Let \(H\) be a subgroup of a finite group \(G\), and let \((\rho, W)\) be a representation of \(H\) over \(\mathbb{C}\). How is the induced representation \(\operatorname{Ind}_H^G(W)\) defined? State its dimension, character formula (Frobenius formula), and Frobenius Reciprocity.
Solution
Regard \(W\) as a left \(\mathbb C[H]\)-module. The induced representation is \[ \operatorname{Ind}_H^G W =\mathbb C[G]\otimes_{\mathbb C[H]}W, \] with \(G\) acting by left multiplication on the first factor.
If \(g_1,\dots,g_m\) are representatives for the left cosets of \(H\) in \(G\), then \[ \mathbb C[G]=\bigoplus_{i=1}^m g_i\mathbb C[H] \] as a right \(\mathbb C[H]\)-module. Hence \[ \dim\operatorname{Ind}_H^GW=[G:H]\dim W. \]
If \(\chi\) is the character of \(W\), extended by \(0\) off \(H\), then for \(g\in G\) the induced character is \[ \chi_{\operatorname{Ind}}(g) =\frac1{|H|}\sum_{x\in G}\chi(x^{-1}gx). \] Equivalently, \[ \chi_{\operatorname{Ind}}(g) =\sum_{\substack{i\\g_i^{-1}gg_i\in H}} \chi(g_i^{-1}gg_i). \]
Frobenius reciprocity is the natural adjunction \[ \operatorname{Hom}_G(\operatorname{Ind}_H^GW,U) \cong \operatorname{Hom}_H(W,\operatorname{Res}_H^GU), \] and at the character level \[ \langle\operatorname{Ind}_H^G\chi,\psi\rangle_G = \langle\chi,\operatorname{Res}_H^G\psi\rangle_H. \]