Break any representation into irreducible building blocks
Every representation of a finite group breaks uniquely into irreducible pieces. Here you find the pieces, watch them separate, and check the count.
Pick a group and a representation to decompose. Each multiplicity comes from one inner product of characters, and it always lands on a whole number.
The multiplicity of each irreducible is computed via the inner product formula. Every representation decomposes uniquely (up to isomorphism) into irreducibles.
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.
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!
Press Decompose into Irreducibles and watch the regular representation's big matrix split into blocks along the diagonal. Each block appears as many times as 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.
Watch the regular representation (a big matrix) split into irreducible blocks along the diagonal. Each block appears 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.
Pick a group and check that the squares of its irreducible dimensions add up to its order.
Each colored square grid represents dim(ρ)^2. The total area always equals the group order |G|.
The dimension of the original representation must equal the sum of dimensions of the irreducible components (with multiplicity). This provides a quick sanity check.
Spot a mistake or have an idea?