If a partial differential operator commutes with a symmetry group (permutations, rotations, reflections, etc.) then it can be decoupled by discretising with a so-called symmetry-adapted basis built from irreducible representations, the basic building blocks of representation theory. In this talk we explore this phenomena using symmetry-adapted multivariate orthogonal polynomials to discretise Schrödinger equations with potentials invariant under permutations or the octohedral symmetry group for the cube.