Let A be a finite-dimensional commutative symmetric Frobenius algebra over a field k, let ₖ A=2n, and let U A be an n-dimensional subspace with orthogonal complement V=U^. For a multiplier subspace L A, write LU for the span of all products u and put U (L) =ₖ (LU). When 1 L, we express U (L) and V (L) through the two nontrivial flattening ranks of the restricted multiplication tensor on (L/k1) U V. It follows that U (L) =V (L) for every two-dimensional multiplier subspace containing 1. We prove that this forced equality is sharp: in the purely inseparable extension ₂ (s², t², u², v²) ₂ (s, t, u, v) there is a nonsimilar orthogonal-dual pair of eight-dimensional subspaces and a three-dimensional multiplier subspace L for which the two product dimensions are 15 and 16. We also describe the failure loci as determinantal strata on Grassmannians and show that multiplicative closure of a separating multiplier space can erase the defect. Thus unrestricted multiplier-subspace geometry and intermediate-field profiles sample genuinely different information.
Ueoka et al. (Sun,) studied this question.