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

TL;DR

This work investigates a partition-based description of the level-5 standard module $L(5\Lambda_0)$ for the twisted affine algebra $A_2^{(2)}$ in the context of a duality with $A_1^{(1)}$. By employing principal-picture vertex operator relations, it derives 34 leading-term difference conditions on partitions and constructs a partial generating function that reproduces the principally specialized character up to high order. The results show a substantial divergence between the sum-side identity obtained for $L_{A_2^{(2)}}(5\Lambda_0)$ and the Borcea dual on the $A_1^{(1)}$ side, $L_{A_1^{(1)}}(2\Lambda_0)$. The paper also connects to a level-2 $\mathfrak{sl}(2,\mathbb{C})^\sim$ basis via duality, discusses the incompleteness of the leading-term list, and outlines computational approaches and future directions for completing the data and understanding the duality's implications.

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 $L(5Λ_0)$ 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 $L_{A_2^{(2)}}(5Λ_0)$ drastically differs from the one for the Borcea dual $L_{A_1^{(1)}}(2Λ_0)$.

Theorems & Definitions (9)

• Remark 3.1
• Theorem 4.1
• Theorem 4.2
• Proposition 4.3
• Proposition 4.4
• proof
• Proposition 4.5
• Proposition 4.6
• Theorem 7.1