Sums of triangular numbers from the Frobenius determinant

TL;DR

This paper links affine superalgebra identities to classical determinant evaluations, deriving formulas for counting representations of numbers as sums of specific triangular numbers, extending prior mathematical results.

Contribution

It establishes a connection between affine superalgebra identities and classical Frobenius determinants, providing new formulas for sums of triangular numbers.

Findings

Derived exact formulas for representations as sums of 4m^2/d triangles

Extended results of Getz, Mahlburg, Milne, and Zagier

Connected algebraic identities to classical determinant evaluations

Abstract

We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4m^2/d triangles, whenever d divides 2m, and 4m(m+1)/d triangles, when d divides 2m or d divides 2m+2. This extends recent results of Getz and Mahlburg, Milne, and Zagier.