Combine representations and decompose the result
Given two representations V and W, their tensor product V tensor W is a new representation where the group acts diagonally: g acts on v tensor w as g(v) tensor g(w). The character of the tensor product is the pointwise product of the individual characters.
The tensor product is usually not irreducible -- it decomposes into a direct sum of irreducibles. Computing this decomposition is a central problem in representation theory.
Select two irreducible representations of S3 and see how their tensor product decomposes. The character is computed by multiplying characters pointwise, then decomposed using inner products with each irreducible character.
The tensor product character is the pointwise product of individual characters. Decompose the result by taking inner products with each irreducible character.
Key insight: The most interesting case is standard tensor standard, which decomposes as trivial + sign + standard. This shows that tensoring can both increase and decrease complexity.
The tensor square V tensor V splits naturally into the symmetric squareSym^2(V) (symmetric tensors) and the exterior square Lambda^2(V) (antisymmetric tensors). Each is a representation in its own right.
Any tensor product V \u2297 V splits into symmetric and antisymmetric parts. The highlighted cells show which tensor basis elements contribute to each part.
Key insight: For the standard representation of S3, Sym^2 = trivial + standard (dim 3) and Lambda^2 = sign (dim 1). The exterior square of a 2D representation is always 1-dimensional and equals the determinant character.
The set of all (virtual) representations with addition (direct sum) and multiplication (tensor product) forms a ring -- the representation ring. This multiplication table shows the complete ring structure for S3.
The representation ring of S3 with multiplication given by tensor product. Each tensor product decomposes uniquely into a direct sum of irreducibles.
Key insight: The representation ring is commutative, with the trivial representation as the identity. Tensoring with the sign representation permutes the irreducibles (it is an involution on the set of irreps).