{
  "abstract": "The main aim of this article is to obtain explicit formulas for the root multiplicities of a class of finite-rank symmetrizable Borcherds-Kac-Moody algebras g(A) whose real part is of type A. The Dynkin diagram of each such algebra is obtained from the disjoint union of a Dynkin diagram of type A and a finite simple graph G_im, whose vertices correspond to imaginary simple roots, by choosing one vertex in each and joining the chosen vertices by a single edge of multiplicity c>0. More generally, we consider the same construction with the type A diagram replaced by the Dynkin diagram of an arbitrary finite-rank symmetrizable Kac-Moody algebra g_re. The resulting symmetrizable Borcherds-Kac-Moody algebra g(A) has real part g_re and imaginary part represented by G_im. Our starting point is a graph-theoretic description of the quotient of the denominator products of g(A) and g_re. This quotient is a multivariate independence polynomial of G_im, with the variable corresponding to the attachment vertex weighted by normalized highest-weight characters of g_re. We prove that the coefficients of powers of this quotient factor into a generalized chromatic polynomial and a tensor-power weight multiplicity. Using this, we obtain explicit formulas for all root multiplicities with nonzero imaginary component in terms of generalized chromatic polynomials of G_im and weight multiplicities in tensor products of highest-weight modules of g_re. When G_im is chordal, these formulas specialize to finite divisor sums involving products of binomial coefficients; paths and complete graphs provide particular examples. In particular, we obtain explicit formulas for the root multiplicities of g(A) when its real part g_re is of type A.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04325",
  "author_total": 3,
  "authors": [
    {
      "name": "Priyanshu Chakraborty"
    },
    {
      "name": "Supriya Saha"
    },
    {
      "name": "R. Venkatesh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46q4d5s79c9wc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04325",
  "fulltext_url": null,
  "id": "46q4d5s79c9wc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04325",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7220692896",
  "pdf_url": "https://arxiv.org/pdf/2610.04325",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46q4d5s79c9wc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04325",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04325"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Chromatic characters and root multiplicities for Borcherds--Kac--Moody algebras",
  "title_text": "Chromatic characters and root multiplicities for Borcherds--Kac--Moody algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/46q4d5s79c9wc",
  "venue": "arXiv",
  "versions": []
}