An example of $A_2$ Rogers-Ramanujan bipartition identities of level 3

TL;DR

This paper presents new positive series and bipartition identities for level 3 modules of affine Lie algebra A^{(1)}_2, using advanced combinatorial and algebraic techniques with representation theory interpretations.

Contribution

It introduces explicit positive series and bipartition identities for affine Lie algebra modules, expanding the combinatorial understanding of these structures.

Findings

Positive Andrews-Gordon type series for level 3 modules

Bipartition identities with representation theoretic interpretations

Proofs based on advanced combinatorial formulas and automata techniques

Abstract

We give manifestly positive Andrews-Gordon type series for the level 3 standard modules of the affine Lie algebra of type $A^{(1)}_2$. We also give corresponding bipartition identities, which have representation theoretic interpretations via the vertex operators. Our proof is based on the Borodin product formula, the Corteel-Welsh recursion for the cylindric partitions, a $q$-version of Sister Celine's technique and a generalization of Andrews' partition ideals by finite automata due to Takigiku and the author.