Proofs and Reductions of Kanade and Russell's partition identities

Abstract

We prove seven of the Rogers-Ramanujan type identities modulo \(12\) that were conjectured by Kanade and Russell. Included among these seven are the two original modulo \(12\) identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level \(2\) modules of \(A_9^{(2)}\). We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.