A vertex operator reformulation of the Kanade-Russell conjecture modulo 9

TL;DR

This paper reformulates the Kanade-Russell conjecture modulo 9 using vertex operators for certain algebraic modules, leading to new partition theorems related to affine Lie algebra types.

Contribution

It introduces a vertex operator approach to the conjecture and derives new partition theorems, expanding the algebraic framework of the problem.

Findings

Reformulation of the conjecture via vertex operators

Derivation of three new partition theorems

Connections to existing results by Andrews-van Ekeren-Heluani

Abstract

We reformulate the Kanade-Russell conjecture modulo 9 via the vertex operators for the level 3 standard modules of type $D^{(3)}_{4}$. Along the same line, we arrive at three partition theorems which may be regarded as an $A^{(2)}_{4}$ analog of the conjecture. One had been proven by Andrews-van Ekeren-Heluani and we point out that the others are easily proved from their results.