Singular lines of trilinear forms

✍ Scribed by Jan Draisma; Ron Shaw

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 147 KB
• Volume: 433
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2010.03.040

✦ Synopsis

We prove that an alternating e-form on a vector space over a quasialgebraically closed field always has a singular (e -1)-dimensional subspace, provided that the dimension of the space is strictly greater than e. Here an (e -1)-dimensional subspace is called singular if pairing it with the e-form yields zero. By the theorem of Chevalley and Warning our result applies in particular to finite base fields.

Our proof is most interesting in the case where e = 3 and the space has odd dimension n; then it involves a beautiful equivariant map from alternating trilinear forms to polynomials of degree n-1 2 -1. We also give a sharp upper bound on the dimension of subspaces all of whose two-dimensional subspaces are singular for a non-degenerate trilinear form. In certain binomial dimensions the trilinear forms attaining this upper bound turn out to form a single orbit under the general linear group, and we classify their singular lines.