Induced representation

Remarks

Up to isomorphism, the result does not depend on the choice of the $t_i$. Equivalently, $\operatorname{Ind}_H^G W = F[G] \otimes_{F[H]} W$, using the group algebra. Its degree is $[G : H] \cdot \dim W$. For finite $G$, its character is $$\operatorname{Ind}_H^G \chi(g) = \frac{1}{\abs{H}} \sum_{x \in G,\ x^{-1} g x \in H} \chi(x^{-1} g x).$$ Example: let $G = S_3$, $H = A_3$ and $\theta$ trivial of degree $1$. Then $V$ has dimension $2$, and the formula gives the values $2$ at $1$, $0$ at transpositions and $2$ at $3$-cycles. This is the sum of the trivial and sign characters. Non-example: for $H = \set{0}$ in $G = \Z$, the index is infinite, so the definition does not apply; the space $\bigoplus_{t \in \Z} W$ with the shift action is a different construction, sometimes called compact induction.