Leading terms of relations on a level 5 module over the twisted affine Lie algebra $A_2^{(2)}$

TL;DR

This work explores the combinatorial structure of level 5 modules over the twisted affine Lie algebra $A_2^{(2)}$, revealing unique partition identities and relations that differ from dual cases.

Contribution

It introduces a new set of 34 difference conditions for partitions related to level 5 modules over $A_2^{(2)}$, expanding understanding of their combinatorial bases.

Findings

Partial generating series match the character up to $q^{41}$

The combinatorial identity for $A_2^{(2)}$ at level 5 differs from the dual case

34 difference conditions for partitions were identified

Abstract

One of the starting points of this work was the duality of Borcea relating standard level $k$ representations of $A_1^{(1)}$ and level $2k+1$ of $A_2^{(2)}$. For $k=1$ the combinatorial bases in both cases yield the two Capparelli identities and we wanted to see if there is a correspondence between the bases in terms of partitions for all $k\in\mathbb N$. By using the vertex operator relations in the principal picture for level $5$ standard $A_2^{(2)}$-modules we reduce a spanning set of Poincare-Birkhoff-Witt-type vectors in $L(5\Lambda_0)$ by removing the leading terms of relations and rendering a list of 34 ``difference'' conditions for partitions.We have with computer programs sorted out the sets of partitions satisfying these conditions and formed the partial generating series which agrees with the principally specialized character for all powers of $q$ up to $41$. Although our list of leading terms is incomplete, our results show that the corresponding combinatorial identity for $L_{A_2^{(2)}}(5\Lambda_0)$ drastically differs from the one for the Borcea dual $L_{A_1^{(1)}}(2\Lambda_0)$.