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.

翻译：将对称性纳入微分方程数值解的研究在过去四十年中一直是主流方向，然而有一个方面较少被了解且未得到充分利用：若偏微分方程的离散化与对称作用（如旋转、反射或置换）可交换，则可通过引入表示理论的知识将其解耦为可并行求解的独立系统。本文通过聚焦于正方形与立方体对称群实践案例的表示理论速成课程，介绍这一优美主题，并阐述其在构建所谓对称适配基中的应用。在应用数学领域尚未广为人知的舒尔引理，在证明所得离散化矩阵的稀疏性方面发挥着关键作用，从而证实偏微分方程确实可被解耦。以薛定谔方程作为示例，我们证明对称适配基能显著增加独立线性系统的数量。反直觉的是，该方法对于对称性较少的偏微分方程反而更有效，例如势函数仅对置换不变、而对旋转或反射非不变的薛定谔方程。我们还探讨了当偏微分方程维度增大时该现象的表现，暗示了在高维情形下可能实现显著的计算节约。