{
  "abstract": "Let $F$ be a non-archimedean local field. For any irreducible component $C$ of a nilpotent variety of type $A$ let $π(C)$ be the irreducible representation of a general linear group over $F$ attached to it. We show that if the multisegment of $C$ is multiplicity free, $C$ is rigid if and only if $π(C)$ is $\\square$-irreducible. In this case, \\[ π(C*D)=soc\\bigl(π(C)\\timesπ(D)\\bigr),\\qquad π(D*C)=soc\\bigl(π(D)\\timesπ(C)\\bigr) \\] for every irreducible component $D$, where $*$ is the binary operation defined by Aizenbud and Lapid, in accordance with a conjecture of Lapid and Mínguez. We also give a computer-assisted example in $A_7$ of a non-simple rigid module $z\\ne0$ which does not admit a proper rigid submodule $y\\ne0$ such that $Ext_Π^1(z,z/y)=0$, answering negatively a question of Aizenbud and Lapid.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26764",
  "author_total": 2,
  "authors": [
    {
      "name": "Erez Lapid"
    },
    {
      "name": "Alberto M\\'ınguez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1qmp71naafv82.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26764",
  "fulltext_url": null,
  "id": "1qmp71naafv82",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26764",
  "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.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7223021780",
  "pdf_url": "https://arxiv.org/pdf/2610.26764",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1qmp71naafv82.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.26764",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26764"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Socle formulas for rigid irreducible components of Lusztig's nilpotent varieties in type $A$",
  "title_text": "Socle formulas for rigid irreducible components of Lusztig's nilpotent varieties in type A",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/1qmp71naafv82",
  "venue": "arXiv",
  "versions": []
}