Build character tables interactively for small groups
A character table squeezes everything about a group's representations into a small grid of numbers. Build a few and look for the patterns.
Pick a group, then hover over rows (irreducibles) and columns (conjugacy classes). Notice the table is always square, then press Show Orthogonality Relations to check the rows.
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) | 1 | 1 | 1 |
| sign(dim 1) | 1 | -1 | 1 |
| standard(dim 2) | 2 | 0 | -1 |
Dimension check: 12 + 12 + 22 = 6 = |G| = 6 ✓
A character of a representation is the trace of the representing matrix. Characters are constant on conjugacy classes, so we can organize them into a compact character table that completely determines the representation up to isomorphism. The character table is the Rosetta Stone of representation theory -- it encodes all essential information about a group's representations in a single matrix.
In this lesson, you will build character tables interactively, verify orthogonality relations, and see the dimension formula in action.
Select a group to see its complete character table. Each row is an irreducible representation, each column is a conjugacy class. Hover to highlight rows and columns, and verify the dimension formula and orthogonality relations.
Key insight: The number of irreducible representations always equals the number of conjugacy classes. The first column gives the dimension of each irreducible, and the sum of squares of dimensions equals the group order.
Choose two irreducibles from the menus and watch the class-weighted products add up. Two different irreducibles give 0; the same one twice gives 1.
The bars show each conjugacy class's contribution to the inner product for S3. Orthogonality means different irreducible characters always sum to zero.
The rows of the character table are orthogonal with respect to the inner product weighted by conjugacy class sizes. This is a deep fact: different irreducible representations are "perpendicular" in a precise sense.
Key insight: Orthogonality means that you can uniquely decompose any representation by taking inner products with each irreducible character -- like projecting a vector onto an orthonormal basis.
Pick a group and add up the squares of its irreducible dimensions. The total always equals the order of the group.
Each colored square grid represents dim(ρ)^2. The total area always equals the group order |G|.
The sum of squares of the dimensions of all irreducible representations equals the order of the group. This powerful constraint severely limits which character tables are possible.
Key insight: For S4 (order 24), the only way to write 24 as a sum of squares with 5 terms (one for each conjugacy class) is 1 + 1 + 4 + 9 + 9. This tells us the dimensions of the irreducibles before we even compute them!
Spot a mistake or have an idea?