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.

翻译：一个鲜为人知但基本的事实是：如果使用与所谓不可约表示（表示理论中的基本构建模块）相对应的对称适配基对偏微分方程进行离散化，则所得的线性系统可以完全解耦为多个更小的独立线性系统。也就是说，表示理论可用于轻松实现偏微分方程数值解的并行化。本文通过针对实际应用的表示理论速成课程介绍这一优美理论，阐述其与将多项式展开分解为不同对称类之间的联系，并以在正方形和立方体等简单对称几何结构上求解薛定谔方程为例进行说明——在这些案例中，独立线性系统的数量最多可增加四倍，且每个系统的维度均显著小于标准基所得结果。