Decomposing a tensor product space into direct sums I'm trying to understand how to decompose certain symmetric and anti-symmetric tensor products of vector spaces into direct summands.
Let $V$ be a complex finite dimensional vector space and denote by $\Lambda^2 V$ and $S^2 V$ the spaces generated by $\frac{1}{2}(v_1 \otimes v_2 - v_1 \otimes v_2)$ and $\frac{1}{2}(v_1 \otimes v_2 + v_1 \otimes v_2)$ respectively (for all $v_1,v_2 \in V$).
At the moment I'm just trying to understand one concrete example. If we take $V = \mathbb{C}^5$ and we decompose $V = \mathbb{C}^3 + \mathbb{C}^2$ my guess is that the tensor product decomposes as:
$$\Lambda^2 V = \Lambda^2 (\mathbb{C}^3 \oplus \mathbb{C}^2) = \Lambda^2\mathbb{C}^3 \oplus \Lambda^2\mathbb{C}^2 \oplus (\mathbb{C}^3 \oplus \mathbb{C}^2)$$
I've arrived at that by thinking of the $\Lambda^2 V$ as an anti-symmetric matrix and then decomposing it blockwise. I'm struggling to prove the above statement in general and I'm trying to understand how $\Lambda^2$ and $S^2$ behave with direct sum decompositions generally.
Thanks
 A: I am guessing that your question is: What is $\bigwedge^k (V \oplus W)$? (This is the wedge product, not the tensor product.)
And also the same question for $S^k = Sym^k$, which is the symmetric product.
Extended hint:
The thing to observe is that there is a natural basis for $V \oplus W$. Namely, you take a basis for $V$ and union it with a basis for $W$.
Let's call our natural basis for $V$ $\{v_1, \ldots, v_n\}$, and our basis for $W$ $\{w_1, \ldots, w_m\}$.
Now, given a basis for a vector space $X$, there is a natural basis for $\bigwedge^k X$ and for $Sym^k X$:
Suppose that $\{x_1, \ldots, x_r\}$ is a basis for $X$. 
1) Then a basis for $\bigwedge^k X$ is given by all of the $\{x_{i_1} \wedge \ldots \wedge x_{i_k} \}$ for all $i_1 < \ldots < i_k$, $i_j \in \{1, 2, \ldots r\}$. 
2) A natural basis for $Sym$ is given similarly, only now you are allowed to take repeated vectors (so $<$ will be replaced by $\leq$) - another description is the set of monomials of degree k in $\mathbb{Z}[x_1, \ldots, x_r]$.
Finally: Given that $\{x_1, \ldots, x_r\} = \{v_1, \ldots, v_n, w_1, \ldots, w_m\}$ is a basis for $V \oplus W = X$, you can now play a combinatorial game to divide up $\bigwedge^k X$ into direct sums of smaller exterior powers of $V$ and $W$. Similarly for the symmetric product. Do you see how to proceed? Please feel free to ask if you have questions.
