Induced representations and Frobenius reciprocity

Let H be a subgroup of a finite group G and W a finite-dimensional complex left representation of H. Define induction using the group algebra of G tensored over the group algebra of H with W. The balancing relation identifies a subgroup factor on the right of a group element with that subgroup element acting on the vector. Left multiplication by G on the first factor is well defined because it respects this relation. It supplies the induced representation without any arbitrary choice of extended matrices.

Choose representatives for left cosets of the form tH. Every group element is uniquely a representative followed by an element of H, so the balanced tensor product has one copy of W for each representative. To act by g on the copy indexed by t, write gt as another representative times h and apply h to the vector in that destination copy. This shows the dimension is the index times the original dimension. A different representative list changes coordinates but not the equivalence class of the representation.

An equivariant map from the induced space to a G-representation V is determined by what it does on tensors whose first factor is the identity. Restricting there gives an H-equivariant map from W to the restriction of V. Conversely, given such an H-map, send a general pure tensor to the group element acting on its image under that map. The balancing relation makes this well defined precisely because the original map respects H. The two procedures undo one another and are linear.

This proves Frobenius reciprocity, an isomorphism of Hom spaces rather than merely an equality of dimensions. For irreducible V over the complex numbers, their common dimension is its multiplicity in the induced representation. Taking characters expresses the same identity through normalized inner products on the two different groups. Each group uses its own order in the denominator. The construction and adjunction also exist over arbitrary fields, but our conversion into ordinary irreducible multiplicities uses the semisimple complex setting.

To compute the induced character at a group element, only coset copies sent to themselves contribute diagonal-block traces. A representative t contributes the subgroup character at the conjugate t inverse times g times t, provided that conjugate lies in H. Summing over the fixed cosets gives the character. Equivalently sum over all group elements satisfying that membership condition and divide by the subgroup order, since each fixed coset has exactly that many representatives. Conjugating within H does not change its character value.

When W is trivial, the induced representation is the permutation action on left cosets, recovering fixed-coset counts. A nontrivial W retains information about the subgroup action within a returning coset and can produce negative or complex traces. Induction does not usually preserve irreducibility: distinct G-types can restrict to overlapping H-types. Reciprocity measures exactly that overlap and often computes a decomposition much faster than building all induced matrices. A dimension check remains essential because confusing index with subgroup order changes the constructed space.