Evaluating SU(3) Verlinde sums using spectral graph theory
Abstract
We realize the SU(3) Verlinde sums Vn(m), up to an explicit factor, as the values at n of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on m2 vertices. For fixed m, we express their generating function in terms of the logarithmic derivative of an associated even spectral polynomial and express this polynomial as an explicit iterated resultant. Exploiting permutation symmetry, we prove that this polynomial is a cube over ℚ, apart from an explicit quadratic factor when 3| m. This factorization yields shorter linear recurrences satisfied by Vn(m) with constant coefficients. We also derive Binet-type formulas expressing Vn(m) as finite linear combinations of powers of rescaled inverse squares of the roots of the spectral polynomial. These results provide an efficient algorithm for computing these Verlinde sums. Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.
Related
- Resultants, Recursive Formulas and Binet Formulas for SU(N) Verlinde Sums
- A general theorem in spectral extremal graph theory
- Lemmas and Conjectures in Spectral Graph Theory
- The Spectral Theory of Cographs: A Survey, Organized by Graph Matrix
- Block-count constrained harmonic sums: spectral expansion and block-directed Euler-Maclaurin
BibTeX
@misc{jorgenson2026evaluating,
title = {{Evaluating $\mathrm SU(3)$ Verlinde sums using spectral graph theory}},
author = {Jay Jorgenson and Anders Karlsson and Lejla Smajlović},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28265},
eprint = {2609.28265},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28265},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Evaluating SU(3) Verlinde sums using spectral graph theory
AU - Jay Jorgenson
AU - Anders Karlsson
AU - Lejla Smajlović
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28265
UR - https://arxiv.org/abs/2609.28265
AB - We realize the SU(3) Verlinde sums V_n(m), up to an explicit factor, as the values at n of the spectral zeta function of a higher-order Laplace operator on the triangular discrete torus on m^2 vertices. For fixed m, we express their generating function in terms of the logarithmic derivative of an associated even spectral polynomial and express this polynomial as an explicit iterated resultant. Exploiting permutation symmetry, we prove that this polynomial is a cube over ℚ, apart from an explicit quadratic factor when 3| m. This factorization yields shorter linear recurrences satisfied by V_n(m) with constant coefficients. We also derive Binet-type formulas expressing V_n(m) as finite linear combinations of powers of rescaled inverse squares of the roots of the spectral polynomial. These results provide an efficient algorithm for computing these Verlinde sums. Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.
ER -