Cayley form, comass, and triality isomorphisms

Mikhail G. Katz, Steve Shnider

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

Following an idea of Dadok, Harvey and Morgan, we apply the triality property of Spin(8) to calculate the comass of self-dual 4-forms on ℝ8. In particular, we prove that the Cayley form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.5)) is SO(8)-conjugate to the Cayley form. We also use triality to prove that the stabilizer in SO(8) of the Cayley form is Spin(7). The results have applications in systolic geometry, calibrated geometry, and Spin(7) manifolds.

Original languageEnglish
Pages (from-to)187-208
Number of pages22
JournalIsrael Journal of Mathematics
Volume178
Issue number1
DOIs
StatePublished - 2010

Bibliographical note

Funding Information:
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 188 2. The Cayley form . . . . . . . . . . . . . . . . . . . . . . . 191 3. Triality for D4 . . . . . . . . . . . . . . . . . . . . . . . . 194 4. Weight spaces in symmetric matrices and self-dual 4-forms 196 5. Proofs of Theorem 1.1 and Theorem 1.2 . . . . . . . . . . 200 6. Stabilizer of the Cayley form . . . . . . . . . . . . . . . . 203 7. A counterexample . . . . . . . . . . . . . . . . . . . . . . 204 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 ∗ Supported by the Israel Science Foundation (grants no. 84/03 and 1294/06) and the BSF (grant 2006393) Received December 24, 2007 and in revised form October 24, 2008

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