Representation Playground

Explore representations of any finite group interactively

Representation Playground

A sandbox for exploring representations of finite groups. Select any group, examine its character table, and decompose representations by computing inner products. Use this space to build intuition and test your understanding.

Character Table Explorer

Select a group and explore its full character table with orthogonality checks and dimension verification.

The symmetric group on 3 elements. The smallest non-abelian group. Three conjugacy classes yield three irreducible representations.

S3
{e}
|C| = 1
(ab)
|C| = 3
(abc)
|C| = 2
trivial(dim 1)111
sign(dim 1)1-11
standard(dim 2)20-1

Dimension check: 12 + 12 + 22 = 6 = |G| = 6 ✓

Decomposition Workbench

Choose a representation and watch the decomposition algorithm compute multiplicities step by step.

The multiplicity of each irreducible is computed via the inner product formula. Every representation decomposes uniquely (up to isomorphism) into irreducibles.

Exercises

  • 1. Verify that the character table of D4 satisfies both row and column orthogonality
  • 2. Decompose the permutation representation of S3 acting on 3 elements
  • 3. Compute the tensor product of the standard representation of S3 with itself
  • 4. Find all irreducible representations of Z/2Z x Z/2Z and verify they are all 1-dimensional
  • 5. Use the hook length formula to compute dim of the [3,2] Young diagram of S5