Diagonal Specht ideals and their varieties
Abstract
We study the diagonal Specht ideals Iλ⊆ℛm,n:=ℂ[x1,…,xm], xr=(xr,1,…,xr,n), generated by the λ-isotypic component of ℛm,n for the diagonal action of Sn. We characterize the set of zeros of Iλ as Vλ=⋃μnottrianglelefteqλHμ and prove that λ↦ Vλ is an isomorphism of posets from (𝒫n,trianglelefteq) to ({Vλ:λ∈ 𝒫n},⊇) for all m, while λ↦ Iλ is an isomorphism of posets from (𝒫n,trianglelefteq) to ({Iλ:λ∈ 𝒫n},⊆) if and only if m=1 or n≤3. Unlike the case m=1, where Specht ideals are always radical, radicality in the diagonal setting depends on λ and m. We prove radicality for hook partitions of length at most three by providing explicit Gröbner bases, and we develop three criteria for non-radicality via content, multidegree, and total degree, showing in particular that Iλ is not radical for any non-hook partition λ when m≥len(λ). Using the GLm(ℂ)-action on ℛm,n, we show that Iλ is radical for all m if and only if it is radical for m=n. Finally, for m≥2, we prove that ℛm,n/Iλ is Cohen-Macaulay if and only if λ=(n) or λ=(n−1,1), and the same is true for ℛm,n/rad(Iλ).
Related
- Specht ideals for the imprimitive complex reflection group G(r,1,n): geometric orbit types and a dominance theorem
- Diagonal F-threshold of binomial edge ideals
- Spectral comparison on Specht modules and mixing of the interchange process
- Nonvanishing higher Specht polynomials and a construction for three row and hook shape Garsia--Procesi modules
- Concept lattices of Q-relations via diagonal and back-diagonal categories
BibTeX
@misc{escofet2026diagonal,
title = {{Diagonal Specht ideals and their varieties}},
author = {Anna Escofet and Cordian Riener and Hugues Verdure},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07955},
eprint = {2610.07955},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07955},
}
RIS
TY - UNPB
TI - Diagonal Specht ideals and their varieties
AU - Anna Escofet
AU - Cordian Riener
AU - Hugues Verdure
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07955
UR - https://arxiv.org/abs/2610.07955
AB - We study the diagonal Specht ideals I_λ⊆ℛ_m,n:=ℂ[x_1,…,x_m], x_r=(x_r,1,…,x_r,n), generated by the λ-isotypic component of ℛ_m,n for the diagonal action of S_n. We characterize the set of zeros of I_λ as V_λ=⋃_μnottrianglelefteqλH_μ and prove that λ↦ V_λ is an isomorphism of posets from (𝒫_n,trianglelefteq) to ({V_λ:λ∈ 𝒫_n},⊇) for all m, while λ↦ I_λ is an isomorphism of posets from (𝒫_n,trianglelefteq) to ({I_λ:λ∈ 𝒫_n},⊆) if and only if m=1 or n≤3. Unlike the case m=1, where Specht ideals are always radical, radicality in the diagonal setting depends on λ and m. We prove radicality for hook partitions of length at most three by providing explicit Gröbner bases, and we develop three criteria for non-radicality via content, multidegree, and total degree, showing in particular that I_λ is not radical for any non-hook partition λ when m≥len(λ). Using the GL_m(ℂ)-action on ℛ_m,n, we show that I_λ is radical for all m if and only if it is radical for m=n. Finally, for m≥2, we prove that ℛ_m,n/I_λ is Cohen-Macaulay if and only if λ=(n) or λ=(n−1,1), and the same is true for ℛ_m,n/rad(I_λ).
ER -