On permutation-invariant construction of glued lattices
Abstract
Given a permutation τ on n letters, we consider lattices spanned by an orbit of one vector x in ℝn under the action of τ by permutation of the coordinates. Such lattices generalize the important class of cyclic lattices and have previously been studied in~\cite{perm}, where a bound on their rank was established. We prove a sufficient condition on x for this bound to be achieved. We further investigate the structure of such permutation-invariant lattices, proving that they are glued by the permuted vector from the orthogonal cyclic blocks and giving a determinant formula for the lattice in terms of determinants of these blocks and the norm of the permuted vector. In the case x is an integer vector, these blocks are sublattices of the root lattices Ak in respective dimensions with root lattices themselves and their glued direct sums also realizable by this construction. We also exhibit a glued construction of permutation-invariant algebraic integral lattices from collections of cyclic number fields. Finally, we prove a strengthened version of a previous result of~\cite{lf_ek} on a related construction of well-rounded lattices spanned by sets of algebraic conjugates.
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BibTeX
@misc{bonini2026permutationinvariant,
title = {{On permutation-invariant construction of glued lattices}},
author = {Maria Fernanda Zordan Bonini and Lenny Fukshansky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20952},
eprint = {2609.20952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20952},
}
RIS
TY - UNPB
TI - On permutation-invariant construction of glued lattices
AU - Maria Fernanda Zordan Bonini
AU - Lenny Fukshansky
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20952
UR - https://arxiv.org/abs/2609.20952
AB - Given a permutation τ on n letters, we consider lattices spanned by an orbit of one vector x in ℝ^n under the action of τ by permutation of the coordinates. Such lattices generalize the important class of cyclic lattices and have previously been studied in~\cite{perm}, where a bound on their rank was established. We prove a sufficient condition on x for this bound to be achieved. We further investigate the structure of such permutation-invariant lattices, proving that they are glued by the permuted vector from the orthogonal cyclic blocks and giving a determinant formula for the lattice in terms of determinants of these blocks and the norm of the permuted vector. In the case x is an integer vector, these blocks are sublattices of the root lattices A_k in respective dimensions with root lattices themselves and their glued direct sums also realizable by this construction. We also exhibit a glued construction of permutation-invariant algebraic integral lattices from collections of cyclic number fields. Finally, we prove a strengthened version of a previous result of~\cite{lf_ek} on a related construction of well-rounded lattices spanned by sets of algebraic conjugates.
ER -