An $\mathrm{A}_{2}$ Bailey tree and $\mathrm{A}_{2}^{(1)}$ Rogers–Ramanujan-type identities

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} .