{
  "abstract": "We give a complete characterization of equivariant multi-head self-attention (MHSA): if an MHSA layer is equivariant to a symmetry group $G$, then $G$ can only act by permuting head-clusters, with QK and OV matrices satisfying an equivariance constraint tied to the group action. As a consequence, we prove that any fixed MHSA architecture that achieves exact equivariance by polynomially parameterizing unconstrained MHSA parameters inevitably leads to expressivity loss within the class of equivariant maps: the equivariance locus of unconstrained MHSA forms a union of extremely many Zariski-irreducible components in a reduced parameter space, and any single architecture covers at most one. For $G=D_4$ acting on $C$ copies of the regular representation as the token feature space, we show that there are $Ω(C^{64})$ components for eight attention heads. We illustrate this with experiments on artificial tasks and show that the predicted head-cluster structures can arise naturally when the attention layer is pressured to become equivariant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30417",
  "author_total": 2,
  "authors": [
    {
      "name": "Tīkun Ông"
    },
    {
      "name": "Georg Bökman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/340xvsyv0n606.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30417",
  "fulltext_url": null,
  "id": "340xvsyv0n606",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30417",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "1707|Advanced Neural Network Applications"
    ]
  ],
  "openalex": "W7204883366",
  "pdf_url": "https://arxiv.org/pdf/2608.30417",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/340xvsyv0n606.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2608.30417",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30417"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Inherent Trade-offs when Enforcing Correct Data Symmetries in Multi-head Self-attention",
  "title_text": "Inherent Trade-offs when Enforcing Correct Data Symmetries in Multi-head Self-attention",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/340xvsyv0n606",
  "venue": "arXiv",
  "versions": []
}