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arXiv · 2303.09069

An $\mathrm{A}_2$ Bailey tree and $\mathrm{A}_2^{(1)}$ Rogers-Ramanujan-type identities

S. Ole Warnaar
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Abstract

The $\mathrm{A}_2$ Bailey chain of Andrews, Schilling and the author is extended to a four-parameter $\mathrm{A}_2$ Bailey tree. As main application of this tree, we prove the Kanade-Russell conjecture for a three-parameter family of Rogers-Ramanujan-type identities related to the principal characters of the affine Lie algebra $\mathrm{A}_2^{(1)}$. Combined with known $q$-series results, this further implies an $\mathrm{A}_2^{(1)}$-analogue of the celebrated Andrews-Gordon $q$-series identities. We also use the $\mathrm{A}_2$ Bailey tree to prove a Rogers-Selberg-type identity for the characters of the principal subspaces of $\mathrm{A}_2^{(1)}$ indexed by arbitrary level-$k$ dominant integral weights $\lambda$. This generalises a result of Feigin, Feigin, Jimbo, Miwa and Mukhin for $\lambda=k\Lambda_0$.

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S. Ole Warnaar. 2023-03-16. An $\mathrm{A}_2$ Bailey tree and $\mathrm{A}_2^{(1)}$ Rogers-Ramanujan-type identities. https://arxiv.org/abs/2303.09069

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