Lifting the associativity relations in orbifold theories

Associated to a complex manifold with the action of a finite group (or an orbifold) one can associate various algebras with so-called inertial products, the prototypical example of which is the Chen-Ruan orbifold cohomology and its K-theoretic variant. The nontrivial part of the construction of the orbifold product is the proof of associativity which follows from an identity in the representation ring of a finite group. We explain how this identity can be lifted, in some cases, to the category of representations and this can be applied towards describing orbifold products in singularity theory.