tensor product of representations

Given two representations of the same group, you can combine them into a new one on the tensor product of their spaces, with the group acting on both factors at once. This 'multiplication' of representations is what gives the collection of representations a ring-like structure: you can add (direct sum) and multiply (tensor) them, and characters turn this product into ordinary pointwise multiplication of functions.

Let (V, rho) and (W, sigma) be representations of G over k. Their tensor product is the space V tensor W with the diagonal action g.(v tensor w) = rho(g)v tensor sigma(g)w, extended linearly — the same element g acts on both legs simultaneously. This is genuinely a representation of G (not of G times G), and its dimension is dim(V) times dim(W). Its character is the product of characters: chi_{V tensor W}(g) = chi_V(g) times chi_W(g).

Tensoring is rarely irreducible even when the factors are: decomposing V tensor W into irreducibles is the Clebsch-Gordan problem, central in physics and in the representation theory of Lie groups. The character formula makes the bookkeeping tractable — multiplicities are inner products of the product character with the irreducible characters. Special cases include the symmetric and exterior squares of V, the G-invariant subspaces of V tensor V under the swap.