$$ \begin{aligned} \Lambda^{r} \mathbb{C}^{n} &=\bigoplus_{i=0}^{r}\left(\Lambda^{(r-i)} V \otimes \Lambda^{i} U\right) \\ &=\left(\Lambda^{r} V \otimes U\right) \oplus\left(\Lambda^{r-1} V \otimes U\right) \\ &=\Lambda^{r} V \oplus \Lambda^{r-1} V \end{aligned} $$
Let $C^n=U\oplus V$ where $U$ is the trivial representation of $S_n$. The second line is trivial as $Λ^r U = 0 , \forall r\ge 2$ as U is of dimension 1.
I have problem understanding the 3rd line.
$(Λ^r V ⊗ U) = Λ^rV $ seems dubious to me. Is it an isomorphism rather than an equality?
Thank you for your help.