Proof of $Af = \sum_{\sigma \in S_{k}} (Sgn \sigma) \sigma f$ is an alternating function.

Question

If $f$ is a $k-$linear function over a vector space $V$, then we need to prove that

$Af = \sum_{\sigma \in S_{k}} (Sgn \sigma) \sigma f$ is an alternating function.

While proving I am facing problem in a step.

For $\tau \in S_{k}$, $\tau(Af) = \sum_{\sigma \in S_{k}} (Sgn \sigma) \tau(\sigma f) $

Now how the above cn be written as $Sgn(\tau) \sum_{\sigma \in S_{k}} (Sgn \tau \sigma) (\tau \sigma) f $?

Which is then equated to $(Sgn \tau) Af$ proving that it is an alternating function.

Answer 1

Just notice that:

• $\tau(\sigma f)=(\tau\sigma)f$;

• $\text{Sgn}$, being a group morphism, you have $\text{Sgn}(\tau\sigma)=\text{Sgn}(\tau)\text{Sgn}(\sigma)$;

• As $\text{Sgn}:S_n\to\{-1,1\}$, you have $\text{Sgn}(\tau)^2=1$.