PySymmetry: A Sage/Python Framework for the Symmetry Reduction of Linear G-Equivariant Systems

Abstract: Despite the prevalence of symmetry in scientific linear systems, these structural properties are often underutilized by standard computational software. This paper introduces PySymmetry, an open-source Sage/Python framework that implements classical representation theory to simplify G-equivariant linear systems. PySymmetry uses projection operators to generate symmetry-adapted bases, transforming equivariant operators into a more efficient block-diagonal form. Its functionalities include defining and reducing representations, calculating multiplicities, and obtaining the explicit block structure. We demonstrate PySymmetry's versatility through three case studies: a chemistry application, a numerical benchmark on the non-Hermitian Schr\"odinger equation that achieved a performance increase of over 17x compared to standard methods, and a symbolic investigation that enabled the first complete analytical classification of a challenging problem in celestial mechanics. Designed for seamless integration with libraries like NumPy and SciPy, PySymmetry offers a powerful, user-friendly tool for exploring symmetries in theoretical and applied contexts. ```