Let $ G=S_3 $ and $ H=S_2 $. Show that $ Res^G_H(sgn_G)=sgn_H $
The symmetric group $G=S_3$ has three irreducible representations $ 1_G, sgn_G $ and $ V$ where $ 1_G $ denotes the trivial representation, $ sgn_G $ is the sign representation and $dimV=2 $.
$ S_2 $ is a subgroup of $ S_3 $, and $ 1_H $ and $ sgn_H$ are the irreducible representations of $ S_2 $ where $ 1_H $ is, of course, the trivial representation
I know that the restriction of a representation takes the representation of $G$ and maps it to a new representation of its subgroup $H$
The restriction of $sgn_G$ will go to either $1_H$ or $sgn_H$. How can we prove it goes to $sgn_H$?
Thank you