Many identities involving multiplicative structure constants originate in representation theory. There are plenty of positivity statements about such structure constants but no combinatorial proof is known, or appeared much later.See the related question here about unsolved problems in representation theory.
One example is the Schubert structure constants (the constants appearing when a product of Schubert polynomials is again expanded in Schubert polynomials). For representation-theoretical reasons, these are non-negative integers, but it is completely mysterious why.
There is also certain symmetries in the Macdonald polynomials and diagonal harmonics, which are evident from the definition, but completely unclear from the current combinatorial models.