{
  "abstract": "Let $\\mathfrak{g}$ be a complex semisimple Lie algebra and, for a dominant integral weight $λ$, let $Θ_λ$ be the multiplicity-free sum of the weights of the irreducible module $V(λ)$. The $Θ_μ$ form a $\\mathbb{Z}$-basis of the $W$-invariants, so there are unique integers $a(μ,λ)$, the atomic numbers, with $\\mathrm{ch}\\,V(λ) = \\sum_μa(μ,λ)Θ_μ$ over dominant $μ$. We ask when they are nonnegative. They are the Moebius transform of the weight-multiplicity function on the dominant-weight poset, which the crosscut theorem turns into an alternating sum of at most $2^{\\mathrm{rank}\\,Φ}$ multiplicities indexed by subsets of the covers of $μ$ in $[μ,λ]$. We prove that $a(μ,λ)\\ge 0$ whenever every connected component of the Dynkin diagram is a path, and identify the coefficient with the dimension of an explicit weight space: it is cut out of $V(λ)$ by alternately taking kernels of raising operators and cokernels of lowering operators, one per cover, in order along the path. Type $D_4$ shows the hypothesis is necessary. Put $β= α_1 + 2α_2 + 2α_3 + 2α_4$; for every dominant $μ$ with $\\langleμ,α_2^\\vee\\rangle = 1$ and $μ+β$ dominant, $a(μ,μ+β)$ is $-2$ if $\\langleμ,α_1^\\vee\\rangle = 0$ and $-1$ otherwise. Restriction to the support of $λ-μ$ carries this family into every irreducible type with a trivalent node. Thus an irreducible finite root system has all atomic numbers nonnegative exactly when its Dynkin diagram is a path, namely in types $A_n$, $B_n$, $C_n$, $F_4$, $G_2$. Deep in the dominant chamber, in an explicit range, $a(μ,λ)$ is the Kostant partition number of $λ-μ$ for the nonsimple positive roots; negative atomic numbers are therefore confined to boundary slabs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08024",
  "author_total": 2,
  "authors": [
    {
      "name": "Felipe de Mattos Chafik Hindi"
    },
    {
      "name": "Waldeck Schutzer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9c8bek7hwd497.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08024",
  "fulltext_url": null,
  "id": "9c8bek7hwd497",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08024",
  "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.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212040301",
  "pdf_url": "https://arxiv.org/pdf/2609.08024",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9c8bek7hwd497.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08024",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08024"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Atomic Decompositions of Lie Characters and the Dominant Weight Poset",
  "title_text": "Atomic Decompositions of Lie Characters and the Dominant Weight Poset",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9c8bek7hwd497",
  "venue": "arXiv",
  "versions": []
}