Visualize irreducible representations as symmetry-breaking animations
Irreducible representations are the atoms of the subject: every representation breaks into them, the way integers break into primes. Here you can watch a few of them act.
Pick a representation — trivial, sign, or standard — then apply an element of S3 to the triangle and watch its matrix. Only the 2D standard representation shows the triangle's real geometry.
Left: watch S3 permute triangle vertices. Right: the corresponding matrix representation. The trace of the matrix is the character value.
A representation is irreducible if it has no proper, nonzero invariant subspaces. Irreducibles are the atoms of representation theory -- every representation decomposes uniquely into a direct sum of irreducibles, just as every integer factors uniquely into primes.
In this lesson, you will see groups act on geometric objects through their irreducible representations, watching how abstract algebra becomes concrete linear algebra.
Watch S3 (the symmetric group on 3 elements) act on a triangle. On the left, see the geometric action permuting vertices. On the right, see the corresponding matrix in each of the three irreducible representations: trivial (dim 1), sign (dim 1), and standard (dim 2).
Key insight: The standard representation captures the "interesting" part of how S3 acts -- it strips away the trivial action (everything fixed) and reveals the genuine 2-dimensional symmetry of the triangle.
Click through the eight symmetries of the square and watch its 2×2 matrix update. Rotations have determinant 1; reflections have determinant -1.
Watch D4 (symmetries of the square) act via its standard 2D representation. Rotations have det = 1, reflections have det = -1.
The dihedral group D4 has 8 elements: 4 rotations and 4 reflections. Its standard 2D representation sends rotations to rotation matrices (det = 1) and reflections to reflection matrices (det = -1). Watch the square transform and see the corresponding matrix update.
Key insight: The determinant distinguishes rotations from reflections, while the trace (character) separates the rotation classes. They don't determine the conjugacy class completely — both reflection classes share determinant -1 and trace 0.
Press Decompose into Irreducibles and watch the regular representation split into blocks. Each irreducible 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.
The regular representation (where G acts on the vector space with basis indexed by G itself) contains every irreducible representation, each appearing with multiplicity equal to its dimension. Click to animate the decomposition.
Key insight: The regular representation is the "universal" representation. Maschke's theorem guarantees it splits into irreducibles, and character orthogonality shows each irreducible appears with multiplicity equal to its dimension — the finite-group analogue of the Peter-Weyl theorem for compact groups.
Spot a mistake or have an idea?