An $\mathrm{A}_2$ Bailey tree and $\mathrm{A}_2^{(1)}$ Rogers-Ramanujan-type identities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912242716377088 |
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| author | Warnaar, S. Ole |
| author_facet | Warnaar, S. Ole |
| contents | 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 $λ$. This generalises a result of Feigin, Feigin, Jimbo, Miwa and Mukhin for $λ=kΛ_0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_09069 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An $\mathrm{A}_2$ Bailey tree and $\mathrm{A}_2^{(1)}$ Rogers-Ramanujan-type identities Warnaar, S. Ole Combinatorics Mathematical Physics Number Theory Representation Theory 05A19, 11P84, 17B10, 33D15, 81R10 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 $λ$. This generalises a result of Feigin, Feigin, Jimbo, Miwa and Mukhin for $λ=kΛ_0$. |
| title | An $\mathrm{A}_2$ Bailey tree and $\mathrm{A}_2^{(1)}$ Rogers-Ramanujan-type identities |
| topic | Combinatorics Mathematical Physics Number Theory Representation Theory 05A19, 11P84, 17B10, 33D15, 81R10 |
| url | https://arxiv.org/abs/2303.09069 |