Cyclic group tensors and the volume ratio of projective tensor products
Abstract
We revisit the problem of estimating the norm of the natural embedding from the Euclidean space ℓ2n3 into the projective tensor product ℓpn⊗πℓqn⊗πℓrn in the range 1≤ p≤ q≤ 2≤ r≤∞, studied by Giladi, Prochno, Schütt, Tomczak-Jaegermann and Werner [J. Funct. Anal. 273 (2017), 471-495]. Using convolution on finite cyclic groups, we establish new lower bounds that improve their estimates and determine the exact asymptotic order of the embedding norm whenever 1/p+1/q+1/r≥ 2. We then apply an estimate of Boedihardjo for the injective norm of random tensors to a randomly signed cyclic convolution tensor and use multilinear interpolation to determine the optimal exponent throughout the entire parameter range, up to logarithmic factors. As consequences, we obtain corresponding improvements and asymptotically sharp estimates, up to logarithmic factors, for the volume ratio of these projective tensor products.
Related
- Characterizing multigraded regularity and virtual resolutions on products of projective spaces
- Cyclic ABC Massey Products
- Direct proofs for spaces with cyclic parallel Ricci tensor
- Nef Hattori classes and products of projective lines
- Resultant multiplicity via projective degrees and applications to tensor eigenvalues
BibTeX
@misc{prochno2026cyclic,
title = {{Cyclic group tensors and the volume ratio of projective tensor products}},
author = {Joscha Prochno},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29524},
eprint = {2610.29524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29524},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cyclic group tensors and the volume ratio of projective tensor products
AU - Joscha Prochno
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29524
UR - https://arxiv.org/abs/2610.29524
AB - We revisit the problem of estimating the norm of the natural embedding from the Euclidean space ℓ_2^n^3 into the projective tensor product ℓ_p^n⊗_πℓ_q^n⊗_πℓ_r^n in the range 1≤ p≤ q≤ 2≤ r≤∞, studied by Giladi, Prochno, Schütt, Tomczak-Jaegermann and Werner [J. Funct. Anal. 273 (2017), 471-495]. Using convolution on finite cyclic groups, we establish new lower bounds that improve their estimates and determine the exact asymptotic order of the embedding norm whenever 1/p+1/q+1/r≥ 2. We then apply an estimate of Boedihardjo for the injective norm of random tensors to a randomly signed cyclic convolution tensor and use multilinear interpolation to determine the optimal exponent throughout the entire parameter range, up to logarithmic factors. As consequences, we obtain corresponding improvements and asymptotically sharp estimates, up to logarithmic factors, for the volume ratio of these projective tensor products.
ER -