A Sharp Multiplier-Dimension Threshold for Orthogonal-Dual Product Spaces

Abstract

Let $A$ be a finite-dimensional commutative symmetric Frobenius algebra over a field $k$, let $\dim_k A=2n$, and let $U\subset A$ be an $n$-dimensional subspace with orthogonal complement $V=U^\perp$. For a multiplier subspace $L\subset A$, write $LU$ for the span of all products $\ell u$ and put $\rho_U(L)=\dim_k(LU)$. When $1\in L$, we express $\rho_U(L)$ and $\rho_V(L)$ through the two nontrivial flattening ranks of the restricted multiplication tensor on $(L/k1)\times U\times V$. It follows that $\rho_U(L)=\rho_V(L)$ for every two-dimensional multiplier subspace containing $1$. We prove that this forced equality is sharp: in the purely inseparable extension $\F_2(s^2,t^2,u^2,v^2)\subset \F_2(s,t,u,v)$ 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.