Secant dimensions and bilinear decomposition widths for finite Heisenberg groups
Abstract
Let G=UT3(Fq) 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 Mq2(C). A single channel is represented by ATdiag(d)B, with A and B in prescribed twisted-convolution algebras. We prove that its affine image closure has dimension 3q2−2 for every central-character pair and that its affine second-secant closure has the expected dimension for every prime power q. For q≥3, 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.
Related
- Bilinear Korányi spherical maximal function on the Heisenberg group
- Bilinear maximal functions associated with some degenerate surfaces on the Heisenberg group
- Effective syzygies of secant varieties
- Multicrossing complex of knot and secant classes
- Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type
BibTeX
@misc{van2026secant,
title = {{Secant dimensions and bilinear decomposition widths for finite Heisenberg groups}},
author = {Nguyen Bich Van and Nguyen Trung Kien},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01104},
eprint = {2611.01104},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01104},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Secant dimensions and bilinear decomposition widths for finite Heisenberg groups
AU - Nguyen Bich Van
AU - Nguyen Trung Kien
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01104
UR - https://arxiv.org/abs/2611.01104
AB - 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^Tdiag(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≥3, 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.
ER -