Parallelisation of partial differential equations via representation theory

Incorporating symmetries into the numerical solution of differential equations has been a mainstay of research over the last 40 years, however, one aspect is less known and under-utilised: discretisations of partial differential equations that commute with symmetry actions (like rotations, reflections or permutations) can be decoupled into independent systems solvable in parallel by incorporating knowledge from representation theory. We introduce this beautiful subject via a crash course in representation theory focussed on hands-on examples for the symmetry groups of the square and cube, and its utilisation in the construction of so-called symmetry-adapted bases. Schur's lemma, which is not well-known in applied mathematics, plays a powerful role in proving sparsity of resulting discretisations and thereby showing that partial differential equations do indeed decouple. Using Schr\"odinger equations as a motivating example, we demonstrate that a symmetry-adapted basis leads to a significant increase in the number of independent linear systems. Counterintuitively, the effectiveness of this approach is in fact greater for partial differential equations with less symmetries, for example a Schr\"odinger equation where the potential is only invariant under permutations, but not under rotations or reflections. We also explore this phenomenon as the dimension of the partial differential equation becomes large, hinting at the potential for significant savings in high-dimensions.