Visualize irreducible representations as symmetry-breaking animations
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).
Left: watch S3 permute triangle vertices. Right: the corresponding matrix representation. The trace of the matrix is the character value.
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.
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.
Watch D4 (symmetries of the square) act via its standard 2D representation. Rotations have det = 1, reflections have det = -1.
Key insight: The determinant distinguishes rotations from reflections, while the trace (character) distinguishes conjugacy classes. Together, they classify all elements up to conjugation.
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.
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: The regular representation is the "universal" representation. Its decomposition into irreducibles is the content of the Peter-Weyl theorem (for compact groups) and Maschke's theorem (for finite groups over C).