Chromatic characters and root multiplicities for Borcherds--Kac--Moody algebras

@misc{chakraborty2026chromatic,
  title = {{Chromatic characters and root multiplicities for Borcherds--Kac--Moody algebras}},
  author = {Priyanshu Chakraborty and Supriya Saha and R. Venkatesh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04325},
  eprint = {2610.04325},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04325},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Chromatic characters and root multiplicities for Borcherds--Kac--Moody algebras
AU  - Priyanshu Chakraborty
AU  - Supriya Saha
AU  - R. Venkatesh
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04325
UR  - https://arxiv.org/abs/2610.04325
AB  - 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.
ER  -