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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.12284 |
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| _version_ | 1866917367665131520 |
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| author | Capparelli, Stefano Meurman, Arne Primc, Mirko |
| author_facet | Capparelli, Stefano Meurman, Arne Primc, Mirko |
| contents | 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 $L_{A_1^{(1)}}(2Λ_0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12284 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Leading terms of relations on a level 5 module over the twisted affine Lie algebra $A_2^{(2)}$ Capparelli, Stefano Meurman, Arne Primc, Mirko Combinatorics Representation Theory 17B10 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 $L_{A_1^{(1)}}(2Λ_0)$. |
| title | Leading terms of relations on a level 5 module over the twisted affine Lie algebra $A_2^{(2)}$ |
| topic | Combinatorics Representation Theory 17B10 |
| url | https://arxiv.org/abs/2511.12284 |