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Diagonal Specht-ideal radicality conjecture

Let m,n≥1m,n\geq 1, let λ=(n−k,1k)\lambda=(n-k,1^k) be a hook partition of nn with 0≤k≤n−10\leq k\leq n-1, and let Rm,n=C[xr,j:1≤r≤m, 1≤j≤n]\mathcal{R}_{m,n}=\mathbb{C}[x_{r,j}:1\leq r\leq m,\ 1\leq j\leq n] carry the diagonal action of SnS_n. If Iλ(m)⊆Rm,nI_{\lambda}^{(m)}\subseteq\mathcal{R}_{m,n} is the ideal generated by the λ\lambda-isotypic component, then Iλ(m)I_{\lambda}^{(m)} is radical; equivalently, Iλ(m)=Iλ(m)I_{\lambda}^{(m)}=\sqrt{I_{\lambda}^{(m)}}.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Reduction to the square diagonal case

    For every partition λ⊢n\lambda\vdash n, the diagonal Specht ideal is radical for every number of blocks mm if and only if Iλ(n)⊆Rn,nI_{\lambda}^{(n)}\subseteq\mathcal{R}_{n,n} is radical.

    source: Escofet, Riener, and Verdure, “Diagonal Specht ideals and their varieties”

References

Primary source

arXiv

Additional references

commutative-algebrainvariant-theory
Posted · edited

Progress summary

Refreshed
Claimed progress

A new preprint narrows the cases where the conjecture may hold or fail, but does not settle it in general.

The conjecture concerns when diagonal Specht ideals are radical. No proposer or original date is identified in the retrieved material.

October 2026 structural results

Anna Escofet, Cordian Riener, and Hugues Verdure report explicit Gröbner bases for short hooks, and content, multidegree, and total-degree criteria proving non-radicality in sufficiently many variable blocks for non-hook cases. This is claimed progress from a preprint, not a complete resolution.

Current status (as of October 2026): Structural progress is claimed in a new preprint, while radicality for all hook partitions and parameters remains open.

Sources

Solutions 0

No solutions have been posted yet.