Break any representation into irreducible building blocks
Maschke's theorem guarantees that every representation of a finite group over C decomposes as a direct sum of irreducible representations, and this decomposition is unique (up to ordering). The multiplicity of each irreducible can be computed using the character inner product formula.
This page brings together all the tools: character tables, inner products, and the dimension formula, applied to decompose concrete representations.
Given a representation with known character values, compute the multiplicity of each irreducible by taking inner products. Select a group and a representation to decompose.
The multiplicity of each irreducible is computed via the inner product formula. Every representation decomposes uniquely (up to isomorphism) into irreducibles.
Key insight: The decomposition formula m_i = (1/|G|) sum |C| chi(g) chi_i(g) works because the irreducible characters form an orthonormal basis for the space of class functions. Decomposition is just an orthogonal projection!
Watch the regular representation (a big matrix) split into irreducible blocks along the diagonal. Each block appears with multiplicity equal to its dimension.
The regular representation (where G acts on itself) decomposes into a direct sum of irreducibles, each appearing with multiplicity equal to its dimension.
Key insight: After a suitable change of basis, every representation matrix becomes block-diagonal, with each block being an irreducible representation. The decomposition is basis-independent -- only the multiplicities matter.
The dimension of the original representation must equal the sum of dimensions of the irreducible components (with multiplicity). This provides a quick sanity check.
Each colored square grid represents dim(\u03C1)^2. The total area always equals the group order |G|.