Parallelisation of partial differential equations via representation theory

Abstract: A little utilised but fundamental fact is that if one discretises a partial differential equation using a symmetry-adapted basis corresponding to so-called irreducible representations, the basic building blocks in representational theory, then the resulting linear system can be completely decoupled into smaller independent linear systems. That is, representation theory can be used to trivially parallelise the numerical solution of partial differential equations. This beautiful theory is introduced via a crash course in representation theory aimed at its practical utilisation, its connection with decomposing expansions in polynomials into different symmetry classes, and give examples of solving Schr\"odinger's equation on simple symmetric geometries like squares and cubes where there is as much as four-fold increase in the number of independent linear systems, each of a significantly smaller dimension than results from standard bases.