5
$\begingroup$

I need references for the following two results. Let $G$ be a finite group and let $H$ be a subgroup of $G$. Let $V$ be a finite-dimensional representation of $H$.

  1. $(\text{Ind}_H^G V)^{\ast} \cong \text{Ind}_H^G V^{\ast}$.

  2. Given a finite-dimensional representation $W$ of $G$, we have $(\text{Ind}_H^G V) \otimes W \cong \text{Ind}_H^G (V \otimes \text{Res}_H^G W)$.

I know references for these that prove them with character theory, but I need them for general fields (not just fields of characteristic $0$).

  • 0
    @QiaochuYuan : Thanks. I just spent 15-20 minutes trying to prove these with that version of Frobenius reciprocity. I was unsuccessful, but that doesn't mean it can't be done.2012-08-08

0 Answers 0