Secant dimensions and bilinear decomposition widths for finite Heisenberg groups

Abstract

Let $G=UT_3(\F_q)$ be the finite Heisenberg group. We study the image closures and secant varieties of a family of equivariant bilinear maps between representations induced from central characters. Evaluation at a basepoint identifies the ambient space with $M_{q^2}(\C)$. A single channel is represented by $A^T\operatorname{diag}(d)B$, with $A$ and $B$ in prescribed twisted-convolution algebras. We prove that its affine image closure has dimension $3q^2-2$ for every central-character pair and that its affine second-secant closure has the expected dimension for every prime power $q$. For $q\ge3$, the proof constructs two differential images with zero intersection. We also prove a subgroup-lifting theorem for channel parameterizations on finite abelian groups with bicharacter coefficients: a parameter point with differential rank $r(3|H|-2)$ on a subgroup $H$ extends by zero to a parameter point with differential rank $r(3|K|-2)$ on the ambient group $K$, provided that every parameter vector is nonzero. Exact Jacobian certificates determine all secant dimensions for $q=2,3,4,5$, with generic widths $2,4,6,9$. The lifting theorem yields specified higher-secant dimensions on infinite families of extension fields. Attainment of the dimension lower bound for arbitrary $q$ remains a conjecture.