979.11025.
• Hoffmann, Detlev W.; Tignol, Jean-Pierre. On 14-dimensional quadratic forms in I³, 8-dimensional forms in I², and the common value property. Doc. Math. 3 (1998), 189-214. MR 1647511, Zbl 0914.11014.

Related 2:

About the isotropy group G₂×G₂ of a generic 14-dimensional spinor:

• [page 281] Igusa, Jun-ichi. On a certain class of prehomogeneous vector spaces. J. Pure Appl. Algebra 47 (1987), no. 3, 265-282. MR 910424, Zbl 0633.14030.
• Generic stabilizer in Spin14. Note (March 31, 2019) by Skip Garibaldi.

Related 3:

For detailed expositions of the main results of this and the next paper see:

• Garibaldi, Skip. Cohomological invariants: exceptional groups and spin groups. With an appendix by Detlev W. Hoffmann. Mem. Amer. Math. Soc. 200 (2009), no. 937, xii+81 pp. MR 2528487, Zbl 1191.11009. (Notes for the Lens mini-cours, June 2006)

On the Galois cohomology of Spin(14)

by Markus Rost (Preprint, March 99, 17 pages)

With corrections/additions of May/June 2006

For i = 6, 7 we define invariants h_i: H¹(k,Spin₁₄) → Hⁱ(k,Z/2) (if -1 is a square).
Further we show that the natural map H¹(k,H) → H¹(k,Spin₁₄) is surjective, where H is the normalizer of G₂×G₂ in Spin₁₄.
One concludes that the essential dimension of Spin₁₄ is equal to 7.
Similar considerations are done for Spin₁₂.
For Spin₁₃ we use the results about Spin₁₄ and some special computations.
We present the list of essential dimensions of the split groups Spin_n for n less or equal to 14.

Full text (June 6, 2006): [tex] [pdf]

Go to: Publications and Preprints · Markus Rost's Web Page