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 $\lambda=k\Lambda_0$.