Schur's lemma

Schur's lemma is a fundamental result in mathematics, particularly in the study of symmetry and algebra. It states that if you have a linear transformation that commutes with all transformations of an irreducible representation (a way of expressing symmetry), then that transformation must be a scalar multiple of the identity, essentially just scaling every element equally. In simpler terms, for the most "basic" symmetrical structures, only uniform scaling transformations can coexist while maintaining the structure's integrity. This helps mathematicians understand and classify symmetrical objects and their properties more deeply.