induced representation

Induction is a machine for building big representations from small ones: you start with a representation of a subgroup H and 'spread it out' over the whole group G, making copies indexed by the cosets of H. It is the natural way to manufacture representations of G when you only understand a piece, H — and it is the partner of restriction, the operation that goes the other way.

Let H be a subgroup of G and W a representation of H. The induced representation Ind_H^G W is the k[G]-module k[G] tensor over k[H] with W. Concretely, choosing coset representatives g_1, ..., g_n for G/H (where n = [G : H]), the underlying space is a direct sum of n copies of W, one per coset, and G permutes the cosets while acting through H within each block. Its dimension is [G : H] times dim W. There is a parallel notion using Hom rather than tensor (coinduction), which coincides with induction when [G : H] is finite.

Induced representations are usually reducible, and decomposing them is a central computational problem; the answer is governed by Frobenius reciprocity, which relates the multiplicity of an irreducible of G in Ind_H^G W to the multiplicity of W in the restriction of that irreducible to H. Inducing the trivial representation of H gives the permutation representation of G on the cosets G/H, tying induction back to group actions.