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

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 $\lambda =k{\Lambda }_0$.