TITLE: Space of morhisms of representations, its dimension in special case
QUESTION [7 upvotes]: The symmetric group $S_n$ acts linearly on $\mathbb{C}^n$, hence it brings up to the representation in $\Lambda^m\mathbb{C}^n$. The goal is to evaluate the dimension of morphisms $\mathrm{Hom}_{S_n}(\Lambda^k\mathbb{C}^n,\Lambda^m\mathbb{C}^n)$.
There is a general way to do it: given two representations $\rho_1$ and $\rho_2$ in $V_1$ and $V_2$ respectively of some group $G$ we can build a representation $T$ in $\mathrm{Hom}_{\mathbb{C}}(V_1,V_2)$ by $$T_g(\phi)=\rho_1(g)\circ\phi\circ\rho_2(g^{-1})$$
So, morphisms of $\rho_1$ and $\rho_2$ are exactly $G$-invariants in this representation of $G$ in $\mathrm{Hom}_{\mathbb{C}}(V_1,V_2)$. But the dimension of the space of $G$-invariants is the trace of $\frac{1}{|G|}\sum T_g$, i.e. the sum of traces $\frac{1}{|G|}\sum \operatorname{tr}(T_g)$. So, I do not believe we should evaluate each trace in $\mathrm{Hom}_{\mathbb{C}}(V_1,V_2)$... Could you help? Perhaps, there is another way?

REPLY [2 votes]: We have $\mathbb{C}^n = triv \oplus V_{std}$,so $$\Lambda^k \mathbb{C}^n \simeq \oplus_{i=0}^k  \Lambda^{i} triv \otimes \Lambda^{k-i} V_{std} = \Lambda^{k-1} V_{std} \oplus \Lambda^k V_{std}$$As the exterior powers of the standard representation are irreducible, it follows that for WLOG $i \leqslant j$ $$\dim \; \text{Hom}(\Lambda^i \mathbb{C}^n, \Lambda^j \mathbb{C}^n) = \begin{cases} 1 & i = j-1 \\ 2 & i = j \\ 0 & o.w.\end{cases}$$by schurs lemma. 
Edit: tried adding 1 + 1 again