Refinement of partition identities

A partition of a positive integer n is a non-increasing sequence of positive integers whose sum is n. A Rogers-Ramanujan type partition identity is a theorem stating that for all n, the number of partitions of n satisfying some difference conditions equals the number of partitions of n satisfying some congruence conditions. In 1993 Alladi and Gordon introduced the method of weighted words to find refinements of Schur's theorem and other partition identities. After explaining their original method which relies on q-series identities, I will present a new version using q-difference equations and recurrences, and apply it to refine two identities from representation theory.