The sign representation of $S_n$ and the restriction ${\mathrm{Res}}_H^G(V)$ of a representation to a subgroup

Let $k$ be a field.

For the symmetric group $S_n$ (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition), the sign representation over $k$ is the one-dimensional representation on $k$ in which $\sigma \cdot a:=\mathrm{sgn}(\sigma )a\mathrm{.}$ This is a representation because $\mathrm{sgn}:S_n\to \{\pm 1\}$ is a group homomorphism (The sign is a homomorphism $S_n\to \{+1,-1\}$, surjective exactly when $n\ge 2$).

Now let $\rho :G\to \mathrm{GL}(V)$ be a representation of a group $G$ over $k$, and let $H\le G$ be a subgroup (Subgroup). The restriction of $\rho$ to $H$ is the representation ${\mathrm{Res}}_H^G(V):=\rho {\mathrm{\mid }}_H:H⟶\mathrm{GL}(V),h⟼\rho (h)\mathrm{.}$ It has the same underlying vector space $V$ and only forgets the action of the elements outside $H$.

• In characteristic $2$, the sign representation coincides with the trivial representation because $-1=1$ in the field.

• Restriction changes the acting group, not the underlying vector space.