Specht ideals for the imprimitive complex reflection group G(r,1,n): geometric orbit types and a dominance theorem
Abstract
Moustrou, Riener, and Verdure related the dominance order on partitions to Specht ideal and variety inclusion for Sn; we previously extended this to direct products of symmetric groups. Here we take a first step toward the imprimitive complex reflection group G(r,1,n)=(ℤ/rℤ)wr Sn. The correct geometric invariant of a point x∈ Kn is a pair (μ,z): a classical partition μ on the r-th powers of the nonzero coordinates, together with the number z of zero coordinates. We construct a G(r,1,n)-invariant ideal I(μ,z) for each such pair, and show that a dominance order on (μ,z), with z taking strict priority over μ, exactly characterizes both ideal inclusion and reverse variety inclusion. We settle a monomial criterion in three regimes, recover the classical Bn-Specht ideal theory at r=2 when z=0, and establish radicality of I(μ,z) at the two extremes of the z=0 level. Four computational examples, checked exhaustively for group invariance, illustrate the construction throughout.
Related
- Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties
- Walk/Zeta correspondence for quantum walks on path graphs: a pre-geometric structure implied by a generalized reflection principle
- Brown-Halmos type theorems and complex reflection groups
- Amitsur groups of Imprimitive Fano Threefolds
- Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities
BibTeX
@misc{nonkane2026specht,
title = {{Specht ideals for the imprimitive complex reflection group $G(r,1,n)$: geometric orbit types and a dominance theorem}},
author = {Ibrahim Nonkané and Joel Tossa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29683},
eprint = {2609.29683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29683},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Specht ideals for the imprimitive complex reflection group G(r,1,n): geometric orbit types and a dominance theorem
AU - Ibrahim Nonkané
AU - Joel Tossa
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29683
UR - https://arxiv.org/abs/2609.29683
AB - Moustrou, Riener, and Verdure related the dominance order on partitions to Specht ideal and variety inclusion for S_n; we previously extended this to direct products of symmetric groups. Here we take a first step toward the imprimitive complex reflection group G(r,1,n)=(ℤ/rℤ)wr S_n. The correct geometric invariant of a point x∈ K^n is a pair (μ,z): a classical partition μ on the r-th powers of the nonzero coordinates, together with the number z of zero coordinates. We construct a G(r,1,n)-invariant ideal I_(μ,z) for each such pair, and show that a dominance order on (μ,z), with z taking strict priority over μ, exactly characterizes both ideal inclusion and reverse variety inclusion. We settle a monomial criterion in three regimes, recover the classical B_n-Specht ideal theory at r=2 when z=0, and establish radicality of I_(μ,z) at the two extremes of the z=0 level. Four computational examples, checked exhaustively for group invariance, illustrate the construction throughout.
ER -