SIGMA 22 (2026), 055, 15 pages      arXiv:2511.12284      https://doi.org/10.3842/SIGMA.2026.055
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

Leading Terms of Relations on a Level 5 Module over the Twisted Affine Lie Algebra $A_2^{(2)}$

Stefano Capparelli ^a, Arne Meurman ^b and Mirko Primc ^c
a) Dipartimento SBAI, Università La Sapienza, Rome, Italy
^b) Department of Mathematics, Lund University, Lund, Sweden
^c) Faculty of Science, University of Zagreb, Zagreb, Croatia

Received November 18, 2025, in final form May 18, 2026; Published online June 02, 2026

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 Poincaré-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. Using computer programs, we enumerated the partitions satisfying these conditions and obtained a truncated generating series agreeing 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 $L_{A_1^{(1)}}(2\Lambda_0)$.

Key words: leading term; integer partition; affine algebra; standard module.

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