Representation theory has a reputation for being “clean”: decompose a representation into irreducibles, read structure from characters, and move on. That picture is accurate in some regimes, but it can hide the real backbone of the subject: the algebra is doing the work, and the hypotheses matter. A single counterexample can teach this faster than […]

Examples are the oxygen of representation theory. The definitions are compact, but the objects they name are not. A “representation” can be a handful of matrices, a module over a group algebra, a symmetry action on solutions of an equation, or a functor into vector spaces. If you do not build examples on purpose, the […]

Representation theory rewards precision, but it also punishes casual habits more sharply than many neighboring subjects. The reason is structural: the objects live at the intersection of algebra, geometry, and linear algebra, and the meaning of a statement can change when you move even slightly across that intersection. This article collects common mistakes that appear […]