{
  "abstract": "We study the diagonal Specht ideals $I_λ\\subseteq\\mathcal{R}_{m,n}:=\\mathbb{C}[\\mathbf{x}_1,\\dots,\\mathbf{x}_m]$, $\\mathbf{x}_r=(x_{r,1},\\dots,x_{r,n})$, generated by the $λ$-isotypic component of $\\mathcal{R}_{m,n}$ for the diagonal action of $S_n$. We characterize the set of zeros of $I_λ$ as $V_λ=\\bigcup_{μ\\not\\trianglelefteqλ}H_μ$ and prove that $λ\\mapsto V_λ$ is an isomorphism of posets from $(\\mathcal{P}_n,\\trianglelefteq)$ to $(\\{V_λ:λ\\in \\mathcal{P}_n\\},\\supseteq)$ for all $m$, while $λ\\mapsto I_λ$ is an isomorphism of posets from $(\\mathcal{P}_n,\\trianglelefteq)$ to $(\\{I_λ:λ\\in \\mathcal{P}_n\\},\\subseteq)$ if and only if $m=1$ or $n\\leq3$. 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 $m\\geq\\operatorname{len}(λ)$. Using the $\\operatorname{GL}_m(\\mathbb{C})$-action on $\\mathcal{R}_{m,n}$, we show that $I_λ$ is radical for all $m$ if and only if it is radical for $m=n$. Finally, for $m\\geq2$, we prove that $\\mathcal{R}_{m,n}/I_λ$ is Cohen-Macaulay if and only if $λ=(n)$ or $λ=(n-1,1)$, and the same is true for $\\mathcal{R}_{m,n}/\\operatorname{rad}(I_λ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07955",
  "author_total": 3,
  "authors": [
    {
      "name": "Anna Escofet"
    },
    {
      "name": "Cordian Riener"
    },
    {
      "name": "Hugues Verdure"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46zz9f2dp8s7m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07955",
  "fulltext_url": null,
  "id": "46zz9f2dp8s7m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07955",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7220959163",
  "pdf_url": "https://arxiv.org/pdf/2610.07955",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46zz9f2dp8s7m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07955",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07955"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Diagonal Specht ideals and their varieties",
  "title_text": "Diagonal Specht ideals and their varieties",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/46zz9f2dp8s7m",
  "venue": "arXiv",
  "versions": []
}